Vibrations of Elastic Systems: With Applications to MEMS and NEMS

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Vibrations of Elastic Systems

SOLID MECHANICS AND ITS APPLICATIONS Volume 184

Series Editor:

G.M.L. GLADWELL Department of Civil Engineering University of Waterloo Waterloo, Ontario, Canada N2L 3GI

Aims and Scope of the Series The fundamental questions arising in mechanics are: Why?, How?, and How much? The aim of this series is to provide lucid accounts written by authoritative researchers giving vision and insight in answering these questions on the subject of mechanics as it relates to solids. The scope of the series covers the entire spectrum of solid mechanics. Thus it includes the foundation of mechanics; variational formulations; computational mechanics; statics, kinematics and dynamics of rigid and elastic bodies: vibrations of solids and structures; dynamical systems and chaos; the theories of elasticity, plasticity and viscoelasticity; composite materials; rods, beams, shells and membranes; structural control and stability; soils, rocks and geomechanics; fracture; tribology; experimental mechanics; biomechanics and machine design. The median level of presentation is the first year graduate student. Some texts are monographs defining the current state of the field; others are accessible to final year undergraduates; but essentially the emphasis is on readability and clarity.

For further volumes: http://www.springer.com/series/6557

Edward B. Magrab

Vibrations of Elastic Systems With Applications to MEMS and NEMS

123

Prof. Edward B. Magrab Department of Mechanical Engineering University of Maryland College Park, MD 20742 USA [email protected]

ISSN 0925-0042 ISBN 978-94-007-2671-0 e-ISBN 978-94-007-2672-7 DOI 10.1007/978-94-007-2672-7 Springer Dordrecht Heidelberg London New York Library of Congress Control Number: 2011941768 © Springer Science+Business Media B.V. 2012 No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

For June Coleman Magrab Still my muse after all these years

Preface

Vibrations occur all around us: in the human body, in mechanical systems and sensors, in buildings and structures, and in vehicles used in the air, on the ground, and in the water. In some cases, these vibrations are undesirable and attempts are made to avoid them or to minimize them; in other cases, vibrations are controlled and put to beneficial uses. Until recently, many of the application areas of vibrations have been largely concerned with objects having one or more of its dimensions being tens of centimeters and larger, a size that we shall denote as the macro scale. During the last decade or so, there has been a large increase in the development of electromechanical devices and systems at the micrometer and nanometer scale. These developments have lead to new families of devices and sensors that require consideration of factors that are not often important at the macro scale: viscous air damping, squeeze film damping, viscous fluid damping, electrostatic and van der Waals attractive forces, and the size and location of proof masses. Thus, with the introduction of these sub millimeter systems, the range of applications and factors has been increased resulting in a renewed interest in the field of the vibrations of elastic systems. The main goal of the book is to take the large body of material that has been traditionally applied to modeling and analyzing vibrating elastic systems at the macro scale and apply it to vibrating systems at the micrometer and nanometer scale. The models of the vibrating elastic systems that will be discussed include single and two degree-of-freedom systems, Euler-Bernoulli and Timoshenko beams, thin rectangular and annular plates, and cylindrical shells. A secondary goal is to present the material in such a manner that one is able to select the least complex model that can be used to capture the essential features of the system being investigated. The essential features of the system could include such effects as in-plane forces, elastic foundations, an appropriate form of damping, in-span attachments and attachments to the boundaries, and such complicating factors as electrostatic attraction, piezoelectric elements, and elastic coupling to another system. To assist in the model selection, a very large amount of numerical results has been generated so that one is also able to determine how changes to boundary conditions, system parameters, and complicating factors affect the system’s natural frequencies and mode shapes and how these systems react to externally applied displacements and forces.

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The material presented is reasonably self-contained and employs only a few solution methods to obtain the results. For continuous systems, the governing equations and boundary conditions are derived from the determination of the contributions to the total energy of the system and the application of the extended Hamilton’s principle. Two solution methods are used to determine the natural frequencies and mode shapes for very general boundary conditions, in-span attachments, and complicating factors such as in-place forces and elastic foundations. When possible, the Laplace transform is used to obtain the characteristic equation in terms of standard functions. For these systems, numerous special cases of the very general solutions are obtained and tabulated. Many of these analytically obtained results are new. For virtually all other cases, the Rayleigh-Ritz method is used. Irrespective of the solution method, almost all solutions that are derived in this book have been numerically evaluated by the author and presented in tables and annotated graphs. This has resulted in a fair amount of new material. The book is organized into seven chapters, six of which describe different vibratory models for micromechanical systems and nano-scale systems and their ranges of applicability. In Chapter 2, single and two degree-of-freedom system models are used to obtain a basic understanding of squeeze film damping, viscous fluid loading, electrostatic and van der Waals attractive forces, piezoelectric and electromagnetic energy harvesters, enhanced piezoelectric energy harvesters, and atomic force microscopy. In Chapters 3 and 4, the Euler-Bernoulli beam is introduced. This model is used to determine: the effects of an in-span proof mass and a proof mass mounted at the free boundary of a cantilever beam; the applicability of elastically coupled beams as a model for double-wall carbon nanotubes; its use as a biosensor; the frequency characteristics of tapered beams and the response of harmonically base-driven cantilever beams used in atomic force microscopy; the effects of electrostatic fields, with and without fringe correction, on the natural frequency; the power generated from a cantilever beam with a piezoelectric layer; and to compare the amplitude frequency response of beams for various types of damping at the macro scale and at the sub millimeter scale. Also determined in Chapter 3 is when a single degree-of-freedom system can be used to estimate the lowest natural frequency a beam with a concentrated mass and when a two degree-of-freedom system can be used to estimate the lowest natural frequency of a beam with a concentrated mass to which a single degree-of-freedom system is attached. In Chapter 5, the Timoshenko theory is introduced, which gives improved estimates for the natural frequency. One of the objectives of this chapter is to numerically show under what conditions one can use the Euler-Bernoulli beam theory and when one should use the Timoshenko beam theory. Therefore, many of the same systems that are examined in Chapter 3 are re-examined in this chapter and the results from each theory are compared and regions of applicability are determined. The transverse and extensional vibrations of thin rectangular and annular circular plates are presented in Chapter 6. The results of extensional vibrations of circular plates have applicability to MEMS resonators for RF devices. In the last chapter, Chapter 7, the Donnell and Flügge shell theories are introduced and used to

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obtain approximate natural frequencies and mode shapes of single-wall and doublewall carbon nanotubes. The results from these shell theories are compared to those predicted by the Euler-Bernoulli and Timoshenko beam theories. I would like to thank my colleagues Dr. Balakumar Balachandran for his encouragement to undertake this project and his continued support to its completion and Dr. Amr Baz for his assistance with some of the material on beam energy harvesters. I would also like to acknowledge the students in my 2011 spring semester graduate class where much of this material was “field-tested.” Their comments and feedback led to several improvements. College Park, Maryland

Edward B. Magrab

Contents

1 Introduction . . . . . . . . . . . . 1.1 A Brief Historical Perspective 1.2 Importance of Vibrations . . . 1.3 Analysis of Vibrating Systems 1.4 About the Book . . . . . . . . Reference . . . . . . . . . . . . . .

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2 Spring-Mass Systems . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Some Preliminaries . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 A Brief Review of Single Degree-of-Freedom Systems 2.2.2 General Solution: Harmonically Varying Forcing . . . 2.2.3 Power Dissipated by a Viscous Damper . . . . . . . . 2.2.4 Structural Damping . . . . . . . . . . . . . . . . . . . 2.3 Squeeze Film Air Damping . . . . . . . . . . . . . . . . . . . 2.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Rectangular Plates . . . . . . . . . . . . . . . . . . . . 2.3.3 Circular Plates . . . . . . . . . . . . . . . . . . . . . . 2.3.4 Base Excitation with Squeeze Film Damping . . . . . 2.3.5 Time-Varying Force Excitation of the Mass . . . . . . 2.4 Viscous Fluid Damping . . . . . . . . . . . . . . . . . . . . . 2.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Single Degree-of-Freedom System in a Viscous Fluid . 2.5 Electrostatic and van der Waals Attraction . . . . . . . . . . . 2.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 2.5.2 Single Degree-of-Freedom System with Electrostatic Attraction . . . . . . . . . . . . . . . . . 2.5.3 van der Waals Attraction and Atomic Force Microscopy 2.6 Energy Harvesters . . . . . . . . . . . . . . . . . . . . . . . . 2.6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 2.6.2 Piezoelectric Generator . . . . . . . . . . . . . . . . . 2.6.3 Maximum Average Power of a Piezoelectric Generator 2.6.4 Permanent Magnet Generator . . . . . . . . . . . . . .

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2.6.5 Maximum Average Power of a Permanent Magnetic Generator . . . . . . . . . . . . . . . 2.7 Two Degree-of-Freedom Systems . . . . . . . . . . . 2.7.1 Introduction . . . . . . . . . . . . . . . . . . . 2.7.2 Harmonic Excitation: Natural Frequencies and Frequency-Response Functions . . . . . . 2.7.3 Enhanced Energy Harvester . . . . . . . . . . . 2.7.4 MEMS Filters . . . . . . . . . . . . . . . . . . 2.7.5 Time-Domain Response . . . . . . . . . . . . . 2.7.6 Design of an Atomic Force Microscope Motion Scanner . . . . . . . . . . . . . . . . . . . . . Appendix 2.1 Forces on a Submerged Vibrating Cylinder . . References . . . . . . . . . . . . . . . . . . . . . . . . . . .

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3 Thin Beams: Part I . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Derivation of Governing Equation and Boundary Conditions 3.2.1 Contributions to the Total Energy . . . . . . . . . . . 3.2.2 Governing Equation . . . . . . . . . . . . . . . . . . 3.2.3 Boundary Conditions . . . . . . . . . . . . . . . . . 3.2.4 Non Dimensional Form of the Governing Equation and Boundary Conditions . . . . . . . . . . . . . . . 3.3 Natural Frequencies and Mode Shapes of Beams with Constant Cross Section and with Attachments . . . . . 3.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Solution for Very General Boundary Conditions . . . 3.3.3 General Solution in the Absence of an Axial Force and an Elastic Foundation . . . . . . . . . . . . . . . 3.3.4 Numerical Results . . . . . . . . . . . . . . . . . . . 3.3.5 Cantilever Beam as a Biosensor . . . . . . . . . . . . 3.4 Single Degree-of-Freedom Approximation of Beams with a Concentrated Mass . . . . . . . . . . . . . . . . . . . 3.5 Beams with In-Span Spring-Mass Systems . . . . . . . . . . 3.5.1 Single Degree-of-Freedom System . . . . . . . . . . 3.5.2 Two Degree-of-Freedom System with Translation and Rotation . . . . . . . . . . . . . . . . . . . . . . 3.6 Effects of an Axial Force and an Elastic Foundation on the Natural Frequency . . . . . . . . . . . . . . . . . . . 3.7 Beams with a Rigid Extended Mass . . . . . . . . . . . . . 3.7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 3.7.2 Cantilever Beam with a Rigid Extended Mass . . . . 3.7.3 Beam with an In-Span Rigid Extended Mass . . . . . 3.8 Beams with Variable Cross Section . . . . . . . . . . . . . . 3.8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 3.8.2 Continuously Changing Cross Section . . . . . . . .

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3.8.3 Linear Taper . . . . . . . . . . . . . . . . . . . . . 3.8.4 Exponential Taper . . . . . . . . . . . . . . . . . . 3.8.5 Approximate Solutions to Tapered Beams: Rayleigh-Ritz Method . . . . . . . . . . . . . . . . 3.8.6 Triangular Taper: Application to Atomic Force Microscopy . . . . . . . . . . . . . . . . . . . . . 3.8.7 Constant Cross Section with a Step Change in Properties . . . . . . . . . . . . . . . . . . . . . 3.8.8 Stepped Beam with an In-Span Rigid Support . . . 3.9 Elastically Connected Beams . . . . . . . . . . . . . . . . 3.9.1 Introduction . . . . . . . . . . . . . . . . . . . . . 3.9.2 Beams Connected by a Continuous Elastic Spring . 3.9.3 Beams with Concentrated Masses Connected by an Elastic Spring . . . . . . . . . . . . . . . . . . . . 3.10 Forced Excitation . . . . . . . . . . . . . . . . . . . . . . 3.10.1 Boundary Conditions and the Generation of Orthogonal Functions . . . . . . . . . . . . . . . . 3.10.2 General Solution . . . . . . . . . . . . . . . . . . 3.10.3 Impulse Response . . . . . . . . . . . . . . . . . . 3.10.4 Time-Dependent Boundary Excitation . . . . . . . 3.10.5 Forced Harmonic Oscillations . . . . . . . . . . . 3.10.6 Harmonic Boundary Excitation . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Thin Beams: Part II . . . . . . . . . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 4.2 Damping . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Generation of Governing Equation . . . . . . . 4.2.2 General Solution . . . . . . . . . . . . . . . . 4.2.3 Illustration of the Effects of Various Types of Damping: Cantilever Beam . . . . . . . . . . . 4.3 In-Plane Forces and Electrostatic Attraction . . . . . . 4.3.1 Introduction . . . . . . . . . . . . . . . . . . . 4.3.2 Beam Subjected to a Constant Axial Force . . . 4.3.3 Beam Subject to In-Plane Forces and Electrostatic Attraction . . . . . . . . . . . 4.4 Piezoelectric Energy Harvesters . . . . . . . . . . . . 4.4.1 Governing Equations and Boundary Conditions 4.4.2 Power from the Harmonic Oscillations of a Base-Excited Cantilever Beam . . . . . . . . . Appendix 4.1 Hydrodynamic Correction Function . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . .

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5 Timoshenko Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Derivation of the Governing Equations and Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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5.2.1 5.2.2 5.2.3 5.2.4 5.2.5

Introduction . . . . . . . . . . . . . . . . . . . . . . . Contributions to the Total Energy . . . . . . . . . . . . Governing Equations . . . . . . . . . . . . . . . . . . Boundary Conditions . . . . . . . . . . . . . . . . . . Non Dimensional Form of the Governing Equations and Boundary Conditions . . . . . . . . . . 5.2.6 Reduction of the Timoshenko Equations to That of Euler-Bernoulli . . . . . . . . . . . . . . . . . . . . 5.3 Natural Frequencies and Mode Shapes of Beams with Constant Cross Section, Elastic Foundation, Axial Force, and In-span Attachments . . . . . . . . . . . . . 5.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Solution for Very General Boundary Conditions . . . . 5.3.3 Special Cases . . . . . . . . . . . . . . . . . . . . . . 5.3.4 Numerical Results . . . . . . . . . . . . . . . . . . . . 5.4 Natural Frequencies of Beams with Variable Cross Section . . 5.4.1 Beams with a Continuous Taper: Rayleigh-Ritz Method 5.4.2 Constant Cross Section with a Step Change in Properties . . . . . . . . . . . . . . . . . . . . . . . 5.4.3 Numerical Results . . . . . . . . . . . . . . . . . . . . 5.5 Beams Connected by a Continuous Elastic Spring . . . . . . . 5.6 Forced Excitation . . . . . . . . . . . . . . . . . . . . . . . . 5.6.1 Boundary Conditions and the Generation of Orthogonal Functions . . . . . . . . . . . . . . . . . . 5.6.2 General Solution . . . . . . . . . . . . . . . . . . . . 5.6.3 Impulse Response . . . . . . . . . . . . . . . . . . . . Appendix 5.1 Definitions of the Solution Functions fl and gl and Their Derivatives . . . . . . . . . . . . . . . . . . . . . . Appendix 5.2 Definitions of Solution Functions fli and gli and Their Derivatives . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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6 Thin Plates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Derivation of Governing Equation and Boundary Conditions: Rectangular Plates . . . . . . . . . . . . . . . . . . 6.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 6.2.2 Contributions to the Total Energy . . . . . . . . . . . . . 6.2.3 Governing Equations . . . . . . . . . . . . . . . . . . . 6.2.4 Boundary Conditions . . . . . . . . . . . . . . . . . . . 6.2.5 Non Dimensional Form of the Governing Equation and Boundary Conditions . . . . . . . . . . . . . . . . . 6.3 Governing Equations and Boundary Conditions: Circular Plates . 6.4 Natural Frequencies and Mode Shapes of Circular Plates for Very General Boundary Conditions . . . . . . . . . . . . . .

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6.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 6.4.2 Natural Frequencies and Mode Shapes of Annular and Solid Circular Plates . . . . . . . . . . . . . . . . . 6.4.3 Numerical Results . . . . . . . . . . . . . . . . . . . . . 6.5 Natural Frequencies and Mode Shapes of Rectangular and Square Plates: Rayleigh-Ritz Method . . . . . . . . . . . . 6.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 6.5.2 Natural Frequencies and Mode Shapes of Rectangular and Square Plates . . . . . . . . . . . . . . 6.5.3 Numerical Results . . . . . . . . . . . . . . . . . . . . . 6.5.4 Comparison with Thin Beams . . . . . . . . . . . . . . 6.6 Forced Excitation of Circular Plates . . . . . . . . . . . . . . . 6.6.1 General Solution to the Forced Excitation of Circular Plates . . . . . . . . . . . . . . . . . . . . . . . 6.6.2 Impulse Response of a Solid Circular Plate . . . . . . . . 6.7 Circular Plate with Concentrated Mass Revisited . . . . . . . . 6.8 Extensional Vibrations of Plates . . . . . . . . . . . . . . . . . 6.8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 6.8.2 Contributions to the Total Energy . . . . . . . . . . . . . 6.8.3 Governing Equations and Boundary Conditions . . . . . 6.8.4 Natural Frequencies and Mode Shapes of a Circular Plate 6.8.5 Numerical Results . . . . . . . . . . . . . . . . . . . . . Appendix 6.1 Elements of Matrices in Eq. (6.100) . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Cylindrical Shells and Carbon Nanotube Approximations 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 7.2 Derivation of Governing Equations and Boundary Conditions: Flügge’s Theory . . . . . . . . . . . . . . 7.2.1 Introduction . . . . . . . . . . . . . . . . . . . 7.2.2 Contributions to the Total Energy . . . . . . . . 7.2.3 Governing Equations . . . . . . . . . . . . . . 7.2.4 Boundary Conditions . . . . . . . . . . . . . . 7.2.5 Boundary Conditions and the Generation of Orthogonal Functions . . . . . . . . . . . . . . 7.3 Derivation of Governing Equations and Boundary Conditions: Donnell’s Theory . . . . . . . . . . . . . . 7.3.1 Introduction . . . . . . . . . . . . . . . . . . . 7.3.2 Contribution to the Total Energy . . . . . . . . 7.3.3 Governing Equations . . . . . . . . . . . . . . 7.3.4 Boundary Conditions . . . . . . . . . . . . . . 7.4 Natural Frequencies of Clamped and Cantilever Shells: Single-Wall Carbon Nanotube Approximations . . . . 7.4.1 Rayleigh-Ritz Solution . . . . . . . . . . . . . 7.4.2 Numerical Results . . . . . . . . . . . . . . . .

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7.5

Natural Frequencies of Hinged Shells: Double-Wall Carbon Nanotube Approximation . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Appendix A Strain Energy in Linear Elastic Bodies . . . . . . . . . . . . . . . . . Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Appendix B Variational Calculus: Generation of Governing Equations, Boundary Conditions, and Orthogonal Functions . . . . . . B.1 Variational Calculus . . . . . . . . . . . . . . . . . . . . B.1.1 System with One Dependent Variable . . . . . . B.1.2 A Special Case for Systems with One Dependent Variable . . . . . . . . . . . . . . . . . . . . . . B.1.3 Systems with N Dependent Variables . . . . . . . B.1.4 A Special Case for Systems with N Dependent Variables . . . . . . . . . . . . . . . . . . . . . . B.2 Orthogonal Functions . . . . . . . . . . . . . . . . . . . B.2.1 Systems with One Dependent Variable . . . . . . B.2.2 Systems with N Dependent Variables . . . . . . . B.3 Application of Results to Specific Elastic Systems . . . . Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix C Laplace Transforms and the Solutions to Differential Equations . . . . . . . . . . . C.1 Definition of the Laplace Transform . C.2 Solution to a Second-Order Equation . C.3 Solution to a Fourth-Order Equation . C.4 Table of Laplace Transform Pairs . . . Reference . . . . . . . . . . . . . . . . . .

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. . . . . .

. . . . . .

. . . . . .

461 463 463 468 473 476

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

477 477 478 479 483 484

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

485

Ordinary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

Chapter 1

Introduction

1.1 A Brief Historical Perspective It is likely that the early interest in vibrations was due to the development of musical instruments such as whistles and drums. In was in modern times, starting around 1583, when Galilei Galileo made his observations about the period of a pendulum, that the subject of vibrations attracted scientific scrutiny. In the 1600’s, strings were analyzed by Marin Mersenne and John Wallis; in the 1700’s, beams were analyzed by Leonhard Euler and Daniel Bernoulli and plates were analyzed by Sophie Germain; in the 1800’s, plates were analyzed by Gustav Kirchhoff and Simeon Poisson, and shells by D. Codazzi and A. E. H. Love. A complete historical development of the subject can be found in (Love, 1927). Lord Rayleigh’s book Theory of Sound, which was first published in 1877, is one of the early comprehensive publications on the subject of vibrations. Since the publication of his book, there has been considerable growth in the diversity of devices and systems that are designed with vibrations in mind: mechanical, electromechanical, biomechanical and biomedical, ships and submarines, and civil structures. Along with this explosion of interest in quantifying the vibrations of systems, came great advances in the computational and analytical tools available to analyze them.

1.2 Importance of Vibrations Vibrations occur all around us. In the human body, where there are low-frequency oscillations of the lungs and the heart and high-frequency oscillations of the larynx as one speaks. In man-made systems, where any unbalance in machines with rotating parts such as fans, washing machines, centrifugal pumps, rotary presses, and turbines, can cause vibrations. In buildings and structures, where passing vehicular, air, and rail traffic or natural phenomena such as earthquakes and wind can cause oscillations. In some cases, oscillations are undesirable. In structural systems, the fluctuating stresses due to vibrations can result in fatigue failure. When performing precision measurements such as with an electron microscope externally caused oscillations E.B. Magrab, Vibrations of Elastic Systems, Solid Mechanics and Its Applications 184, C Springer Science+Business Media B.V. 2012 DOI 10.1007/978-94-007-2672-7_1, 

1

2

1

Introduction

must be substantially minimized. In air, roadway, and railway vehicles, oscillatory input to the passenger compartments must be reduced. In machinery, vibrations can cause excessive wear or cause situations that make a device difficult to control. Vibrating systems can also produce unwanted audible acoustic energy that is annoying or harmful. On the other hand, vibrations also have many beneficial uses in such widely diverse applications as vibratory parts feeders, paint mixers, transducers and sensors, ultrasonic devices used in medicine and dentistry, sirens and alarms for warnings, determining fundamental properties of materials, and stimulating bone growth. During the last decade or so, there has been a increase in the development of electromechanical devices and systems at the micrometer and nanometer scale. The introduction of these artifacts at this sub millimeter scale has created renewed interest in the vibrations of elastic systems. These developments have lead to new families of devices and sensors such as vibrating cantilever beam mass sensors, piezoelectric beam energy harvesters, carbon nanotube oscillators, and vibrating cantilever beam sensors for atomic force microscopes. Along with these devices come additional effects that are important at this scale such as viscous air damping, squeeze film damping, electrostatic attraction, and the size and location of a proof mass. Thus, the range of applications that the vibration of elastic systems has to consider has been increased.

1.3 Analysis of Vibrating Systems The analyses of systems subject to vibrations or designed to vibrate have many aspects. Typically, a system is designed to meet a set of vibration performance criteria such as to oscillate at a specific frequency, avoid a system resonance, operate at or below specific amplitude levels, have its response controlled, and be isolated from its surroundings. These criteria may involve the entire system or only specific portions of it. To determine if the performance criteria have been met, experiments are performed to determine the characteristics of the input to the system, the output from the system, and the system itself. Some of the characteristics of interest could be whether the input is harmonic, periodic, transient, or random and its respective frequency content and magnitude. Some of the characteristics of the output of the system could be the magnitude and frequency content of the force, velocity, displacement, acceleration, or stress at one or more locations. Some of the characteristics of the system itself could be its natural frequencies and mode shapes and its response to a specific input quantity. To design a system to meet its performance criteria, it is often necessary to model the system and then to analyze it in the context of these criteria. The type of model one uses may be a function of its size: the sub micrometer scale, micrometer scale, millimeter scale, or the centimeter scale and greater, which we denote as the macro scale. The model will also be a function of its shape, the way in which it is

1.4

About the Book

3

expected to oscillate, the way it is supported, and how it is constrained. If shape can be ignored, then the system can be modeled as a spring-mass system. If geometry is important, then one must choose an appropriate representation such as a beam, plate, or shell and decide if the geometry can be treated as a constant geometry or if it must be treated as a system with variable geometry. The system’s environment, in conjunction with its size, will determine which type of damping is important and if it must be taken into account. The model may also have to include the effects of any attachments to its interior and to its boundaries and may have to account for externally applied constraints and forces such as an elastic foundation, in-plane forces, and coupling to other elastic systems. Thus, there are many decisions that must be made with regard to what should be included in the model so that it adequately represents the actual system.

1.4 About the Book The main goal of the book is to take the large body of material that has been traditionally applied to modeling and analyzing vibrating elastic systems at the macro scale and apply it to vibrating systems at the micrometer and nanometer scale. The models of the vibrating elastic systems that will be discussed include single and two degree-of-freedom spring-mass systems, Euler-Bernoulli and Timoshenko beams, thin rectangular and annular plates, and cylindrical shells. A second goal is to present the material in such a manner that one is able to select the least complex model that can be used to capture the essential features of the system being investigated. The essential features of the system could include such effects as in-plane forces, elastic foundations, an appropriate form of damping, in-span attachments and attachments to the boundaries, and such complicating factors as electrostatic attraction, piezoelectric elements, and elastic coupling to another system. To assist in the model selection, a very large amount of numerical results has been generated so that one is able compare the various models to determine how changes to boundary conditions, system parameters, and complicating factors affect the natural frequencies and mode shapes and the response to externally applied displacements and forces. In order to be able to cover the wide range of models and complicating factors in sufficient detail, an efficient means of presenting the material is required. The approach employed here has been to obtain an expression for the total energy of each model and then to use the extended Hamilton’s principle to derive the governing equations and boundary conditions. The expression for the total energy of the system includes the effects of any complicating factors. In addition to providing an efficient and consistent way in which to obtain the governing equations and boundary conditions, the expression for the total energy of the system can be used directly as the starting point for the Rayleigh-Ritz method. Another advantage of the energy approach is that the results given here can be extended to systems that include other effects by modifying the expression for the total energy. The expressions used to

Two degree-of-freedom Euler-Bernoulli theory

Beams

Rectangular Circular

Donnell’s theory Flügge’s theory

Thin Plates

Thin Cylindrical Shells

– In-plane force Elastic foundation Extensional oscillations Elastic coupling to another shell

Damping: structural, viscous, squeeze film, viscous fluid Electrostatic force van der Waals force Magnetic force Piezoelectric element Piezoelectric element Damping: structural, viscous, squeeze film, viscous fluid, viscous air Axial force Elastic foundation Electrostatic force Elastic coupling to another beam Layered beams Axial force Elastic foundation Elastic coupling to another beam

Single degree-of-freedom

Spring-Mass

Timoshenko theory

Additional factors

System

– Translation spring Torsion spring Concentrated mass –

Translation spring Torsion spring Concentrated mass

– Translation spring Torsion spring Concentrated mass Extended mass

–

Boundary attachments

–

Translation spring Torsion spring Concentrated mass Single degree-of-freedom system Concentrated mass Concentrated mass

– Translation spring Concentrated mass Single degree-of-freedom system Finite-length rigid mass

–

In-Span attachments

1

Constant

Constant Continuously variable Constant with abrupt change in properties Constant Constant

– Constant Continuously variable Constant with abrupt change in properties

–

Cross section

Table 1.1 The elastic systems considered in this book. Typical MEMS and NEMS applications of these systems are described in Table 1.2

4 Introduction

1.4

About the Book

5

arrive at the governing equations and boundary conditions will be the same. A list of the elastic systems and their additional factors that are considered in this book to model microelectromechanical and nano electromechanical systems are given in Table 1.1 and the corresponding specific applications associated with these elastic systems are given in Table 1.2. To make the application of the energy approach more efficient, an appendix, Appendix B, is provided with a general derivation of the extended Hamilton’s principle for systems with one or more dependent variables and it is shown there the conditions required in order for one to be able to generate orthogonal functions. Since a primary solution method employed in this book is the separable of variables, the generation and use of orthogonal functions is very important. Consequently, the use of energy approach, the application of the extended Hamilton’s principle, and the results of Appendix B provide the basis for a consistent approach to deriving the governing equations and boundary conditions and the basis for two very powerful solution techniques: the generation of orthogonal functions and the separation of variables and the Rayleigh-Ritz method. It will be seen that a major advantage of the use of the extended Hamilton’s principle is that the boundary conditions are a natural consequence of the method. This will prove to be very important when the Timoshenko beam theory, thin plate theory, and thin cylindrical shell theories are considered. In these cases, obtaining the boundary conditions can be quite involved if the force balance and moment balance methods are used. To determine the effects that various parameters and complicating factors have on a system, the following procedure is employed. For each elastic system, a solution for a very general set of boundary conditions and complicating factors as is practical is obtained. Once the general solution has been obtained, many of its special cases are examined in a direct and straightforward manner. This approach, while

Table 1.2 Table 1.1

Typical MEMS and NEMS application areas of the elastic systems described in

System Spring-Mass

Typical MEMS and NEMS Applications Single degree-of-freedom Two degree-of-freedom

Beams

Thin Plates Thin Cylindrical Shells

Euler-Bernoulli theory

Timoshenko theory Rectangular Circular Donnell’s theory Flügge’s theory

Piezoelectric and magnetic energy harvesters Atomic force microscopy Enhanced piezoelectric energy harvester Filters Atomic force microscopy Biosensors Effects of proof mass Piezoelectric energy harvester Atomic force microscopy Electrostatic devices Single- and double-wall carbon nanotubes – RF devices Single- and double-wall carbon nanotubes

6

1

Introduction

introducing a little more algebraic complexity at the outset, is a very efficient way of obtaining a solution to a class of systems and greatly reduces the need to re-solve and/or re-derive the equations each time another combination of factors is examined. In most cases, many of the systems’ special cases are listed in tables. As a consequence, in several cases, new analytical results have been obtained. In order to be able to use the least complex model to represent a system, each subsequent system is compared to a simpler model. For example, the conditions under which a beam with a concentrated mass can be modeled as a single degreeof-freedom system are determined. Other examples are the determination of the conditions when a beam can be used to model a narrow thin plate and when the Euler-Bernoulli beam theory can be used instead of the Timoshenko beam theory. An underlying aspect that allows one to present the large amount of material given in this book is the availability of the modern computer environments such as R R and Mathematica . These programs permit one to devote less space MATLAB to presenting special numerical solution techniques and more space to the development of the governing equations and boundary conditions, obtaining the general solutions, and presenting and discussing the numerical results. Consequently, virtually all solutions that are derived in this book have been numerically evaluated. This has produced a substantial amount of annotated graphical and tabular results that illustrate the influence that the various system parameters have on their respective responses. Many of these numerical results are new. In addition, the numerical results are presented in terms of non dimensional quantities making them applicable to a wide range of systems.

Reference Love AEH (1927) A treatise of the mathematical theory of elasticity, 4th edn. Dover, New York, NY, pp 1–31

Chapter 2

Spring-Mass Systems

The single degree-of-freedom system subject to mass and base excitation is used to model an elastic system to determine the frequency-domain effects of squeeze film air damping and viscous fluid damping. This model is also used to determine the important response characteristics of electrostatic attraction and van der Waals forces, the maximum average power from piezoelectric and electromagnetic coupling, and to illustrate the fundamental working principle of an atomic force microscope. The two degree-of-freedom system is introduced to examine microelectromechanical filters, atomic force microscope specimen control devices, and as a means to increase the input to piezoelectric energy harvesters. An appendix gives the details of the derivation of a hydrodynamic function that expresses the effects of a viscous fluid on a vibrating cylinder.

2.1 Introduction In determining the response of structural systems in the subsequent chapters, it will be seen that the different models frequently reduce to that of a set of single degree of freedom systems. Thus, a basic understanding of the response of single degree-offreedom systems in general and its response when the system is subjected to various complicating factors such as squeeze film damping, viscous fluid loading, electrostatic attraction, and piezoelectric and electromagnetic coupling is required. In this chapter, we shall analyze such systems in the absence of the structural aspects; in the subsequent chapters, the structure will be taken into account.

2.2 Some Preliminaries 2.2.1 A Brief Review of Single Degree-of-Freedom Systems A single degree-of-freedom system is shown in Fig. 2.1. The static displacement of the mass is δ st . The mass undergoes a displacement x (t) and the rigid container a

E.B. Magrab, Vibrations of Elastic Systems, Solid Mechanics and Its Applications 184, C Springer Science+Business Media B.V. 2012 DOI 10.1007/978-94-007-2672-7_2, 

7

8

2 Spring-Mass Systems

y k

Fs

c

Fd

δst mg

m

m

mg

x fo + f (t)

fo + f (t)

(a)

(b)

Fig. 2.1 (a) Vertical vibrations of a spring-mass-damper system (b) Free-body diagram

known displacement y (t). Both of these displacements are with respect to an inertial frame. The relationship between these two displacements is z (t) = x (t) − y (t) .

(2.1)

The mass is subjected to an externally applied constant force fo , a time-varying force f (t), and a reaction force Fr (z, z˙, z¨). This reaction force has been introduced so that forces that are produced by such phenomena as squeeze film damping, electrostatic attraction, and viscous fluids can be straightforwardly incorporated. When the rigid container is stationary, y (t) = 0 and z (t) = x(t). Referring to Fig. 2.1b, a summation of forces on the mass m in the vertical direction gives

m

d2x + Fs + Fd + Fr (z, z˙, z¨) = mg + fo + f (t) dt2

(2.2)

where the over dot indicates the derivative with respect to the time t and g = 9.81 m/s2 is the acceleration of gravity. The constant force fo can be caused, for example, by an electrostatic attraction (see Section 2.5.2), by a pressure difference between the top and bottom surfaces of the mass, and by a magnetic force if the mass were composed of a magnetic material. When the spring is linear, Fs = k (z + δst ), where k is the spring constant (N/m). The spring constant k is sometimes referred to as the derivative of the spring force since dFs /dz = k. When the damper is a linear viscous damper, Fd = c˙z, where c is the damper constant (Ns/m). For this case, Eq. (2.2) becomes

m

dz d2 x + c + k (z + δst ) + Fr (z, z˙, z¨) = mg + fo + f (t). 2 dt dt

(2.3)

2.2

Some Preliminaries

9

At the MEMS and NEMS scale, viscous damping arises from different phenomena that are functions of ambient pressure and temperature, amplitude and frequency of oscillation, viscosity, and geometric characteristics. Consideration of these effects and the computation of c can be found in (Martin and Houston 2007; Bhiladvala and Wang 2004; Keskar et al. 2008; Li et al. 2006). It is seen from Eq. (2.3) that δst =

1 (mg + fo ) m. k

(2.4)

From Eq. (2.1), x (t) = z (t) + y (t) and, therefore, using Eq. (2.4), Eq. (2.3) can be written as m

d2 z dz d2 y + c . + kz + F z ˙ , z ¨ = f (t) − m (z, ) r dt2 dt dt2

(2.5)

This equation represents the motion of the mass about the static equilibrium position. When y (t) = 0, Eq. (2.5) becomes m

d2 x dx + c + kx + Fr (x, x˙ , x¨ ) = f (t) dt2 dt

(2.6)

and when the reaction force is not present, Fr = 0 and Eq. (2.6) simplifies to m

dx d2 x + c + kx = f (t). 2 dt dt

(2.7)

Equation (2.7) can be used to model torsional oscillations. If kt is the torsional spring constant (Nm/rad), ct the torsional viscous damping constant (Nsm/rad), θ the angular rotation of the mass (rad), J the mass moment of inertia (kg m2 ), and M (t) the applied external moment (Nm), then Eq. (2.7) can be written as J

d2 θ dθ + kt θ = M(t). + ct dt dt2

(2.8)

Before proceeding, the following definitions are introduced. Natural Frequency—ωn For translating systems  ωn = 2π fn =

k rad/s m

(2.9)

where fn is the natural frequency in Hz. For torsional oscillations  ωn = 2π fn =

kt rad/s. J

(2.10)

10

2 Spring-Mass Systems

Damping Factor—ζ For translating systems ζ =

c cωn c = √ = . 2mωn 2k 2 km

(2.11)

When 0 < ζ < 1 the system is called an underdamped system, when ζ = 1 it is critically damped, and when ζ > 1 it is overdamped system. When ζ = 0, the system is undamped. For torsional oscillations ζ =

ct ct = √ . 2 Jωn 2 kt J

(2.12)

Period of Undamped Oscillations—T T=

1 2π = s. fn ωn

(2.13)

We return to Eq. (2.5) and set Fr = 0 to obtain m

d2 y dz d2 z + c + kz = f (t) − m 2 2 dt dt dt

(2.14)

We now introduce Eqs. (2.9) and (2.11) into Eq. (2.14) to arrive at the following governing equation of motion in terms of the natural frequency and damping factor f (t) d2 y d2 z dz 2 + 2ζ ω z = + ω − 2. n n dt m dt2 dt

(2.15)

If we let τ = ωn t, then Eq. (2.15) becomes f (τ ) d2 y dz d2 z + z = − + 2ζ . dτ k dτ 2 dτ 2

(2.16)

It is mentioned that when f (τ ) = 0, Eq. (2.16) can be used to describe the motion of an accelerometer, where d2 y/dτ 2 is the acceleration of the base (Balachandran and Magrab 2009, p. 237).

2.2.2 General Solution: Harmonically Varying Forcing We assume that ζ < 1, the initial conditions are zero, and the applied force and base displacement are of the form f (t) = Fo cos (τ ) y(t) = Yo cos (τ )

(2.17)

2.2

Some Preliminaries

11

where  = ω/ωn . It is seen that when ω = ωn ,  = 1,. To obtain a solution to Eq. (2.16), we assume x (τ ) = Xo cos (τ )

(2.18)

z (τ ) = Zo cos (τ ) and find that (Balachandran and Magrab 2009, pp. 671–673)  z (τ ) = H()

 Fo + 2 Yo cos (τ − θ ()) k

(2.19)

where 1 H() =  2 1 − 2 + (2ζ )2 θ () = tan−1

(2.20)

2ζ  . 1 − 2

The quantity H() is the amplitude response and the quantity θ () is the phase response. It is seen from Eq. (2.20) that the frequency at which the maximum value of the amplitude response occurs is a function of ζ , as will be demonstrated subsequently. A plot of H() and θ () is shown in Fig. 2.2. It is seen that for viscous damping, the phase angle is 90◦ when  = 1, irrespective of the value of ζ . When Yo = 0, there are three frequency regions of interest based on Eq. (2.20). The first region is when  > 1, where H () ∼ = 1/2 and z (τ ) ∼ 1/m. This region is called the mass controlled region and is important in the design of vibration isolators. We now use Eq. (2.20) to define the quality factor Q. Quality Factor—Q A quantity that is often used to define the band pass portion of H() when ζ is small is the quality factor Q, which is given by Q=

c . Bw

(2.21)

The quantity c is the center frequency and is defined as the geometric mean frequency c =

 cu cl

(2.22)

and Bw is the bandwidth given by Bw = cu − cl .

(2.23)

where cu and cl , respectively, are the upper and lower cutoff frequencies that satisfy Hmax H (cl ) = H (cu ) = √ . 2

(2.24)

The quantity Hmax is given by (Balachandran and Magrab, 2009, p. 211) Hmax =

1  2ζ 1 − ζ 2

√ ζ ≤1 2

(2.25)

√ ζ ≤ 1 2.

(2.26)

and occurs at max =

 1 − 2ζ 2

When an explicit relation for determining the value of Hmax does not exist, one determines its value numerically. Using Eqs. (2.24) and (2.25), it can be shown that (Balachandran and Magrab, 2009, p. 212)   cu = 1 − 2ζ 2 + 2ζ 1 − ζ 2   cl = 1 − 2ζ 2 − 2ζ 1 − ζ 2 .

(2.27)

2.2

Some Preliminaries

13

When ζ < 0.1, Eq. (2.27) can be approximated by √

cu ≈

√

cl ≈ Thus,

1 + 2ζ ≈ 1 + ζ

(2.28)

1 − 2ζ ≈ 1 − ζ .  1 − ζ2 ≈ 1

c ≈

Bw ≈ 2ζ and Eqs. (2.21) and (2.25) give Q≈

1 ≈ Hmax 2ζ

(2.29)

which overestimates the value of Q. The error made in using Eq. (2.29) relative to Eq. (2.21) is less than 3% for ζ < 0.1 and when ζ < 0.01 the error is less than 0.03%. The quality factor has been shown to be of fundamental importance in the determination of the noise floor in MEMS sensors and plays a role in determining the sensitivity of certain MEMS devices (Gabrielson 1993; Levinzon 2004).

2.2.3 Power Dissipated by a Viscous Damper The average power that is dissipated in the viscous damper per period of oscillation T = 2π/ω = 2π/(ωn ) is

Pavg

1 = T

T

 Pi dt = 2π

0

2π/

Pi dτ

(2.30)

0

where Pi is the instantaneous power given by  Pi = Fd z˙ = c

dz dt



2

2ζ k 2 = H ()2 ωn

= cωn2 

dz dτ

2

ωn Fo + 2 ωn Yo k

2

(2.31) sin (τ − θ ()) W. 2

In obtaining Eq. (2.31), we have used Eqs. (2.11) and (2.19). Upon substituting Eq. (2.31) into Eq. (2.30) and performing the integration, we obtain Pavg =

ζk 2 H ()2 ωn



ωn Fo + 2 ωn Yo k

2 W.

(2.32)

14

2 Spring-Mass Systems

The average dissipated power is a maximum at the value of  = max that makes dPavg = 0. d

(2.33)

We shall determine the maximum dissipated power for two separate cases: Fo = 0 and Yo = 0 and Yo = 0 and Fo = 0. For the first case, we perform the operation indicated by Eq. (2.33) and find that max = 1. Thus, the maximum average power dissipated into the viscous damper by the external force is Pavg,max

ζk 2 = H (max ) 2max ωn PF Q PF = W = 4ζ 2



ωn Fo k

2 (2.34)

where PF =

ωn Fo2 W. k

(2.35)

For the second case, we employ Eq. (2.33) and find  max 2,1 =

  2 1 − 2ζ 2 ±



 2 4 1 − 2ζ 2 − 3

ζ < 0.2588.

(2.36)

Thus, the maximum average power dissipated into the viscous damper by the rigid container’s displacement is Pavg,max = ζ PY H 2 (max 1 ) 6max 1 W

(2.37)

PY = kωn Yo2 W.

(2.38)

where

When ζ 1, Eq. (2.81) can be approximated by 4 4 − j√ cir ≈ 1 + √ 2Re 2Re

Re >> 1.

It has been found numerically that this approximation gives the following errors: for Real [ cir ], the error is less than 1% when Re > 4 and less than 5% when Re > 0.6; for Imag [ cir ], the error is less than 1% when Re > 6000 and less than 5% when Re > 200.

2.4.2 Single Degree-of-Freedom System in a Viscous Fluid Consider a cylindrical rod of mass m and length L that is completely submerged in a viscous fluid. The mass is attached to a spring with constant k and a viscous damper c. It is assumed that the system’s container is stationary; thus, y(t) = 0. The system is undergoing forced harmonic vibrations of frequency ω and magnitude Fo . In Eq. (2.79), the negative sign indicates that force is in the same direction as Fr shown in Fig. 2.1; that is, it is in a direction opposite to x. Hence, Fr = −Ff and Eq. (2.6) becomes (m + ma )

d2 x dx + (c + cv ) + kx = Fo ejωt dt dt2

(2.83)

where ma and cv are given by Eq. (2.80). The quantity ma and cv are assumed to be reasonably represented by Eq. (2.80) if L/b >> 1. Before obtaining the solution to Eq. (2.83), we introduce the following non dimensional quantity Ren =

ρf ωn b2 4μf

(2.84)

ρf ρf ALωn = c 2ζρm

(2.85)

and note that ρf ρf AL = , m ρm

 where m = ρm AL, ρm is the density of the cylindrical rod, and ωn = k m is the natural frequency of the system in a vacuum. Then, with τ = ωn t, Eq. (2.83) can be written as me ()

Fo jτ d2 w dw +w= e + ce () dτ k dτ 2

(2.86)

2.4

Viscous Fluid Damping

29

where, from Eqs. (2.85), ρf Real ((Ren )) ρm ρf Imag ((Ren )) ce () = 2ζ − ρm  √ 4K1 j Ren  . √ (Ren ) = 1 + √ j Ren K0 j Ren  me () = 1 +

(2.87)

It is noted that ( (Ren )) < 0 and, therefore, ce ≥ 1. To solve Eq. (2.86), we assume a solution of the form x (t) = Xo ejτ and obtain Xo =

Fo Hf ()e−jψ() k

(2.88)

where the amplitude response and phase response, respectively, are Hf () =

−1/2  2 1 − 2 me () + (ce ())2

ψ() = tan−1

(2.89)

ce () . 1 − 2 me ()

When the fluid medium is absent, ρf = 0 and, therefore, me = 1 and ce = 2ζ and Eq. (2.89) reduces to Eq. (2.20). The values of Hf as a function of  for Ren = 10, 100, and 1000, and ζ = 0, are shown in Fig. 2.11a for ρf /ρm = 0.5 and in Fig. 2.12a for ρf /ρm = 1.5. Also appearing in each of these figures is the quality factor Q corresponding to each value of Ren . The maximum value of Hf and the frequency at which it occurs, max , are shown in Fig. 2.11b for ρf /ρm = 0.5 and in Fig. 2.12b for ρf /ρm = 1.5. From 2

35

10

Ren = 101, Q = 2.63 Ren = 102, Q = 9.29 Ren = 103, Q = 30

10 × Ωmax

30

Q

Hf ( Ωmax)

25

1

Hf(Ω)

Magnitude

10

0

10

20 15 10 5

10–1

0

0.5

1

1.5

0

10

1

2

10

Ω

Ren

(a)

(b)

3

10

Fig. 2.11 Changes in the resonant frequency ratio and magnitude of Hf for ρf /ρm = 0.5: (a) Hf for Ren = 10, 100, and 1000; (b) Magnitude of the maximum values of Hf , values of max at which maximum values of Hf occur, and Q as a function of Ren

30

2 Spring-Mass Systems 16

102 Ren = 101, Q = 1.2

14

Ren = 102, Q = 4.64

10 × Ωmax

Q Hf ( Ωmax)

Ren = 103, Q = 14.9

12

Hf(Ω)

Magnitude

101

0

10

10 8 6 4 2

10–1

0

0

Ω

102 Ren

(a)

(b)

0.5

1

1.5

101

103

Fig. 2.12 Changes in the resonant frequency ratio and magnitude of Hf for ρf /ρm = 1.5: (a) Hf for Ren = 10, 100, and 1000; (b) Magnitude of the maximum values of Hf , values of max at which maximum values of Hf occur, and Q as a function of Ren

these quantities, we have used Eq. (2.21) to compute the quality factor Q, which is also plotted in Figs. 2.11b and 2.12b. It is seen that Hf is strongly influenced by the values of the ratio ρf /ρm and by Ren . As observed in these figures, the viscous damping decreases as Ren increases, which results in an increase in the maximum value of Hf . From Fig. 2.10 and Eq. (2.89), it is seenthat at large values of Ren  the resonant frequency can be estimated as  ≈ 1 1 + ρf ρm . The estimates are very close to what is shown in Figs. 2.11a and 2.12a when Ren = 1000. On the other hand, as Ren decreases, Real (()) increases and, therefore, so does me . Hence, since a resonance occurs in the vicinity of  = 1/me , a decrease in Ren will cause a decrease in the value of  at which Hf is a maximum. This shift is indicated in the figures. It is seen in Figs. 2.11b and 2.12b that the quality factor and the maximum values of Hf vary in an almost identical manner as a function of Ren . This is a direct consequence of the approximation given by Eq. (2.29).

2.5 Electrostatic and van der Waals Attraction 2.5.1 Introduction Electrostatics is a well-established sensing and actuating technique. An electrostatically actuated device is a conductor that is elastically suspended above a stationary conductor. The opposing surfaces of the conductors are usually flat and parallel to each other. A dielectric medium such as air fills the gap between them. The overall system behaves as a variable gap capacitor. An applied DC voltage causes the elastically-suspended conductor to displace, which results in a change in the capacitance. Typical applications for these devices are switches, micro-mirrors, pressure sensors, micro-pumps, moving valves, and micro-grippers. When an AC component is superimposed on the steady voltage to excite harmonic motions of the system, resonators are obtained. These devices are used in signal filtering and

2.5

Electrostatic and van der Waals Attraction

31

chemical and mass sensing. A review of these and other applications can be found in (Batra et al. 2007). It will be shown that the applied electrostatic voltage has an upper limit beyond which the electrostatic force is not balanced by the restoring force of the elasticallysuspended conductor and the suspended conductor eventually touches the lower conductor and the device no longer functions. This phenomenon is called pull-in instability and the value of the voltage at which this occurs is an important design parameter in the development of many electrostatic devices. The van der Waals force exhibits properties similar to that of an electrostatic force. As will be shown in Section 2.5.3, the nature of this attractive force is the basis of the non contacting atomic force microscope (Martin et al. 1987), which is an instrument that can be used to measure intermolecular and inter-surface forces, to characterize the mechanical properties of various materials, to measure the local viscoelastic properties of films and cells, and to visualize artifacts in molecular biology. Some advantages of an atomic force microscope (AFM) are that it provides a three dimensional image of the surface, the surface does not need any special treatment or preparation, and it works in a fluid. The latter two advantages are what make it possible for an AFM to examine biological materials. A more detailed discussion of various aspects of atomic force microscopy can be found in the literature (Binnig et al. 1986; Butt et al. 2005; Jalili and Laxminarayana 2004; Raman et al. 2008; Abramovitch et al. 2007).

2.5.2 Single Degree-of-Freedom System with Electrostatic Attraction We shall consider a single degree of freedom system in which the bottom surface of the mass is flat and parallel to an opposing stationary flat surface. The surfaces have an area A and are separated by an air gap of distance do . A voltage of magnitude Vo is applied across this gap to form a capacitor. The electrostatic attractive force created by the voltage differential is given by1 Fr = −

1

εo AVo2 2 (do − x)2

(2.90)

If i is the current, e is the voltage, and q the charge, then the potential energy of a capacitor is



1 Ue = iedt = edq = C ede = Ce2 2

since and i = dq/dt and q = Ce, where C is the capacitance. For parallel plates separated by a distance go , C = εo A/go and we have Ue = The force is obtained from F=−

εo Ae2 . 2go

∂Ue εo Ae2 = . ∂go 2g2o

32

2 Spring-Mass Systems

where the minus sign indicates that the electrostatic force is in a direction opposite to that shown in Fig. 2.1 and εo = 8.854 × 10−12 F/m is the permittivity of free space. When one of the plates is the shape of a beam and the fixed plate is the same size and shape of the beam, the right hand side of Eq. (2.90) often includes a fringe correction factor due to the finite width and thickness of the beam. The fringe correction factor is introduced in Section 3.2.1. With f (t) = y (t) = 0 and using Eq. (2.90), Eq. (2.6) becomes m

εo Vo2 A d2 x dx + kx = + c . dt dt2 2 (do − x)2

(2.91)

The non dimensional quantities τ = ωn t and the damping factor ζ defined in Eq. (2.11) are introduced, and Eq. (2.91) becomes e21 Vo2 d2 w dw + 2ζ + w = dτ 2 dτ (1 − w)2

(2.92)

where e21 =

εo A −2 V 2kdo3

and

w=

x < 1. do

(2.93)

Since the electrostatic force draws the mass towards the fixed plate, it is necessary to determine the static deflection of the mass and the conditions that are required so that the mass never comes in contact with the fixed plate. That is, one needs to know the magnitude of V so that this condition can be avoided. The static equilibrium position is determined from Eq. (2.92) by ignoring the time-varying terms. Thus, w=

e21 Vo2 (1 − w)2

(2.94)

or w3 − 2w2 + w − e21 Vo2 = 0.

(2.95)

Equation (2.95) has three roots. The types of roots, real or complex, can be determined from the sign of the quantity (Tuma 1979, p. 7; Zhang and Zhao 2006)  D=

e21 Vo2

e21 Vo2 1 − 4 27

 (2.96)

or when this quantity equals zero. The quantity D = 0 when e21 Vo2 = 4/27. This value of e21 Vo2 results in two equal roots and one root in which w > 1, which is inadmissible. The equal roots have a value wst = 1/3. When e21 Vo2 > 4/27 there is one real root, the value of which is always greater than one and is, therefore,

2.5

Electrostatic and van der Waals Attraction

33

inadmissible. When e21 Vo2 < 4/27, there are three real, unequal roots. Two of these roots are always larger than 1/3 and, therefore, are ignored. Thus, the maximum voltage that can be applied to the electrostatic system is determined from  4 1 Vs,max = V (2.97) e1 27 which results in a maximum static displacement wmax = 1/3. It is seen that the ability to resist this attractive force is a function of the spring stiffness k. For a fixed geometry and gap do , the larger the value of k the larger the value of Vs,max . Then, from Eqs. (2.94) and (2.97), we obtain w(1 − w)2 =

4 Vo2 , 2 27 Vs,max

Vo ≤ Vs,max and w ≤ 1/3.

(2.98)

This relationship ensures that for the model given, the solution will always be stable; that is, the mass will not come in contact with the stationary plate. When the system is subjected to a step voltage Vo u(τ ), where u(τ ) is the unit step function, then, depending on the value of the damping factor ζ , the mass will overshoot its static equilibrium position. One should expect in this case that the magnitude of Vs,max will be lower than that obtained by only considering the static displacement. To determine the amount that this value will change, we use an energy balance (Nielson and Barbastathis 2006). At any instance of time, the total energy of the system Ein is Ein = Ek + Ep + Ed

(2.99)

where Ek is the kinetic energy, Ep is the potential energy, and Ed is the energy dissipated in the viscous damper. The maximum overshoot occurs when the damping is zero; thus, we set Ed = 0. At the instant in time when the kinetic energy is zero, the potential energy is a maximum. In this situation, Ek = 0 and Ep =

1 2 kx . 2

Thus, Eq. (2.99) reduces to Ein = Ep =

1 2 1 kx = kdo2 w2 2 2

(2.100)

where we have used Eq. (2.93). The energy input to the system comes from the electrostatic system, which is obtained by integrating the electrostatic force over the distance traveled. Thus, from Eq. (2.90)

x Ein = 0

εo Vo2 A Fr dx = 2

x 0

dx (do − x)2

=

εo Vo2 Aw εo Vo2 Ax = . 2do (do − x) 2do (1 − w)

(2.101)

34

2 Spring-Mass Systems

Then, from Eqs. (2.100) and (2.101), it is found that w(1 − w) = 2e21 Vo2 .

(2.102)

Solving Eq. (2.102) for w, it is found that the maximum value of the voltage is Vd,max =

1 √ V e1 2 2

(2.103)

which results in a maximum displacement w = wmax = 1/2. Consequently, Eq. (2.102) can be written as w(1 − w) =

1 Vo2 , 2 4 Vd,max

Vo ≤ Vd,max and w ≤ 1/2.

(2.104)

To determine the amount by which the maximum voltage changes when overshoot is taken into account, we take the ratio of Vd,max to Vs,max . Thus, Vd,max = Vs,max



27 = 0.9186. 32

(2.105)

Consequently, when the damping is zero, the maximum dynamic voltage is 8% less than the maximum static voltage, but the maximum value of w has increased from 1/3 to 1/2. This maximum value of the displacement ratio is a function of the damping factor; as the damping factor decreases, the maximum permissible value of V decreases and the maximum permissible value of wmax increases. These ideas are depicted graphically in Fig. 2.13, where we have displayed Eqs. (2.98) and (2.104). It is noted that the long-time response of the case of critical damping, ζ = 1, roughly corresponds the static case. Also, as ζ varies from 0 to 1, the relationship between maximum voltage and the maximum displacement is approximated by the dotted line in the figure. We now return to Eq. (2.92) and determine qualitatively some of the effects that the electrostatic force has on the system. The electrostatic term can be expanded as a series so that Eq. (2.92) can be written as

 dw d2 w 2 2 2 3 1 + 2w + 3w + 2ζ V + 4w + . . . + w = e 1 o dτ dτ 2

(2.106)

which, upon rearrangement, yields 

 d2 w dw 2 2 2 2 2 3 w − e 3w + 2ζ V V + 4w + . . . = e21 Vo2 . + 1 − 2e o o 1 1 dτ 2 dτ

(2.107)

It is seen that the electrostatic field tends to reduce the magnitude of the linear   stiffness of the system as represented by the term 1 − 2e21 Vo2 , which will decrease the natural frequency of the system as discussed further in Section 2.5.3. In addition,

2.5

Electrostatic and van der Waals Attraction

35

1 0.9 0.8

Vo /Vmax

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

Quasi-static (ζ = 1) Dynamic (ζ = 0)

0

0.1

0.2

0.3

0.4

0.5

w (= x/do)

Fig. 2.13 Value of allowable voltage Vo as a function of w for stable behavior of the single degreeof-freedom system. When Vo = Vmax , then w = wmax for a given value of ζ

the constant term e21 Vo2 has the same effect as fo in Eqs. (2.2) and (2.4); that is, it causes a static deflection of the spring. We shall now examine the case of an applied voltage of the form Vo = Vo u (t) where u(t) is the unit step function. The application of a shock load can be found in (Younis et al. 2006). The results of the numerical solution of Eq. (2.92) for this step application of voltage are shown in Fig. 2.14 for two values of the damping factor. From Fig. 2.13, it is seen that as the damping factor decreases, the maximum displacement increases, but the maximum voltage for stable operation decreases. These characteristics are exhibited in Fig. 2.14. We also see that the responses confirm the observations that we made with respect to Eq. (2.107). As the magnitude of e21 Vo2 decreases the equivalent spring stiffness increases, which results in a decrease in the response’s period of oscillation. Lastly, the static displacement is determined from Eq. (2.95). Its value is a function of e21 Vo2 so that as the damping factor decreases the maximum allowable value of e21 Vo2 decreases and, consequently, the static displacement decreases. The influence of the static voltage can be further demonstrated by considering the application of an external force to the mass. If this force is a half sine wave of frequency ωo , duration to = π/ωo , and magnitude Fo , then Eq. (2.92) is modified as e21 Vo2 dw d2 w + w = + 2ζ + fo sin (o τ ) [u(τ ) − u (τ − τo )] dτ dτ 2 (1 − w)2

(2.108)

36

2 Spring-Mass Systems 1

1

0.9

0.9 e12 Vo2 = 0.14815 e12 Vo2 = 0.13865

0.7

0.7

0.6

0.6

0.5 wmax = 0.429

0.4 0.3

e12 Vo2 = 0.13845

0.2

0.5 0.3 0.1

0

0

10

15

20

25

30

35

wmax = 0.472

e12 Vo2 = 0.13001

0.2

0.1 5

e12 Vo2 = 0.13011

0.4

e12 Vo2 = 0.13855

wstatic = 0.24

0

e12 Vo2 = 0.13019

0.8

w(τ)

w(τ)

0.8

wstatic = 0.207

0

5

10

15

20

τ

τ

(a)

(b)

25

30

35

Fig. 2.14 Displacement response of a single degree-of-freedom system subjected to a suddenly applied electrostatic field for several values of e21 Vo2 (a) ζ = 0.15 and (b) ζ = 0.05. Note that e21 Vo2 = 0.14815 = 4/27 1 2 2

e1Vo = 0.1, fo = 0.12509

0.9

2 2

e1Vo = 0.0357, fo = 0.35

0.8 0.7

w(τ)

0.6 0.5 0.4 0.3 0.2 0.1 0

wstatic = 0.0387

0

10

20

30 τ

40

50

60

Fig. 2.15 Displacement response of the mass of a single degree-of-freedom system to a half sine wave pulse of duration τo = 44.88 when ζ = 0.07. The product fo e21 Vo2 is the same for each case

where τo = ωn to and fo = Fo / (kdo ). The numerical evaluation of Eq. (2.108) is shown in Fig. 2.15, where it is seen that depending on the magnitude of fo with respect to magnitude of e21 Vo2 the displacement response either remains stable or after a short period of time it becomes unstable. For both cases shown in Fig. 2.15, the product fo e21 Vo2 is constant.

2.5

Electrostatic and van der Waals Attraction

37

Natural Frequency To determine the change in natural frequency as a function of the magnitude of the electrostatic force, the small undamped oscillations about the mass’s static equilibrium position are examined. Thus, we assume that w = ws + wd

(2.109)

where ws is a solution to Eq. (2.94); that is, ws =

e21 Vo2 (1 − ws )2

.

(2.110)

Substituting Eq. (2.109) into Eq. (2.92) with ζ = 0, we obtain e21 Vo2 d 2 wd + wd + ws = 2 dτ (1 − ws − wd )2

(2.111)

since, by assumption, ws is independent of time. We express the right hand side of Eq. (2.111) in a series and then neglect all terms containing wnd , n ≤ 2 (recall Eq. (2.106)). Then, Eq. (2.111) becomes e21 Vo2 2e21 Vo2 wd d2 wd + w + w = + d s dτ 2 (1 − ws )2 (1 − ws )3

(2.112)

or, upon using Eq. (2.110), 2e21 Vo2 wd d 2 wd + w = . d dτ 2 (1 − ws )3

(2.113)

To determine the natural frequency coefficient, we assume that wd = Wo e j τ

(2.114)

where  = ω/ωn . Substituting Eq. (2.114) into Eq. (2.113), we obtain the following expression for the natural frequency coefficient  n =

1−

2e21 Vo2 (1 − ws )3

.

(2.115)

Thus, for a given value of e21 Vo2 , 0 ≤ e21 Vo2 < 4/27, we determine ws from Eq. (2.110) and then use both of these values in Eq. (2.115) to determine n . The results are shown in Fig. 2.16, where it is seen that as the magnitude of e21 Vo2 increases, the natural frequency coefficient approaches zero. This is expected, since we have previously shown that e21 Vo2 “softens” the spring k and when e21 Vo2 = 4/27, the effective spring constant is zero.

38

2 Spring-Mass Systems 1 0.9 0.8 0.7

Ωn

0.6 0.5 0.4 0.3 0.2 0.1 0

0

0.05

0.1

0.15

e21V2o

Fig. 2.16 Change in the natural frequency coefficient of a single degree-of-freedom system subject to an electrostatic force as a function of e21 Vo2 . Pull-in occurs at e21 Vo2 = 4/27 = 0.1481

2.5.3 van der Waals Attraction and Atomic Force Microscopy The response of the mass of a single degree-of-freedom system to an attractive force at the atomic scale is now examined. This model can be used to represent a reducedorder model of the cantilever arm and sensing tip of a non contacting mode atomic force microscope. In this mode of operation, the cantilever is excited at its base. The cantilever provides the stiffness k (N/m) and the sensing tip is represented by the mass m (kg). We also include some viscous damping c to represent air damping. The van der Waals force for a sphere of radius R (m) separated by a distance d from a large plate is given by (Butt et al. 2005) FW =

ˆ HR N 6d2

(2.116)

ˆ is the Hamaker constant (Bergström 1997) that has a value on the order where H −19 of 10 J. Expressions for the van der Waals force for other interacting geometries can be found in (Butt et al. 2005, Table 3). If the mass at rest is a distance do from the flat surface, then d = do − x, where x is the instantaneous displacement of the mass. Under these assumptions, Eq. (2.116) becomes FW =

ˆ HR 6 (do − x)2

N.

(2.117)

2.5

Electrostatic and van der Waals Attraction

39

This model has been extended to include a Lennard-Jones repulsive force that occurs when the mass gets very close to the stationary surface (Zhang et al. 2009; Rützel et al. 2003). Since we are considering a single degree-of-freedom system with a moving base, we use Eq. (2.5) with f (t) = 0 and Fr = −FW to obtain (Burnham et al. 1997; Garcıa and Paulo 1999; Stark et al. 2003) m

ˆ dx dy HR d2 x + c + kx = c + ky + 2 dt dt dt 6 (do − x)2

(2.118)

where y is the displacement of the base. Using Eqs. (2.9) and (2.11), we can rewrite Eq. (2.118) as d2 wˆ dwˆ dˆy h + 2ζ + wˆ = 2ζ + yˆ +  2 dτ 2 dτ dτ 1−w ˆ

(2.119)

where τ = ωn t and h=

ˆ HR , 6kdo3

wˆ =

x , do

yˆ =

y . do

(2.120)

Following the procedure that was used to arrive at Eq. (2.107), Eq. (2.119) can be written as 

d2 wˆ dwˆ dˆy 2 3 + 2ζ + 4w + . . . = h + 2ζ + − 2 h) w ˆ − h 3w + yˆ . (2.121) (1 2 dτ dτ dτ Thus, it is seen that a mass that is acted upon by van der Waals forces behaves in a similar manner as that of a mass subjected to an electrostatic force. If it is assumed that wˆ 0 + B

F

3

2

F 1 P

A − V >0 B +

P = Poling direction 31 mode

2 During manufacture, a piezoelectric material is polarized with a DC electric field with a polarizing voltage that is usually applied at a temperature slightly below the material’s Curie temperature. Prior to the application of the polarizing voltage the dipole moments are randomly oriented. If the negative terminal of the polarizing voltage is at the top of the material, the direction of the polarizing axis is considered to go from the bottom of the material (positive terminal) to the top. After the removal of the polarizing voltage, the dipole moments are permanently aligned in what is called the poling direction. Compression (tension) of the material along the direction of polarization or tension (compression) perpendicular to the polarization direction generates a voltage of the same (opposite) polarity as the poling voltage.

2.6

Energy Harvesters

43

elements that utilize the 33 mode are typically configured in a stack. Those that utilize the 31 mode are typically attached to thin beams undergoing bending. When configured in a stack, the stack is very stiff and, therefore, requires high forces to produce useful strains. When attached to a beam though, the same amount of force that was applied to a stack provides significantly larger strains in the piezoelectric element. Consequently, utilizing the 31 mode on a beam can produce enough output voltage to make up for the lower coupling coefficient. This will be discussed in Section 4.4. Consider the based-excited single degree-of-freedom system shown in Fig. 2.19a. The mass of the single degree-of-freedom system is attached to one end of a piezoelectric material such that a compression or extension of the piezoelectric material creates a voltage. The piezoelectric element obeys the following constitutive relations for one-dimensional motion (ANSI/IEEE Std 1987) T3 = cE33 S3 − e33 E3

(2.126)

S D3 = e33 S3 + ε33 E3

where3 D3 is the total electric charge per area (C/m2 ), E3 is the electric field strength (N/C = V/m), T3 is the stress (N/m2 ), S3 is the total strain, e33 is an electromechanical coupling constant that relates the strain to the applied field (C/m2 ), cE33 is the elastic stiffness of the material measured at constant electric field (for example, short

c

k Zs

x m Piezoelectric

VT VT

h

Zp

ZL

y (a)

Fig. 2.19 Two single degree-of-freedom energy harvesting devices and their connections to an electric circuit (a) Piezoelectric and (b) Permanent magnet, where the magnet is fixed to the frame that undergoes a displacement y

c

k

y Rc

x m

Coil VT

S

N

S

RL

vL

VT

(b)

The following units may prove useful in the subsequent material. If C = coulomb, J = joule, V = volt, A = ampere, W = watt, H = henry, and F = farad, then J = Ws = Nm, V = J/C, A = C/s, F = C/V, ohm = Js/C2 , and H = Js2 /C2 . Also note the distinction in the use of the symbol C: C is coulomb when not italicized and C italicized in this chapter denotes a capacitor. 3

44

2 Spring-Mass Systems

S circuit) (N/m2 ), and ε33 is the dielectric constant (permittivity) measured at constant strain (F/m). Another form of Eq. (2.126) is (ANSI/IEEE Std 1987)

S3 = sE33 T3 + d33 E3

(2.127)

T E D3 = d33 T3 + ε33 3

where d33 is  a coupling  constant that relates strain and stress to the applied field (C/N), sE33 = 1/cE33 is the compliance of the piezoelectric element measured at T is the permitconstant electrical field (for example, short circuit) (m2 /N), and ε33 tivity measured at constant stress (F/m). Equation (2.127) can be inverted to obtain T ε33 S3 − Do d33 D3 = E S3 + s33

T3 =

where

d33 D3 Do Do E3 sE33

(2.128)

 T 2 Do = sE33 ε33 1 − k33 2 = k33

2 d33 E T s33 ε33

(2.129)

1 when k33 > 1 2. In addition, the short circuit compliance is related to the open circuit compliance (ANSI/IEEE Std 1987)

 E 2 sD 33 = s33 1 − k33

(2.148)

or, in terms of the elastic stiffness,

 2 cE33 = cD 33 1 − k33

(2.149)

where the superscript D refers to the open circuit quantity. To reduce the complexity of the results somewhat, it is assumed that the stiffness of the piezoelectric is much greater than that of the spring of the single degreeof-freedom system so that kK can be neglected. It is also assumed that the shunt impedance is infinite and that the stray impedance is zero; thus, Zp → ∞ and Zs = 0. The last assumption is that the load impedance is purely resistive so that ZL = RL . The case where the resistive load is in parallel with an inductance has

48

2 Spring-Mass Systems

also been examined (Renno et al. 2009). From these assumptions, we have from Eq. (2.138) that Z = RL and Eqs. (2.144) and (2.145), respectively, become Yo 2E (1 + jE rL ) m AR + jBR j3 rL Vref V Vo = E AR + jBR

Wo =

(2.150)

where AR = 1 − 2E (1 + 2ζE rL )     BR = E 2ζE + rL 1 + ke2 − 2E rL rL =

(2.151)

S R . ωnE Cpe L

S R is referred to as the time constant of the circuit formed by the The product Cpe L loaded capacitance of the piezoelectric element and the load resistor.

2.6.3 Maximum Average Power of a Piezoelectric Generator To determine the conditions under which one can transfer the maximum average power from the piezoelectric element to the load resistor RL , we use Eq. (2.150) to obtain Pavg

 3 2 2 E rL Vref 6E rL ke2 Pref |Vo |2 Vrms   2   W = = = = RL 2RL 2RL AR + B2R 2 A2R + B2R

where for a harmonically varying voltage Vrms = Vo

√

(2.152)

2 is the rms voltage,

E W Pref = ωnE Yo2 Kpe

(2.153)

and Pref is analogous to PY given by Eq. (2.38). For base excitation, the power output as a function of the number of g’s acceleration input to the base is often of interest. In this case, if g = 9.81 m/s2 is the gravity constant, then Eq. (2.152) can be written as Pa =

Pavg 2 rL k2 =  2E e 2  W 2 ag Pref 2 AR + BR

(2.154)

where ag =

ω 2 Yo g

and

E g2 Kpe P ref =  3 W. ωnE

(2.155)

2.6

Energy Harvesters

49

To find the value of rL that maximizes Pa , we determine the value of rL = ropt that satisfies dPa = 0. drL

(2.156)

Performing the operation indicated in Eq. (2.156), we obtain

ropt

    4E + 4ζE2 − 2 2E + 1 1   =      . E 4 + 4ζ 2 − 2 1 + k2 2 + 1 + k2 2 e e E E E

(2.157)

Since ropt is a function of frequency, there are many values of ropt , but not all of them will maximize Pa . It is worthwhile to investigate two extreme values of ropt when ζE = 0: ropt = 0, the short circuit value and ropt = ∞, the open circuit value. The short circuit value is obtained by setting the numerator of Eq. (2.157) to zero. Then, with ζE = 0, we have

2 4E − 22E + 1 = 2E − 1 = 0.

(2.158)

E = SC = 1

(2.159)

Thus,

or, using Eqs. (2.134) and (2.142),  ωSC = ωnE =

E Kpe

m

 =

AcE33 mh

rad/s.

(2.160)

The open circuit value is obtained by setting the denominator of Eq. (2.157) to zero. Then, with ζE = 0, we have  2 2

4E − 2 1 + ke2 2E + 1 + ke2 = 2E − 1 − ke2 = 0.

(2.161)

The solution of Eq. (2.161) is E = OC =

 1 1 + ke2 =  2 1 − k33

(2.162)

where we have used Eq. (2.147). Upon using Eqs. (2.134), (2.142), and (2.149) in Eq. (2.162), we obtain ωOC = 

ωnE 2 1 − k33

 =



E Kpe

2 m 1 − k33

 =

AcE33



2 mh 1 − k33

 =

AcD 33 rad/s. (2.163) mh

50

2 Spring-Mass Systems

Next, we determine the values of ropt at the open circuit frequency E = OC and short circuit frequency E = SC when ζE = 0. Substituting Eq. (2.159) into Eq. (2.157), we obtain for the optimum value of the short circuit load resistor rSC = 

1  2 1 + ke2 2ζE

at

E = 1.

(2.164)

Substituting Eq. (2.162) into Eq. (2.157), we obtain for the optimum value of the open circuit load resistor rOC



   1 2 1 + k 2ζ 2 = 1 + k e E e 1 + ke2

at

E =

 1 + ke2 .

(2.165)

When ke /(2ζE ) >> 1, Eqs. (2.164) and (2.165), respectively, become rSC ≈

2ζE ke2

rOC ≈

ke2 1 2 1 + ke 2ζE

at

E = 1 at

E =

 1 + ke2 .

(2.166)

Upon substituting Eq. (2.166) into Eq. (2.154), we find that the maximum power can be approximated by 

 Pavg 1 ≈ at E = SC = 1 and rL = rSC 2 ag Pref 16ζE SC    Pavg 1 ≈ at E = OC = 1 + ke2 and rL = rOC . 2 ag Pref 16ζE

(2.167)

OC

The error in using these approximations is less than 1% when ζE < 0.08 and k33 = 0.75 and when ζE < 0.018 and k33 = 0.40. For the latter case, the error exceeds 22% when ζE = 0.1. In all cases, the approximate relations given by Eq. (2.167) overestimate the power. The maximum power given by Eq. (2.154) always occurs at E = SC and E = OC and the corresponding values of rSC and rOC . For any other value of rL , the individual power response will never exceed the optimal power response that is obtained by evaluating Pavg with the optimum value of ropt at each frequency. This observation is illustrated in Fig. 2.20 for ζE = 0.03 and ke = 1.134 (k33 = 0.75), where we have plotted Pavg a2g P ref for different values of rL . It is seen from this figure that Eq. (2.167) indicates that the maximum power ratio is approximately equal to 1/(16 × 0.03) = 2.08, which is in good agreement with that found in Fig. 2.20.

2.6

Energy Harvesters

51

2.5 rL = ropt(1.15 ) rL = ropt(ΩE)

Pavg /(a2gP′ref)

2

1.5

1

0.5 ΩOC

ΩSC 0 0.5

1

1.5 ΩE

2

2.5

Fig. 2.20 Maximum average power response for different values of rL as a function of E for k33 = 0.75 and ζ = 0.03

From Eq. (2.150), the magnitude of the voltage response with respect to the input acceleration in g’s is Vm =

|Vo | E rL = ag Vref A2R + B2R

(2.168)

where ag is given in Eq. (2.155) and ge33 Vref =  2 V. S ωnE ε33

(2.169)

A graph of the normalized voltage given by Eq. (2.168) is shown in Fig. 2.21 for the same values of rL that were used to obtain Fig. 2.20. Here, we see that the maximum voltage for this normalized response occurs at E = SC . As shown in Fig. 2.22, the preceding results are also sensitive to the value of the mechanical damping factor ζE , which significantly affects the amplitude of the maximum power and whether two separate operating frequencies can be identified. It is pointed out that if one is interested in using the normalized power as given by Eq. (2.152), that is, Pavg  6 rL k 2 =  2E e 2  Pref 2 AR + BR

(2.170)

52

2 Spring-Mass Systems 3.5 rL = ropt(Ω(1.15) rL = ropt(ΩE)

3

|Vo|/(a2gV′ref)

2.5

2

1.5

1

0.5 ΩOC

ΩSC 0 0.5

1

1.5 ΩE

2

2.5

Fig 2.21 Voltage response for different values of rL as a function of E for k33 = 0.75 and ζE = 0.03

5 ζE = 0.01

4.5

ζE = 0.025 ζE = 0.05

4

Pavg/(a2gP′ref)

3.5 3 2.5 2 1.5 1 0.5 0 0.5

1 ΩE

1.5

Fig. 2.22 Maximum average power response for rL = ropt (E ) as a function of E for k33 = 0.75 and for three values of ζ E

2.6

Energy Harvesters

53

12 rL = ropt(Ω(1.15) rL = ropt(ΩE)

10

Pavg/Pref

8

6

4

2 ΩSC 0 0.5

1

ΩOC 1.5

2

2.5

ΩE

Fig. 2.23 Maximum average power response for different values of rL as a function of E for k33 = 0.75 and ζ = 0.03 when normalized with respect to a different reference power than used to obtain Fig. 2.20

then, as shown in Fig. 2.23, the results look quite different from those shown in Fig. 2.20. However, the dimensional average power curves represented by Pavg will be the same.

2.6.4 Permanent Magnet Generator Consider the base-excited single degree-of-freedom system model of a permanent magnet generator shown in Fig. 2.19b. A coil is attached to the mass of the single degree-of-freedom system. The velocity of the coil in the magnetic field causes a current i to flow, which, in turn, produces a voltage VT across the terminals as shown in the figure. Thus (Stephen 2006; Spreemann et al. 2008; Cheng et al. 2007; Soliman et al. 2008; Beeby et al. 2007), VT = 

dw V dt

(2.171)

where we have introduced the notational change w (t) = z (t),  = BNl N/A or Vs/m

(2.172)

is the electromagnetic coupling factor, N is the number of turns of the coil, l is the length of the conductor in the magnetic field (m), and B is the flux density (T, where

54

2 Spring-Mass Systems

T = Tesla = Weber/m2 = Vs/m2 ). The force generated by the current in the coil that acts on the mass is Fr = i N.

(2.173)

For the circuit diagram shown in Fig. 2.19b, two simplifying assumptions are made: (1) the inductive part of the coil impedance is small with respect to the resistive part so that Zc ≈ Rc ; and (2) the load is a resistor so that ZL = RL . Then, it is straightforward to show that the current and voltage are related by i=

dw  VT A = Rc + R L Rc + RL dt

(2.174)

where Rc is the coil resistance, RL is the load resistance, and we have used Eq. (2.171). Therefore, the voltage across the load is VL = iRL =

RL  dw V. Rc + RL dt

(2.175)

In view of Eq. (2.173), Eq. (2.5) with f (t) = 0 and w (t) = z (t) becomes m

d2 w dw d2 y +c + kw + i = −m 2 . 2 dt dt dt

(2.176)

Substituting Eq. (2.174) into Eq. (2.176), we obtain   dw d2 w 2 d2 y m 2 + c+ + kw = −m 2 . dt Rc + RL dt dt

(2.177)

If we assume that the system is undergoing harmonic oscillations of the form w = Wo ejωt y = Yo ejωt then the solution to Eq. (2.177) is 2 Yo m 1 − 2 + j2ζme 

(2.178)

2ζme = 2ζ + 2ζe Rref 1 2ζe = = Rc + RL rc + rL Rc RL rc = , rL = Rref Rref  2 ωn Rref = ohm k

(2.179)

Wo = where  = ω/ωn ,

2.6

Energy Harvesters

55

and ωn and ζ , respectively, are given by Eqs. (2.9) and (2.11). Therefore, Eq. (2.175) gives the following voltage across the load VL =

j3 rL Vref   V (rc + rL ) 1 − 2 + j2ζme 

(2.180)

where Vref = Yo ωn V.

(2.181)

2.6.5 Maximum Average Power of a Permanent Magnetic Generator For harmonic oscillations, the average power into the load resistor is Pavg =

2 |VL |2 Vrms 6 r L Po

  = = 2 RL 2RL 2 (rc + rL )2 1 − 2 + (2ζme )2

6 rL Po  W =  2 2 (rc + rL )2 1 − 2 + 2 (2ζ (rc + rL ) + 1)2

(2.182)

where Po = ωn Yo2 k W.

(2.183)

If the average power is normalized with respect to the base acceleration, then Eq. (2.182) becomes, Pavg 2 rL P a  W

=  2 a2g 2 (rc + rL )2 1 − 2 + 2 (2ζ (rc + rL ) + 1)2

(2.184)

where ag is given by Eq. (2.155), P a =

kg2 W ωn3

(2.185)

and g = 9.81 m/s2 is the gravity constant. The maximum average power is determined from the value of rL = ropt that satisfies   Pavg d = 0. (2.186) drL a2g P a

56

2 Spring-Mass Systems

Performing the operation indicated in Eq. (2.186) on Eq. (2.184), we arrive at    2 (4ζ rc + 1) rL = rc2 +  . 2 1 − 2 + (2ζ )2

(2.187)

Upon substituting Eq. (2.187) into Eq. (2.184) and numerically evaluating the result reveals that irrespective of the combination of rc and ζ , Pavg is always a maximum at  = 1. Thus, the optimum value of rL to obtain the maximum average power is determined from Eq. (2.187) with  = 1, which gives rL = ropt = rc +

1 . 2ζ

(2.188)

Substituting Eq. (2.188) into Eq. (2.184) and setting  = 1, we obtain the following maximum average power ratio Pavg 1 = . 16ζ (1 + 2ζ rc ) a2g P a

(2.189)

From Eq. (2.189), it is seen that to obtain the maximum power to the load resistor, one should make the coil resistance as small as possible.

2.7 Two Degree-of-Freedom Systems 2.7.1 Introduction Consider the two degree-of-freedom system shown in Fig. 2.24 where the base of m1 is given a known displacement x3 and a force f1 (t) is applied to m1 and a force f2 (t) is applied to m2 . This general system has been chosen so that the results can be straightforwardly specialized to the cases that are of interest. It is mentioned that when one analyzes systems with a moving base it is usually assumed that the masses are initially at rest and there are no forces applied to the inertial elements. From the free-body diagram given in Fig. 2.25, we arrive at the following pair of coupled ordinary differential equations x3

x1

x2

k1

k2 m1

c1

k3 m2

f1(t)

c2

Fig. 2.24 System with two degrees of freedom and a moving base

f2(t)

c3

2.7

Two Degree-of-Freedom Systems

57 m1x1

Fig. 2.25 Free-body diagrams for masses m1 and m2 along with the respective inertial forces. The over dot indicates the derivative with respect to time

m2x2

x1

k1(x1−x3)

x2

k2(x2−x1)

c1(x1−x3)

c2(x2−x1)

m1

k3x2 m2

f1(t)

m1

c3 x2 f2(t)

d2 x1 dx2 dx3 dx1 + (c1 + c2 ) + (k1 + k2 ) x1 − c2 − k2 x2 − c1 − k1 x3 = f1 (t) 2 dt dt dt dt dx2 d 2 x2 dx1 + (k2 + k3 ) x2 − c2 − k2 x1 = f2 (t). m2 2 + (c2 + c3 ) dt dt dt (2.190)

The following quantities  ωnj =

kj rad/s mj

ωn2 1 = √ ωr = ωn1 mr

m2 , m1

τ = ωn1 t

cj 2ζj = , mj ωnj

k3 = , k2

j = 1, 2,

mr =



(2.191) k2 , k1

k32

c32

c3 = c2

are introduced into Eq. (2.190) to obtain  dx1 d2 x1 dx2 2 x1 − 2ζ2 mr ωr + + 2ζ m ω ω + 1 + m − mr ωr2 x2 (2ζ ) 1 2 r r r r 2 dτ dτ dτ dx3 f1 (τ ) = + 2ζ1 + x3 k1 dτ d2 x2 dx1 f2 (τ ) dx2 + 2ζ2 ωr (1 + c32 ) . + ωr2 (1 + k32 ) x2 − 2ζ2 ωr − ωr2 x1 = 2 dτ dτ dτ k1 mr (2.192) It is noted that ωn1 is the natural frequency of the system uncoupled from the system containing m2 ; that is, when k2 = c2 = m2 = 0. Similarly, ωn2 is the natural frequency of the system uncoupled from the system containing m1 ; that is, when k1 = c1 = m1 = 0 and, in addition, when k3 = c3 = 0. Upon taking the Laplace transforms of the individual terms on each side of Eq. (2.192) using pair 2 of Table C.1 of Appendix C, we arrive at A(s)X1 (s) − B(s)X2 (s) = K1 (s) −C(s)X1 (s) + E(s)X2 (s) = K2 (s)

(2.193)

58

2 Spring-Mass Systems

where A(s) = s2 + 2 (ζ1 + ζ2 mr ωr ) s + 1 + mr ωr2 B(s) = 2ζ2 mr ωr s + mr ωr2

C(s) = 2ζ2 ωr s + ωr2 = B(s) mr

(2.194)

E(s) = s2 + 2ζ2 ωr (1 + c32 ) s + ωr2 (1 + k32 ) and F1 (s) + x˙ 1 (0) + [s + 2ζ1 + 2ζ2 mr ωr ] x1 (0) − 2ζ2 mr ωr x2 (0) k1 + (2ζ1 s + 1) X3 (s) F2 (s) + x˙ 2 (0) + [s + 2ζ2 ωr (1 + c32 )] x2 (0) − 2ζ2 ωr x1 (0). K2 (s) = k1 mr K1 (s) =

(2.195)

In Eqs. (2.193) and (2.195), the transforms K1 (s) and K2 (s) are determined by the externally applied forces, the initial conditions of the masses, and the displacement of the base. The quantities X1 (s) and X2 (s), respectively, are the Laplace transforms of x1 (τ ) and x2 (τ ), and F1 (s) and F2 (s), respectively, are the Laplace transforms of the force inputs f1 (τ ) and f2 (τ ). Furthermore, x1 (0) and x˙ 1 (0), respectively, are the initial displacement and the initial velocity of mass m1 , and x2 (0) and x˙ 2 (0), respectively, are the initial displacement and the initial velocity of mass m2 . In arriving at Eq. (2.195), we have assumed that x3 (0) = 0. Solving for X1 (s) and X2 (s) in Eq. (2.193) yields K1 (s)E(s) K2 (s)B(s) + D(s) D(s) K1 (s)C(s) K2 (s)A(s) X2 (s) = + D(s) D(s)

X1 (s) =

(2.196)

where D(s) = A(s)E(s) − B(s)C(s) = s4 + [2ζ1 + 2ζ2 ωr mr + 2ζ2 ωr (1 + c32 )] s3   + 1 + mr ωr2 + ωr2 + 4ζ1 ζ2 ωr + ωr2 k32 + 4ζ2 ωr c32 (ζ1 + ζ2 ωr mr ) s2    + 2ζ2 ωr + 2ζ1 ωr2 + 2k32 ωr2 (ζ1 + ζ2 ωr mr ) + 2c32 ζ2 ωr 1 + mr ωr2 s    + ωr2 1 + k32 1 + mr ωr2 . (2.197) We shall now define a quantity called the transfer function of the system, which is valid for the case where the initial conditions are zero and the base is fixed; that is, X3 = 0. For a two degree-of-freedom system, there are four transfer functions. One pair of transfer functions is determined by applying an impulse force to mass m1 and

2.7

Two Degree-of-Freedom Systems

59

determining the displacement response of each mass and the other pair of transfer functions is determined by applying an impulse force to mass m2 and determining the displacement response of each mass. Thus, for the first case, f1 (τ ) = Fo δ(τ ) and f2 (τ ) = 0, and for the second case f1 (τ ) = 0 and f2 (τ ) = Fo δ(τ ). From transform pair 5 of Table C.1 of Appendix C, we have for the first case that F1 (s) = Fo and for the second case F2 (s) = Fo . The transfer functions Gln (s), l = 1, 2 and n = 1, 2, where the first subscript denotes the displacement of mass ml and the second subscript denotes the mass to which the impulse force has been applied, are defined for the first case as X1 (s) E(s) X1 (s) = = m/N F1 (s) Fo k1 D(s) X2 (s) X2 (s) C(s) = m/N = G21 (s) = F1 (s) Fo k1 D(s) G11 (s) =

(2.198)

where we have used Eqs. (2.195) and (2.196). Similarly, for the second case, we obtain X1 (s) B(s) X1 (s) = = = G21 (s) m/N F2 (s) Fo k1 mr D(s) X2 (s) A(s) X2 (s) G22 (s) = = = m/N. F2 (s) Fo k1 mr D(s)

G12 (s) =

(2.199)

We now examine several special cases of these results.

2.7.2 Harmonic Excitation: Natural Frequencies and Frequency-Response Functions The solution for harmonic excitation can be obtained directly from Eqs. (2.193) to (2.197) by setting s = j, where  = ω/ωn1 , and by setting x1 (0) = x2 (0) = x˙ 1 (0) = x˙ 2 (0) = 0. This substitution is equivalent to assuming that xl = Xl ejτ, fl = gl ejτ , l = 1, 2, and x3 = X3 ejτ. With these substitutions, Eqs. (2.193) to (2.195), respectively, become 

A(j) −B(j) −C(j) E(j)



X1 X2



=

K1 (j) K2 (j)

 (2.200)

where A(j) = −2 + 2j (ζ1 + ζ2 mr ωr )  + 1 + mr ωr2 B(j) = 2jζ2 mr ωr  + mr ωr2

C(j) = 2jζ2 ωr  + ωr2 = B(j) mr E(j) = −2 + 2jζ2 ωr (1 + c32 )  + ωr2 (1 + k32 )

(2.201)

60

2 Spring-Mass Systems

and g1 + (2jζ1  + 1) X3 k1 g2 K2 (j) = . k1 mr

K1 (j) =

(2.202)

In addition, Eqs. (2.196) and (2.197), respectively, become K1 (j)E(j) K2 (j)B(j) + D(j) D(j) K1 (j)C(j) K2 (j)A(j) + X2 (j) = D(j) D(j) X1 (j) =

(2.203)

where D(j) = 4 − j [2ζ1 + 2ζ2 ωr mr + 2ζ2 ωr (1 + c32 )] 3   − 1 + mr ωr2 + ωr2 + 4ζ1 ζ2 ωr + ωr2 k32 + 4ζ2 ωr c32 (ζ1 + ζ2 ωr mr ) 2    + j 2ζ2 ωr + 2ζ1 ωr2 + 2k32 ωr2 (ζ1 + ζ2 ωr mr ) + 2c32 ζ2 ωr 1 + mr ωr2     + ωr2 1 + k32 1 + mr ωr2 . (2.204) It is seen from Eq. (2.201) that |E (j)| is a minimum when 2 = ωr2 (1 + k32 ) .

(2.205)

We shall see that this observation is the basis for one type of vibration absorber. Natural Frequencies The natural frequencies of the two degree-of-freedom system are obtained by setting ζ1 = ζ2 = ζ32 = 0 in Eq. (2.204) and setting D (j) = 0. Then the natural frequency coefficients can be determined from the solution to  4 − a1  2 + a2 = 0

(2.206)

a1 = 1 + ωr2 (1 + mr + k32 )    a2 = ωr2 1 + k32 1 + mr ωr2 .

(2.207)

where

Thus,  1,2 =

   0.5 a1 ∓ a21 − 4a2 .

(2.208)

2.7

Two Degree-of-Freedom Systems

61

The natural frequency coefficients for the uncoupled systems are n1 = 1 and n2 = ωr . When k32 = 0 and ωr >> 1, it is straightforward to show that Eq. (2.208) gives 1 1 ≈ √ 1 + mr √ 2 ≈ ωr (1 + mr ).

(2.209)

In terms of dimensional quantities, Eq. (2.209) becomes ωn1 rad/s ω1 ≈ √ 1 + mr √ ω2 ≈ ωn2 1 + mr rad/s.

(2.210)

On the other hand, when ωr 0 or compressive force So < 0

The value of the buckling force is determined from the solution of Eq. (4.76) with  = 0 and So = −So and the satisfaction of the boundary conditions given by Eqs. (4.77) and (4.78). Solving the modified Eq. (4.76) subject to these the boundary conditions results in the following equation from which So = Sbuckling is obtained 1

  

 Sbuckling sin Sbuckling + 2 cos Sbuckling − 1 = 0.

4.3

In-Plane Forces and Electrostatic Attraction

243

4.3.3 Beam Subject to In-Plane Forces and Electrostatic Attraction Beam Clamped at Both Ends A beam that is clamped at both ends and subjected to an electrostatic attraction, an externally applied axial tensile force, and an in-plane tensile force due to the stretching of the neutral axis caused by the beam’s deflection from the electrostatic field is now considered (Batra et al. 2008; Krylov 2007; Gorthi et al. 2006; Younis et al. 2003; Abdel-Rahman et al. 2002; Kuang and Chen 2004; Zhang and Zhao 2006; Rhoads et al. 2006; Kafumbe et al. 2005). The geometry of the beam is shown in Fig. 4.7. The governing equation is given by Eq. (4.73), which in anticipation of what is to come, is rewritten as ∂ 2 yˆ ∂ 2 yˆ ∂ 4 yˆ − So 2 − dr 2 4 ∂η ∂η ∂η

1  0

∂ yˆ ∂η

2

  ∂ 2 yˆ dη − E12 Vo2 Gr yˆ + 2 = 0 ∂τ

(4.85)

where   1 c3 Gr yˆ =  2 +  1.24 1 − yˆ 1 − yˆ   1 do 2 dr = 2 r  0.76   0.24   0.76 do do h c3 = c2 = 0.204 + 0.6 . b b b

(4.86)

For a beam with a rectangular cross section of height h, r2 = h2 /12 and, therefore, dr = 6 (do /h)2 . When h >> do , the effects of the in-plane stretching are decreased and vice versa. Since this term indicates a stiffening effect; that is, it makes the beam appear stiffer, we would expect that yˆ would be smaller for a given value of Vo . It is noted that when dr → 0, the effects of the displacementinduced in-plane stretching can be neglected and when c3 = 0 the fringe effects are neglected.

h

Beam

b

Vo do

Fig. 4.7 Geometry of a beam under an electrostatic force

Fixed plate

L

244

4 Thin Beams: Part II

For a beam clamped at each end, the boundary conditions at η = 0 are yˆ (0, τ ) = 0

(4.87)

yˆ (0, τ ) = 0 and those at η = 1 are yˆ (1, τ ) = 0 yˆ (1, τ ) = 0.

(4.88)

To obtain an estimate of the lowest natural frequency of beam after it has been subjected to an electrostatic field of magnitude E12 Vo2 , we proceed in the following manner (Batra et al. 2008). We assume a separable solution of the form yˆ (η, τ ) = ϕ (τ ) Y (η)

(4.89)

where Y (η) satisfies the boundary conditions given by Eqs. (4.87) and (4.88). Substituting Eq. (4.89) into Eq. (4.85), we obtain d4 Y d2 Y d2 Y ϕ 4 − So ϕ 2 − dr ϕ 3 2 dη dη dη

1 

dY dη

2 dη − E12 Vo2 Gr (ϕY) + Y

d2 ϕ = 0. (4.90) dτ 2

0

Multiplying Eq. (4.90) by Y (η) and integrating over the (non dimensional) length of the beam, we arrive at mo

d2 ϕ ˆ r (ϕ) + kϕ + k1 ϕ 3 = e21 Vo2 G dτ 2

where mo = I1 ,

k = I3 + So I2 ,

1 

1 I1 =

Y 2 dη,

I2 =

0

ˆ r (ϕ) = G

k1 = dr I22 , dY dη

1 Y (η) Gr (ϕY) dη 0

1 

2 dη,

I3 =

0

(4.91)

d2 Y dη2

(4.92)

2 dη.

0

In arriving at Eq. (4.92), we have used integration by parts and the fact that Y (η) satisfies the boundary conditions given by Eqs. (4.87) and (4.88) to determine that

1 0

1 0

d4 Y Y 4 dη = dη

1 

d2 Y dη2

2 dη

0

d2 Y Y 2 dη = − dη

1  0

dY dη

2 dη.

4.3

In-Plane Forces and Electrostatic Attraction

245

We assume that Y (η) is the solution to a beam clamped at both ends and subject to a static uniform loading of unit magnitude that is applied over the length of the beam. For this case, Y (η) = η2 (η − 1)2 .

(4.93)

If Eq. (4.93) is substituted into the definitions of I1 , I2 , and I3 given in Eq. (4.92), ˆ r (ϕ) it is found that I1 = 1/630, I2 = 2/105, and I3 = 4/5. The integral for G cannot be obtained in an explicit form and must be evaluated numerically after ϕ is specified. Inserting these results into Eq. (4.92), it is found that mo = 1/630 k = 4/5 + (2/105) So

(4.94)

k1 = dr (2/105)2 . To determine the static displacement at pull-in where yˆ = yˆ PI ; that is, when the system becomes unstable, and the voltage Vo = VPI , which is the value at which this instability occurs, the following procedure has been suggested (Batra et al. 2008). From Eqs. (4.89) and (4.93), yˆ = yˆ PI will be known when ϕ has been determined since, from Eq. (4.93), Y = Ymax = 1/16 at η = 0.5. To determine the displacement under static conditions, the inertia term in Eq. (4.91) is ignored; that is, mo ϕ¨ = 0, and Eq. (4.91) simplifies to ˆ r (ϕ) . kϕ + k1 ϕ 3 = E12 Vo2 G

(4.95)

To obtain a second condition that will enable us to determine the voltage at which this static displacement occurs, Eq. (4.95) is differentiated with respect to ϕ to yield

k + 3k1 ϕ 2 = E12 Vo2

ˆ r (ϕ) dG . dϕ

(4.96)

Solving Eq. (4.96) for E12 Vo2 and then substituting the result into Eq. (4.95) gives

  ˆ ˆ r (ϕ) k + 3k1 ϕ 2 − dGr (ϕ) kϕ + k1 ϕ 3 = 0. G dϕ

(4.97)

It is noted from Eqs. (4.86) and (4.92) that ˆ r (ϕ) d dG = dϕ dϕ

1

1 YGr (ϕY) dη =

0

Y 0

d [Gr (ϕY)] dη dϕ

(4.98)

246

4 Thin Beams: Part II

where d [Gr (ϕY)] = Y dϕ



2 (1 − ϕY)3

+

1.24c3 (1 − ϕY)2.24

 .

(4.99)

Equation (4.97) is solved numerically for ϕ = ϕPI and this value ϕ is substituted into either Eq. (4.95) or Eq. (4.96) to obtain E12 Vo2 = vo ; in other words, at ϕ = ϕPI , √ VPI = vo /E1 . Then, from Eqs. (4.89) and (4.93), the non dimensional pull-in displacement is yˆ PI = ϕPI Ymax = ϕPI /16

(4.100)

which occurs at Vo = VPI . To determine the natural frequency coefficient , the solution to Eq. (4.91) is linearized about its static equilibrium position at a given voltage Vo , 0 ≤ Vo < VPI . Thus, we assume a solution to Eq. (4.91) of the form ϕ = ϕs + ϕd , where ϕ s is a solution to ˆ r (ϕs ) kϕs + k1 ϕs3 = E12 Vo2 G

(4.101)

and ϕ s is independent of time. It is seen from Eq. (4.101) that ϕs = ϕs (Vo ). Thus, with assumption that ϕ = ϕs + ϕd , Eq. (4.91) becomes mo

d 2 ϕd ˆ r (ϕs + ϕd ) . + k (ϕs + ϕd ) + k1 (ϕs + ϕd )3 = E12 Vo2 G dτ 2

(4.102)

However, ˆ r (ϕs + ϕd ) = G

1 YGr (Y [ϕs + ϕd ]) dη 0



1 =

Y 0

1 (1 − Yϕs − Yϕd )2

+

c3 (1 − Yϕs − Yϕd )1.24

(4.103)

 dη.

We can linearize Gr (Y [ϕs + ϕd ]) with respect to ϕ d by expressing this function as a series in ϕ d and then by ignoring terms containing ϕdn , n ≥ 2. Thus, Gr (Y [ϕs + ϕd ]) =

1

+

c3

(1 − Yϕs − Yϕd ) (1 − Yϕs − Yϕd )1.24 1 2Yϕd c3 1.24c3 Yϕd ≈ + + + 2 3 1.24 (1 − Yϕs ) (1 − Yϕs ) (1 − Yϕs ) (1 − Yϕs )2.24 d ≈ Gr (Yϕs ) + ϕd Gr (Yϕs ) dϕs (4.104) 2

4.3

In-Plane Forces and Electrostatic Attraction

247

where we have used Eq. (4.99). Then, using Eq. (4.104), Eq. (4.103) becomes ˆ r (ϕs + ϕd ) ≈ G



1

Y Gr (Yϕs ) + ϕd

d Gr (Yϕs ) dη dϕs

0

1 ≈

1 YGr (Yϕs ) dη + ϕd

0

Y

d Gr (Yϕs ) dη dϕs

(4.105)

0

ˆ r (ϕs ) . ˆ r (ϕs ) + ϕd d G ≈G dϕs In addition, it is noted that (ϕs + ϕd )3 = ϕs3 + 3ϕs2 ϕd + 3ϕd2 ϕs + ϕd3 reduces to (ϕs + ϕd )3 ≈ ϕs3 + 3ϕs2 ϕd

(4.106)

when terms containing ϕdn , n ≥ 2 are ignored. Upon substituting Eqs. (4.105) and (4.106) into Eq. (4.102) and using Eq. (4.101), we obtain  d 2 ϕd d ˆ mo 2 + k + 3k1 ϕs2 − E12 Vo2 Gr (ϕs ) ϕd = 0. dϕs dτ

(4.107)

To determine the lowest natural frequency coefficient , we assume a solution of the form ϕd = d e j

2τ

(4.108)

where 2 = ωto and to is given in Eq. (3.53). Substituting Eq. (4.108) into Eq. (4.107), the following expression for the natural frequency coefficient is obtained   = 2

 1 d ˆ Gr (ϕs ) . k + 3k1 ϕs2 − E12 Vo2 mo dϕs

(4.109)

Graphs of Eq. (4.109) are given in Figs. 4.8 to 4.11, which show the change in the natural frequency coefficient 2 as a function of E1 Vo for several combinations of h/b and h/do and for So = 0. It is mentioned that the curves labeled 4 in these figures are equivalent to the curve shown in Fig. 2.16. It is seen that the fringe effects are very important in all circumstances and, therefore, should be included in these types of analyses. It is also seen that when h/do = 0.5 the displacement-induced in-plane force has a strong effect on 2 . This is not the case when h/do = 5. Furthermore, when h/do = 0.5 it is found for the case represented by curve 1 that

25

20

[1] (E1Vo)PI = 9.887, yPI = 0.581 [2] (E1Vo)PI = 10.76, yPI = 0.56 [3] (E1Vo)PI = 7.596, yPI = 0.415

15

[4] (E1Vo)PI = 8.403, yPI = 0.397

Ω2

3

4

1

2

10 h/b = 0.2, h/do = 0.5

5

0

0

2

4

6 E1Vo

8

10

12

Fig. 4.8 Change in the natural frequency coefficient of a beam clamped at both ends as a function of E1 Vo for h/b = 0.2, h/do = 0.5, and So = 0: [1] in-plane stretching and fringe effects included; [2] in-plane stretching included and fringe effects neglected; [3] in-plane stretching neglected and fringe effects included; and [4] in-plane stretching and fringe effects neglected 25

20

[1] (E1Vo)PI = 8.262, yPI = 0.403 [2] (E1Vo)PI = 8.422, yPI = 0.399 [3] (E1Vo)PI = 8.243, yPI = 0.401

15

[4] (E1Vo)PI = 8.403, yPI = 0.397 Ω2

1,3

2,4

10

h/b = 0.2, h/do = 5

5

0

0

1

2

3

4

5

6

7

8

9

E1Vo

Fig. 4.9 Change in the natural frequency coefficient of a beam clamped at both ends as a function of E1 Vo for h/b = 0.2, h/do = 5, and So = 0: [1] in-plane stretching and fringe effects included; [2] in-plane stretching included and fringe effects neglected; [3] in-plane stretching neglected and fringe effects included; and [4] in-plane stretching and fringe effects neglected

25

[1] (E1Vo)PI = 6.8, yPI = 0.644

20

[2] (E1Vo)PI = 10.76, yPI = 0.56 [3] (E1Vo)PI = 4.96, yPI = 0.47

15 Ω2

[4] (E1Vo)PI = 8.403, yPI = 0.397

3

1

4

6 E1Vo

4

2

10 h/b = 2, h/do = 0.5

5

0

0

2

8

10

12

Fig. 4.10 Change in the natural frequency coefficient of a beam clamped at both ends as a function of E1 Vo for h/b = 2, h/do = 0.5, and So = 0: [1] in-plane stretching and fringe effects included; [2] in-plane stretching included and fringe effects neglected; [3] in-plane stretching neglected and fringe effects included; and [4] in-plane stretching and fringe effects neglected 25

[1] (E1Vo)PI = 7.297, yPI = 0.425

20

[2] (E1Vo)PI = 8.422, yPI = 0.399 [3] (E1Vo)PI = 7.279, yPI = 0.422

15

[4] (E1Vo)PI = 8.403, yPI = 0.397

2,4

Ω2

1,3

10

h / b = 2, h /do = 5

5

0

0

1

2

3

4

5

6

7

8

9

E1Vo

Fig. 4.11 Change in the natural frequency coefficient of a beam clamped at both ends as a function of E1 Vo for h/b = 2, h/do = 5, and So = 0: [1] in-plane stretching and fringe effects included; [2] in-plane stretching included and fringe effects neglected; [3] in-plane stretching neglected and fringe effects included; and [4] in-plane stretching and fringe effects neglected

250

4 Thin Beams: Part II 40 35 30

Ω2

25 20 15 10 So =

0

20

40 60 80

5 0

0

2

4

6

8

10

E1Vo

Fig. 4.12 Change in the natural frequency coefficient of a beam clamped at both ends as a function of E1 Vo for h/b = 2, h/do = 0.5, and for several values of So

there is a range of values of E1 Vo near the pull-in value for which the displacementinduced in-plane tensile force overcomes the electrostatic spring softening effect and 2 increases. When h/do = 5, the displacement-induced in-plane tensile force has very little effect on 2 as shown by the curves labeled 1 and 3. It has been shown that the results exhibited by curve 1 in Figs. 4.8 to 4.11 agree very closely with the solution obtained from a three-dimensional finite element model (Batra et al. 2008). It is noted from the legends of Figs. 4.8 to 4.11 that the maximum displacement for curve 4, which is the case that omits fringe effects and displacement-induced axial stretching, is yˆ PI = 0.397. This value differs from the value of 1/3 predicted by a single degree-of-freedom system. Refer to the discussion following Eq. (2.97). In Fig. 4.12, the change in the natural frequency coefficient as a function of E1 Vo for several values of So when h/b = 2 and h/do = 0.5 is shown. It is seen that the stiffening effect of So increases the value of VPI . Cantilever Beam For the cantilever beam, it is assumed that no axially force is applied; that is, So = 0, and that there is no displacement induced stretching of the neutral axis because the end η = 1 is free. Neglecting the stretching of the neutral axis is equivalent to setting dr = 0. Then, from Eq. (4.92) it is found that k = I3 and k1 = 0. Hence, Eqs. (4.101) and (4.109), respectively, become ˆ r (ϕs ) I3 ϕs = E12 Vo2 G

(4.110)

4.3

In-Plane Forces and Electrostatic Attraction

251

and   = 2

 1 d ˆ I3 − E12 Vo2 Gr (ϕs ) . mo dϕs

(4.111)

The displacement function for a uniformly loaded cantilever beam is

 Y (η) = η2 η2 − 4η + 6 .

(4.112)

If Eq. (4.112) is substituted into the definitions of I1 and I3 given in Eq. (4.92), it is found that I1 = 104/45 and I3 = 144/5 and, therefore, mo = 104/45. Graphs of Eq. (4.111) are given in Fig. 4.13, which show the change in the natural frequency coefficient as a function of E1 Vo for several combinations of h/b and h/do . It is seen from this figure that the values of E1 Vo at which pull-in occurs is substantially less than those for a beam clamped at both ends. 4 3.5 [1] h/b = 0.2, h/do = 0.5 [2] h/b = 0.2, h/do = 5 [3] h/b = 2, h/do = 0.5 [4] h/b = 2, h/do = 5

3

Ω2

2.5 2

3

4

1 2

5

[1] (E1Vo)PI = 1.176, yPI = 0.467

1.5

[2] (E1Vo)PI = 1.277, yPI = 0.451 [3] (E1Vo)PI = 0.7673, yPI = 0.528

1

[4] (E1Vo)PI = 1.127, yPI = 0.475 [5] (E1Vo)PI = 1.302, yPI = 0.447

0.5 0

0

0.2

0.4

0.6

0.8

1

1.2

1.4

E1Vo

Fig. 4.13 Change in the natural frequency coefficient of a cantilever beam as a function of E1 Vo for several combinations of h/b and h/do . Curves 1 to 4 include fringe effects; curve 5 neglects fringe effects and is, therefore, independent of h/b and h/do

252

4 Thin Beams: Part II

4.4 Piezoelectric Energy Harvesters 4.4.1 Governing Equations and Boundary Conditions In this section, we shall consider a beam composed of two layers as shown in Fig. 4.14. Other investigations of two and three layered beams and the determination of their various characteristics are available (Wang and Cross 1999; Dunsch and Breguet 2007; Jiang et al. 2005; Hu et al. 2007; Tabesh and Frechette 2008; Ha et al. 2006; Lee et al. 2005; Leibowitz and Vinson 1993, pp. 257–267; Wang and Xu 2007, pp. 142–146; Lu et al. 2004; Erturk and Inman 2008B). Referring to Fig. 4.14, the top layer of the beam 0 ≤ z ≤ hp is a piezoelectric material and the bottom layer −hs ≤ z ≤ 0 is a structural material; they both have the same length and width. The cross sections of each layer are constant along the length of the beam. To derive the equations of motion of this system, we shall first obtain an expression for the system’s total energy in a manner that was employed in Section 3.2.1. In doing so, we draw on the results of Sections 2.6.2. It is also noted from Fig. 2.18 that for a beam in bending we are interested in 31 mode of actuation of the piezoelectric material. Typical values for a two layer MEMS-scale piezoelectric beam composed of a piezoelectric layer and an aluminum structural layer are given in Table 4.2 along with several calculated quantities that will appear in the development that follows. The total energy of the system is composed of the kinetic energy T, the electric enthalpy of the piezoelectric element Up , the strain energy of the structural element Us , the external work done by an applied force Wa , and the difference between the kinetic energy and potential energy of the attachments to the boundary F (C1 ) . Thus, the total energy of the system is obtained from ET = T − Up − Us + Wa + F (C1 ) .

(4.113)

We shall now determine each of these contributions. The parameters appearing in the expressions that follow are shown in Fig. 4.14.

b

ρs, Es

hs +

y

a Neutral axis

hp

Fig. 4.14 Cross section of a two layer beam composed of a piezoelectric beam and a structural beam

E

T

ρp, s11, ε33, d31 − z

4.4

Piezoelectric Energy Harvesters

253

Table 4.2 Representative values for a beam composed of a piezoelectric layer and an aluminum structural layer and the computed values of several of the parameters appearing in Eqs. (4.157) and (4.189) Quantity (units)

Value

Computed parameters

b (m) L (m) hs (m)   Es  N/m2  ρs kg/m3  sE11 m2 /N  ρp kg/m3 hp (m) T (F/m) ε33 d31 (C/N) k31

0.015 0.050 0.003 70 × 109 2700 1.5 × 10−11 7800 0.0005 2.12 × 10−8 −2.0 × 10−10 0.35

to = 5.52 × 10−4 s a = −0.00126 m (EI)e = 3.699 Nm2 (ρA)e = 0.18 kg/m β = −0.0003 C α = 25.8 Cp = 3.19 × 10−8 F Pref = 3.46 × 106 W

Kinetic Energy The kinetic energy of the two layers is given by 1 T= 2

=



L b/2 0 ρs 0 −b/2 −hs

1 (ρA)e 2

L  0

∂w ∂t

∂w ∂t

2

1 dzdydx + 2



L b/2 hp ρp 0 −b/2 0

∂w ∂t

2

2

dzdydx (4.114)

dx

where (ρA)e = ρs bhs + ρp bhp kg/m.

(4.115)

Mechanical and Electrical Energy The strain energy in the structural layer of the beam is given by Eq. (3.12); that is, 1 Us = 2



L b/2 0 Es 0 −b/2 −hs

∂ 2w ∂x2

2 (z − a)2 dzdydx.

(4.116)

The quantity a is the distance from the coordinate axis to the neutral axis. Its value will be determined subsequently. The electric enthalpy function for the piezoelectric layer of the beam in bending is (Tiersten 1969, pp. 34–36)

254

4 Thin Beams: Part II

1 Up = 2

L b/2 hp (T1 S1 − E3 D3 ) dzdydx

(4.117)

0 −b/2 0

where T1 is the stress in the piezoelectric layer in the x-direction, S1 is the corresponding strain, E3 is the electric field, D3 is the electric charge, and from Eq. (3.8), S1 = − (z − a)

∂ 2w . ∂x2

(4.118)

The piezoelectric constitutive relations for the 31 mode, which are analogous to Eq. (2.127) for the 33 mode, are (ANSI/IEEE Std 1987) S1 = sE11 T1 + d31 E3 T D3 = d31 T1 + ε33 E3

(4.119)

which can be expressed as [recall Eq. (2.128)] S1 d31 − E E3 E s11 s11 d31 T D3 = E S1 + εˆ 33 E3 s11 T1 =

(4.120)

where

 T T 2 εˆ 33 1 − k31 = ε33 2 = k31

2 d31

(4.121)

T sE11 ε33

2 is the piezoelectric material coupling factor for longitudinal deformation of a and k31 thickness polarized piezoelectric beam. Comparing the first equation of Eq. (2.132) with the first equation of Eq. (4.120), it is found that cE11 = 1/sE11 and e31 = d31 /sE11 . The quantity E3 must satisfy the Maxwell equations in the piezoelectric layer. The constants appearing in Eq. (4.119) are defined in Section 2.6.2. Substituting Eq. (4.120) into Eq. (4.117), we obtain

1 Up = 2

L b/2 hp  0 −b/2 0

S12 sE11

 d31 T 2 − 2 E S1 E3 − εˆ 33 E3 dzdydx. s11

(4.122)

An expression for the electric field E3 will now be obtained. We have indirectly assumed that D1 = D2 = 0; therefore, Maxwell’s equation has the simple form (Tiersten 1969, p. 30)

4.4

Piezoelectric Energy Harvesters

255

∂D3 =0 ∂z

(4.123)

∂ϕ ∂z

(4.124)

and the electric field is given by E3 = −

where ϕ = ϕ (x, z, t) is a scalar electric potential with the units of volts (V). Notice that Eq. (4.123) indicates that D3 is independent of z. As will be seen subsequently, we shall also use D3 to determine the electric current flowing into a load connected to the surfaces of the piezoelectric layer. From the second equation of Eq. (4.120) and from Eq. (4.118), it is found that T E3 − (z − a) D3 = εˆ 33

d31 ∂ 2 w . sE11 ∂x2

(4.125)

Using Eq. (4.125) in Eq. (4.123), it is seen that ∂D3 d31 ∂ 2 w T ∂E3 =0 = εˆ 33 − E ∂z ∂z s11 ∂x2 and, therefore, ∂E3 d31 ∂ 2 w . = T E ∂z εˆ 33 s11 ∂x2

(4.126)

Equation (4.126) implies that E3 is a linear function of z. Substituting Eq. (4.124) into Eq. (4.126), we obtain ∂E3 d31 ∂ 2 w ∂ 2ϕ . =− 2 = T E ∂z ∂z εˆ 33 s11 ∂x2

(4.127)

Consequently, from Eq. (4.127), the following form of the electric potential is assumed ϕ (x, z, t) = φ0 (x, t) + φ1 (x, t) z + φ2 (x, t) z2

(4.128)

where φl , l = 0, 1, 2, are to be determined. From Eqs. (4.124), (4.126), and (4.128), E3 = −φ1 (x, t) − 2zφ2 (x, t) d31 ∂ 2 w ∂E3 = −2φ2 (x, t) = T E ∂z εˆ 33 s11 ∂x2 and, therefore,

(4.129)

256

4 Thin Beams: Part II

φ2 (x, t) = −

2 k31 d31 ∂ 2 w ∂ 2w   2. = − T E 2 2 2ˆε33 s11 ∂x 2d31 1 − k31 ∂x

(4.130)

Since ϕ is a potential, it is only defined with respect to a reference value and, consequently, φ 0 cannot be determined. However, the potential differences between the top and bottom surfaces of the piezoelectric layer can be determined. Hence, to determine φ 1 , we consider the potential difference   ϕ = ϕ x, hp , t − ϕ (x, 0, t) = φ0 (x, t) + hp φ1 (x, t) + h2p φ2 (x, t) − φ0 (x, t) = hp φ1 (x, t) + h2p φ2 (x, t) and, therefore, φ1 (x, t) =

ϕ − hp φ2 (x, t) hp

d31 hp ∂ 2 w ϕ + T E . = hp 2ˆε33 s11 ∂x2

(4.131)

In arriving at Eq. (4.131), Eq. (4.130) was used. Substituting Eqs. (4.130) and (4.131) into Eq. (4.129), we obtain k2 ϕ  31 2  E3 = − − hp d31 1 − k31



 2 hp ∂ w V/m. −z 2 ∂x2

(4.132)

From Eqs. (4.125) and (4.132), the electric displacement is εˆ T D3 = − 33 ϕ − hp



 hp d31 ∂ 2 w −a E C/m2 2 s11 ∂x2

(4.133)

and from Eqs. (4.120) and (4.132), the stress is d31 ϕ 1 + E T1 = E s11 hp s11

"

2 k31



  2 1 − k31

#  hp ∂ 2w N/m2 . − z − (z − a) 2 ∂x2

(4.134)

Substituting Eqs. (4.118) and (4.132) into Eq. (4.122), adding the result with Eq. (4.116), and performing the integration with respect to y and z, we obtain

L "



∂ 2w ∂x2

2

b (ϕ)2 hp 0  2 #  ∂ w d31 bϕ hp −a −2 dx E 2 ∂x2 s11

1 Up + Us = 2

(EI)e

T − εˆ 33

(4.135)

4.4

Piezoelectric Energy Harvesters

257

where

0 (EI)e = bEs −hs

−

b (z − a)2 dz + E s11

2bk2  31 2  E s11 1 − k31

bk2 − E  31 2  s11 1 − k31 =

hp (z − a)2 dz + 0



hp (z − a)

 hp − z dz 2

0

hp 

hp −z 2

2

(4.136) dz

0

  3 bEs  b  (hs + a)3 − a3 + E hp − a + a3 3 3s11 +

bh3p

2 k31   2 12 sE11 1 − k31

is the effective bending stiffness. The last term of Eq. (4.136) is the stiffness of the piezoelectric layer about its geometric center and is a result of the electric field E3 being a function the curvature of the beam. Before proceeding, the second and last terms of Eq. (4.136) are examined more  3 closely. If the cubic hp − a is expanded and the term containing h3p is combined with the last expression, the result is bh3p 3sE11

, 1+

2 k31

-

  . 2 4 1 − k31

(4.137)

A typical value for k31 is 0.32; then the second term inside the square brackets of Eq. (4.137) is approximately equal to 0.025 and may be ignored. In this case, Eq. (4.136) simplifies to (EI)e =

  3 b  bEs  hp − a + a3 . (hs + a)3 − a3 + E 3 s11

(4.138)

The location of the neutral axis a due to mechanical loading only; that is, there is no applied voltage (ϕ = 0), is determined from the following relation that determines the axial force Fx on the beam

258

4 Thin Beams: Part II

b/2 0 Fx =

b/2 hp σx dzdy +

−b/2 −hs

∂ 2w = − bEs 2 ∂x

T1 dzdy

−b/2 0

0 −hs

b ∂ 2w (z − a) dz − E s11 ∂x2

k2 b ∂ 2w − E  31 2  2 s11 1 − k31 ∂x

hp 

hp z− 2

hp (z − a) dz

(4.139)

0

 dz.

0

In arriving at Eq. (4.139), Eqs. (3.9) and (4.134) have been used. The value of a is that value for which Fx = 0. Therefore, performing the integrations, setting Fx = 0, and solving for a, it is found that a=

h2p /sE − Es h2s  11 . 2 Es hs + hp /sE11

(4.140)

Referring to Fig. 4.14, it is noted that when a < 0, the neutral axis lies in the structural layer of the beam. Externally Applied Force The work done by an externally applied force of magnitude Fo per unit length is

L Wa =

Fo f (x, t) w (x, t) dx

(4.141)

0

where f (x, t) is a non dimensional temporal and spatial shape function. Charge from the Piezoelectric Beam The charge q flowing from the surface z = hp is given by (Tiersten 1969, pp. 179–181)

L b/2 q (t) = −

L D3 dydx = −b

0 −b/2

D3 dx.

(4.142)

0

On this surface, we ignore possible edge effects and assume that ϕ is independent of position along the length of the beam; that is, ϕ = V (t), where V(t) is the voltage created from the piezoelectric material (ANSI/IEEE Std. 1987). Then, using Eq. (4.133) in Eq. (4.142), we obtain

4.4

Piezoelectric Energy Harvesters

L , q= b

T εˆ 33 ϕ + hp

259



0

 hp d31 ∂ 2 w −a E dx 2 s11 ∂x2

 

L 2 T εˆ 33 hp d31 ∂ w −a E = bL V +b dx hp 2 ∂x2 s11 = bL

T εˆ 33

hp

 V +b

(4.143)

0



 hp d31  − a E w (L, t) − w (0, t) C 2 s11

where the prime denotes the derivative with respect to x. The current I flowing into a resistive load RL is obtained from I=−

∂q V A. = ∂t RL

(4.144)

Attachments to the Boundary Most practical applications of this type of system consider cantilever beams. Therefore, we shall limit our analysis to a cantilever beam that has attached to its free end a translational spring with stiffness kR (N/m) and a mass MR (kg). For this configuration, the difference between the kinetic energy and the potential energy is [recall Eq. (3.36)] F (C1 ) =

1 1 ¨ 2 (L, t) − kR w2 (L, t) . MR w 2 2

(4.145)

Total Energy Substituting Eqs. (4.114), (4.135), (4.141), and (4.145) into Eq. (4.113), we arrive at

1 ET = 2

L

Fdx + F(C1 )

(4.146)

0

where F(C1 ) is given by Eq. (4.145),   2 2 ∂w 2 ∂ w T b 2 − (EI)e + εˆ 33 V ∂t hp ∂x2  2  ∂ w d31 bV hp −a +2 E + Fo f (x, t) w 2 ∂x2 s11 

F = (ρA)e

and we have replaced ϕ with V(t).

(4.147)

260

4 Thin Beams: Part II

Governing Equation and Boundary Conditions The governing equation is obtained by using Eq. (B.123) of Appendix B with u replaced by w and F given by Eq. (4.147) to arrive at (EI)e

∂ 4w ∂ 2w + (ρA)e 2 = Fo f (x, t) . 4 ∂x ∂t

(4.148)

The following general boundary conditions. are obtained from Case 2 of Table B.2 with F (C1 ) given by Eq. (4.145); that is, all aij and all Aij are zero except a21 = MR and A21 = kR . At x = 0 Either w (0, t) = 0 or ! ∂ 3 w !! =0 (EI)e ∂x3 !x=0

(4.149a)

and either ∂w (0, t) /∂x = 0 or (EI)e

!   ∂ 2 w !! d31 bV hp = − a ∂x2 !x=0 2 sE11

(4.149b)

At x = L Either w (L, t) = 0 or ! ! ∂ 3 w !! ∂ 2 w !! = kR w (L, t) + MR 2 ! (EI)e ∂x3 !x=L ∂t x=L

(4.150a)

and either ∂w (L, t) /∂x = 0 or !   ∂ 2 w !! d31 bV hp = E −a (EI)e ∂x2 !x=L 2 s11

(4.150b)

4.4.2 Power from the Harmonic Oscillations of a Base-Excited Cantilever Beam The results of the previous section are illustrated by considering a viscously damped cantilever beam without attachments at the free end. The damping constant is denoted c Ns/m2 . Then, from Eq. (4.6), Eq. (4.148) becomes2 (EI)e 2

∂ 4w ∂ 2w ∂w + (ρA)e 2 = Fo f (x, t) . +c 4 ∂t ∂x ∂t

(4.151)

For another approach to obtaining a solution to a similar system, see (Erturk and Inman 2008A).

4.4

Piezoelectric Energy Harvesters

261

It is further assumed that the clamped end at x = 0 is excited harmonically with an amplitude Wo e jωt and the external force is zero; that is, Fo = 0. Based on the type of excitation that is being applied to the clamped end of the beam, we assume a solution of the form w (x, t) = W (x) e jωt V (t) = Vo e jωt .

(4.152)

Then the governing equation given by Eq. (4.151) becomes (EI)e

 d4 W 2 + jωc − ω (ρA) e W = 0. dx4

(4.153)

From Eq. (4.149), the boundary conditions at x = 0 are W (0) = Wo W (0) = 0

(4.154)

and the boundary conditions at x = L, which are given by Eq. (4.150) with kR = MR = 0, are (EI)e

W (L)

d31 bVo = sE11



 hp −a 2

(4.155)

W (L) = 0

where the prime indicates the derivative with respect to x. The electric boundary condition given by Eqs. (4.143) and (4.144) yields   T εˆ 33 hp 1 d31 Vo = −b bL + − a E W (L) hp jωRL 2 s11

,

(4.156)

where Eq. (4.154) has been used; that is, W (0) = 0. The following quantities are introduced η=

x , L

4 = ω2 to2 ,

Wo W (η) (ρA)e L4 2 s , Y¯ o = , to2 = L L (EI)e εˆ T bL Cp (EI)e Cp = 33 F, α = hp Lβ 2

Y (η) =

Vo Cp cL4 cto , 2ζ = = , Vˆ o = to (EI)e β (ρA)e   hp RL Cp d31 τo = , β=b − a E C or Nm/V. to 2 s11

(4.157)

262

4 Thin Beams: Part II

Then, the governing equation given by Eq. (4.153) becomes  d4 Y 2 4 Y = 0. + 2ζ j −  dη4

(4.158)

The boundary conditions at η = 0, which are given by Eq. (1.154), become Y (0) = Y¯ o Y (0) = 0

(4.159)

and the boundary conditions at η = 1, which are given by Eq. (1.155), are Vˆ o α Y (1) = 0 Y (1) =

(4.160)

where the prime now indicates the derivative with respect to η. The electrical boundary condition given by Eq. (4.156) becomes  1+

1 Vˆ o = −Y (1) . j2 τo

(4.161)

To obtain a solution to Eqs. (4.158) to (4.161), we follow the procedure of Section 3.10.6 and assume a solution of the form Y (η) = ys (η) + yd (η)

(4.162)

d4 ys =0 dη4

(4.163)

where ys is a solution to

and the boundary conditions ys (0) = Y¯ o , y s (0) = 0 y s (1) =

Vˆ o , y (1) = 0. α s

(4.164)

The function yd is a solution to  

d 4 yd 2 4 4 2 y ys + 2ζ j −  =  − 2ζ j d dη4

(4.165)

and the boundary conditions at η = 0 yd (0) = 0 y d (0) = 0

(4.166a)

4.4

Piezoelectric Energy Harvesters

263

and the boundary conditions at η = 1 y d (1) = 0 y d (1) = 0.

(4.166b)

The solution to Eq. (4.163) with the boundary conditions given by Eq. (4.164) is ys (η) = Y¯ o +

Vˆ o 2 η . 2α

(4.167)

Then, substituting Eq. (4.167) in Eq. (4.165), we obtain    

d4 yd Vˆ o 2 2 4 4 2 η . Y¯ o + + 2ζ j −  yd =  − 2ζ j 2α dη4

(4.168)

To obtain the solution to Eq. (4.168) and the boundary conditions given by Eq. (4.166), we assume a solution of the form yd (η) =

∞ 

An Yn (η)

(4.169)

n=1

where An is to be determined, Yn is a solution to d 4 Yn − 4n Yn = 0 dη4

(4.170)

and the boundary conditions η = 0 Yn (0) = 0 Yn (0) = 0

(4.171a)

and the boundary conditions at η = 1 Yn (1) = 0 Yn (1) = 0.

(4.171b)

The solution to Eqs. (4.170) and (4.171) is the orthogonal function given by Case 3 of Table 3.3. Thus, Yn (η) = −

T (n ) T (n η) + S (n η) Q (n )

(4.172)

where n are solutions to R (n ) T (n ) − Q2 (n ) = 0

(4.173)

264

4 Thin Beams: Part II

and R (n ), etc. are given by Eq. (C.19) of Appendix C. The values of n that satisfy Eq. (4.173) are given in Case 3 of Table 3.5. Therefore, from Eqs. (4.162), (4.167), and (4.169) Y (η) = Y¯ o +

∞ Vˆ o 2  An Yn (η). η + 2α

(4.174)

n=1

Differentiating Eq. (4.174) with respect to η and substituting the result into Eq. (4.161), we get Vˆ o = −γ α

∞ 

An Yn (1)

(4.175)

n=1

where the prime denotes the derivative with respect to η, γ =

j2 τo α + (1 + α) j2 τo

(4.176)

and Yn (1) = −

T (n ) n S (n ) + n R (n ) . Q (n )

(4.177)

In arriving at Eq. (4.177), Eq. (C.20) of Appendix C has been used. It is noted that γ has the form of the frequency response function of a high pass RC filter. From Eq. (4.176), it is seen that when τo → ∞; that is, when open circuit is approached, γ → 1/ (1 + α). Conversely, when τo → 0; that is, when short circuit is approached, γ → 0. Upon substituting Eq. (4.169) into the boundary conditions, it is found from Eq. (4.171a) that Eq. (4.166a) is satisfied and from Eq. (4.171b) that Eq. (4.166b) is satisfied. Substituting Eqs. (4.169) and (4.175) into Eq. (4.168) and using Eq. (4.170), we obtain ∞  n=1

 An

 4n − 4 Yn (η) + 2ζ j2 Yn (η)

    γ  4 +  − 2ζ j2 Yn (1) η2 = 4 − 2ζ j2 Y¯ o . 2

(4.178)

Multiplying Eq. (4.178) by Yl (η) and integrating over the range 0 ≤ η ≤ 1, we obtain ⎧ ∞ ⎨  1  4 4 2 ˆAn Yn (η) Yl (η) dη  −  + 2ζ j ⎩ n n=1 0⎫ (4.179) ⎬ 

γ 4 − 2ζ j2 Yn (1) al = bl l = 1, 2, ... + ⎭ 2

4.4

Piezoelectric Energy Harvesters

265

where

1 al =

1 η Yl (η) dη, 2

0

bl =

dl =

Yl (η) dη 0



4 − 2ζ j2 dl ,

(4.180)

An Aˆ n = . Y¯ o

However, from Eqs. (3.356) and (3.357) with a (η) = 1 and mi = mL = mR = jL = jR = 0,

1 Yn (η) Yl (η) dη = δnl Nn 0

(4.181)

1 Yn2 (η) dη = Nn 0

where δ nl is the kronecker delta. Then, Eq. (4.179) can be written as ∞ 

Aˆ n cln = bl

l = 1, 2, ...

(4.182)

n=1

where  

γ 4  − 2ζ j2 Yn (1) al . cln = 4n − 4 + 2ζ j2 Nn δnl + 2

(4.183)

Equation (4.182) can be written in matrix form as ⎤⎧ ⎫ ⎧ ⎫ ˆ c11 c12 · · · ⎪ ⎨ A1 ⎪ ⎬ ⎨ b1 ⎪ ⎬ ⎪ ⎥ Aˆ 2 ⎢ c21 c22 b 2 . = ⎦ ⎣ ⎪ ⎪ .. .. ⎪ .. .. ⎪ ⎩ ⎭ ⎩ ⎭ . . . . ⎡

(4.184)

When damping is neglected ζ = 0 and when the piezoelectric element is shorted γ = 0; then Eq. (4.182) simplifies to bn  . Aˆ n =  4 n − 4 Nn

(4.185)

Substituting Eq. (4.185) into Eq. (4.174) gives Y (η) = Y¯ o + Y¯ o

∞  n=1

bn Yn (η)   . 4n − 4 Nn

(4.186)

266

4 Thin Beams: Part II

After notational differences are taken into account, it is seen that Eq. (4.186) is the same Eq. (3.433). If P is the average power into the load resistance RL , then [recall Eq. (2.152)] P=

2 Vrms 1 |Vo |2 = . RL 2 RL

(4.187)

In terms of the non dimensional quantities given by Eq. (4.157) and the expression for Vˆ o given by Eq. (4.175), Eq. (4.187) can be written as P

P 1 = = 2 2τo Pref Y¯ o

! ∞ !2 !  ! ! ! Aˆ n Yn (1)! !γ ! !

(4.188)

n=1

where Pref =

α2 β 2 W. to Cp

(4.189)

The value of P given by Eq. (4.188) as a function of the non dimensional frequency  in the vicinity of 1 is shown in Fig. 4.15 for ζ = 0.01 and for 40 τo = 0.02 35

τo = 0.2 τo = 2

30

P′

25 20 15 10 5 0 1.85

1.86

1.87

1.88 Ω

1.89

1.9

1.91

Fig. 4.15 Non dimensional power of a piezoelectric beam in the vicinity of its first natural frequency having the properties described in Table 4.2 as a function of frequency for ζ = 0.01 and τo = 0.02, 0.2, and 2

4.4

Piezoelectric Energy Harvesters

267

τo = 0.02, 0.2, and 2. It was found numerically that it was sufficient to use one term in Eq. (4.188) to obtain this figure. It is seen that the resonances occur at or very close to the first natural frequency of the system, which is 1 = 1.875. However, the magnitude of the average power depends on the value of τ o . Therefore, in Fig. 4.16, the value of the average power in the vicinity of  = 1 is shown as a function of τ o for ζ = 0.01 and 0.005. The curves in Fig. 4.16 show that there are two almost equal maximum values of the average power for each value of ζ . The power at the (ζ ) and the power at the smaller of larger of these two values of τ o is denoted POC (ζ ) as the open circuit these two values of τ o is denoted PSC (ζ ). We identify POC (OC) and PSC maximum, which occurs at τo = τo (ζ ) as the short circuit maximum, (SC) which occurs at τo = τo . From a Fig. 4.16, it is found that (0.005) P (0.005) POC = SC ≈ 2. POC (0.01) PSC (0.01)

In other words, when the damping is halved, the maximum peak power is doubled. However, the ratios of τ o at which these maxima occur are very different: when ζ = (OC) (SC) (OC) (SC) ≈ 37, and when ζ = 0.005, τo ≈ 150. From Table 4.2, τo τo 0.01, τo Fig. 4.16, and Eq. (4.157), is it found that for ζ = 0.01, R(OC) ≈ 29.6 kohm L

80

4

ζ = 0.01 3

70

ζ = 0.005

60

P′

50 40

2

1

30 P′SC τ o(SC) 1 35.2 0.0464 3 70.3 0.0231

20

P′OC τo(OC) 2 37.6 1.71 4 75.4 3.43

10 0 10−3

10−2

10−1

100

101

102

τo

Fig. 4.16 Maximum non dimensional power of a beam having the properties described in Table 4.2 as a function of τ o for ζ = 0.01 and 0.005

268

4 Thin Beams: Part II (SC)

(OC)

and RL ≈ 803 ohm. For ζ = 0.005, it is found that RL ≈ 59.4 kohm (SC) ≈ 400 ohm. To get an estimate of the maximum power for ζ = 0.01, and RL it is assumed that the base of the beam is subjected to 0.1 g acceleration at its first natural frequency; that is, at  = 1 = 1.875.   In terms of the non dimensional quantity Y¯ o , this converts to Y¯ o = 0.98to2 L41 = 4.83 × 10−7 . Then, from Table 4.2, Fig. 4.16, and Eq. (4.189) it is found that, P = P Pref Y¯ o2 =  2  37.6 3.46 × 106 4.83 × 10−7 ≈ 30 μW. It is noted that the power increases with the square of the base’s acceleration. One of the drawbacks of the energy harvesters discussed in this section is that their maximum power output is limited to frequencies centered about the system’s natural frequency. It has been found that by making the structural layer a ferromagnetic material and placing magnets near the free end of the cantilever beam or by placing a magnet on the free end of the cantilever beam with magnets placed nearby, the bandwidth and the magnitude of the maximum power of the piezoelectric beam can be increased (Stanton et al. 2010; Xing et al. 2009). The magnets themselves, however, do not generate any power; they are used only to alter the response of the beam.

Appendix 4.1 Hydrodynamic Correction Function The complex-valued hydrodynamic correction function for a rectangular cross section is given by (Sader 1998) corr (ω) = r (ω) + ji (ω) where r (ω) = ar br i (ω) = ai bi and ar = 0.91324 − 0.48274λ + 0.46842λ2 − 0.12866λ3 + 0.044055λ4 − 0.0035117λ5 + 0.00069085λ6 br = 1 − 0.56964λ + 0.48690λ2 − 0.13444λ3 + 0.045155λ4 − 0.0035862λ5 + 0.00069085λ6 ai = − 0.024134, −0.029256λ + 0.016294λ2 − 0.00010961λ3 + 0.000064577λ4 − 0.00004451λ5 bi = 1 − 0.59702λ + 0.55182λ2 − 0.18375λ3 + 0.079156λ4 − 0.014369λ5 + 0.0028361λ6 .

(4.190)

Appendix 4.1 Hydrodynamic Correction Function

269

The quantity λ is defined as λ = log10 Re ωρf b2 = rf 2 4μf ρ f b2 rf = , 2 = ωto 4to μf

Re =

and b is the width of the beam and to is given by Eq. (3.53). The other quantities are defined in Section 2.4.1 and Eq. (3.48). The correction function is approximate and is stated as being within 0.1% for 10−6 ≤ Re ≤ 104 . This correction function is for a beam in an infinite fluid medium. Correction functions for the case where one surface of the beam is a finite distance from an infinite rigid plane parallel to this surface have been obtained (Green and Sader 2005; Basak et al. 2006). The effect of the geometric correction factor given by Eq. (4.190) is shown in Fig. 4.17 by comparing  cir with  rect .

102

Γcir(Re) Γrect(Re)

Γcir, Γrect

101

real(Γcir, Γrect) 100

imag(−Γcir, −Γrect)

10−1

10−2 −1 10

100

101

102

103

104

Re

Fig. 4.17 Hydrodynamic functions for beams with circular and rectangular cross sections in an infinite fluid medium

270

4 Thin Beams: Part II

References Abdel-Rahman EM, Younis MI, Nayfeh AH (2002) Characterization of the mechanical behavior of an electrically actuated microbeam. J Micromech Microeng 12(6):759–766 ANSI/IEEE Std. 176-1987 (1988) IEEE standard on piezoelectricity. The Institute of Electrical and Electronics Engineers, New York, NY Basak S, Raman A, Garimella SV (2006) Hydrodynamic loading of microcantilevers vibrating in viscous fluids. J Appl Phys 99:114906 Batra RC, Porfiri M, Spinello D (2007) Review of modeling electrostatically actuated microelectromechanical systems. Smart Mater Struct 16(6):R23–R31 Batra RC, Porfiri M, Spinello D (2008) Vibrations of narrow microbeams predeformed by an electric field. J Sound Vib 309:600–612 Dunsch R, Breguet J-M (2007) Unified mechanical approach to piezoelectric bender modeling. Sens Actuators A 134(2):436–446 Erturk A, Inman DJ (2008A) A Distributed parameter electromechanical model for cantilevered piezoelectric energy harvesters. ASME J Vib Acoust 130:041002 Erturk A, Inman DJ (2008B) Issues in mathematical modeling of piezoelectric energy harvesters. Smart Mater Struct 17:065016 Gorthi S, Mohanty A, Chatterjee A (2006) Cantilever beam electrostatic MEMS actuators beyond pull-in. J Micromech Microeng 16:1800–1810 Green CP, Sader JE (2005) Frequency response of cantilever beams immersed in viscous fluids near a solid surface with applications to the atomic force microscope. J Appl Phys 98:114913 Ha J-L, Fung R-F, Chang S-H (2006) Quantitative determination of material viscoelasticity using a piezoelectric cantilever bimorph beam. J Sound Vib 289:529–550 Hosaka H, Itao K, Kurada S (1995) Damping characteristics of beam-shaped micro-oscillators. Sens Actuators A 49:87–95 Hu Y, Hu T, Jiang Q (2007) Coupled analysis for the harvesting structure and the modulating circuit in a piezoelectric bimorph energy harvester. Acta Mech Solida Sin 20(4):296–208 Jiang S, Li X, Guo S, Hu Y, Yang J, Jiang Q (2005) Performance of a piezoelectric bimorph for scavenging vibration energy. Smart Mater Struct 14:769–774 Kafumbe SMM, Burdess JS, Harris AJ (2005) Frequency adjustment of microelectromechanical cantilevers using electrostatic pull down. J Micromech Microeng 15:1033–1039 Krylov S (2007) Lyapunov exponents as a criterion for the dynamic pull-in instability of electrostatically actuated microstructures. Int J Non-Linear Mech 42:626–642 Kuang J-H, Chen C-J (2004) Dynamic characteristics of shaped micro-actuators solved using the differential quadrature method. J Micromech Microeng 14:647–655 Lee S-Y, Ko B, Yang W (2005) Theoretical modeling, experiments and optimization of piezoelectric multimorph. Smart Mater Struct 14:1343–1352 Leibowitz M, Vinson JR (1993) The use of Hamilton’s principle in laminated and composite piezoelectric structures. ASME Adaptive Structures and Material Systems, AD-Vol. 35 Lu F, Lee HP, Lim SP (2004) Modeling and analysis of micro piezoelectric power generators for micro-electromechanical-systems applications. Smart Mater Struct 13:57–63 Rhoads JF, Shaw SW, Turner KL (2006) The nonlinear response of resonant microbeam systems with purely-parametric electrostatic actuation. J Micromech Microeng 16:890–899 Sader JE (1998) Frequency response of cantilever beams immersed in viscous fluids with applications to the atomic force microscope. J Appl Phys 84(1):64–76 Sawano S, Arie T, Akita S (2010) Carbon nanotube resonator in liquid. Nano Lett 10:3395–3398 Sharpe WN Jr, Yuan B, Vaidyanathan R, Edwards RL (1997) Measurements of Young’s modulus, Poisson’s ratio, and tensile strength of polysilicon. Proceedings of the Tenth IEEE International Workshop on Microelectromechanical Systems, Nagoya, Japan, pp 424–429 Stanton SC, McGehee CC, Mann BP (2010) Nonlinear dynamics for broadband energy harvesting: investigation of a bistable piezoelectric inertial generator. Physica D 239:64–653

References

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Tabesh A, Frechette LG (2008) An improved small-deflection electromechanical model for piezoelectric bending beam actuators and energy harvesters. J Micromech Microeng 18:104009 Tiersten HF (1969) Linear piezoelectric plate vibrations. Plenum Press, New York, NY Wang Q-M, Cross LE (1999) Constitutive equations of symmetrical triple layer piezoelectric benders. IEEE Trans Ultrason Ferroelectr Freq Control 46(6):1343–1351 Wang Y, Xu R-Q (2007) Free vibration of sandwich beams coupled with piezoelectric layers. In: Wang J, Chen W (eds) Piezoelectricity, acoustic waves and device applications. World Scientific Publishing, Singapore, pp 142–146 Xing X, Lou J, Yang GM, Obi O, Driscoll C, Sun NX (2009) Wideband vibration energy harvester with high frequency permeability magnetic material. Appl Phys Lett 95:134103 Younis MI, Abdel-Rahman EM, Nayfeh A (2003) A reduced-order model for electrically actuated microbeam-based MEMS. J Microelectromech Syst 12(5):671–690 Zhang Y, Zhao Y (2006) Numerical and analytical study on the pull-in instability of micro-structure under electrostatic loading. Sens Actuators A 127:366–380

Chapter 5

Timoshenko Beams

The natural frequencies and mode shapes of Timoshenko beams of constant cross section, continuously variable cross section, and cross sections with step changes in properties for numerous combinations of boundary conditions and boundary and in-span attachments are obtained. The in-span and boundary attachments include springs, concentrated masses, and single degree-of-freedom systems. The responses of these systems to externally applied forces to its interior and the natural frequencies of elastically connected beams, which have been used to model double-wall carbon nanotubes, are determined. For all numeral results, comparisons are made to the numerical results obtained from the Euler-Bernoulli theory and regions of applicability are inferred.

5.1 Introduction In this chapter, an improved beam theory called the Timoshenko beam theory is introduced. This theory includes the effects of transverse shear and rotary inertia of the beam’s cross section. As was done for the Euler-Bernoulli beams in Chapter 3, solutions for the natural frequencies and mode shapes of Timoshenko beams for a wide range of boundary conditions and in-span attachments, axial loading, and tapers are obtained. The responses of the beams to externally applied forces are also determined. One of the objectives of the chapter is to numerically show under what conditions one can use the Euler-Bernoulli beam theory and when one should use the Timoshenko beam theory.

5.2 Derivation of the Governing Equations and Boundary Conditions 5.2.1 Introduction The Euler-Bernoulli beam theory introduced in Chapters 3 and 4 was based on several assumptions: the neutral axis remains unaltered, there is zero strain perpendicular the axis of the beam, and plane sections remain plane and orthogonal to the E.B. Magrab, Vibrations of Elastic Systems, Solid Mechanics and Its Applications 184, C Springer Science+Business Media B.V. 2012 DOI 10.1007/978-94-007-2672-7_5, 

273

274

5 Timoshenko Beams

neutral axis. The Timoshenko beam theory removes the last assumption: the plane sections still remain plane, but they no longer remain orthogonal to the neutral axis. In addition, the theory accounts for shear strain of the cross section and for its rotary inertia. The starting point for the development of the Timoshenko theory is with the assumption that the displacement along the axis of the beam is u = − (z − a) ψ (x, t)

(5.1)

where ψ is the rotation of the cross section due to bending and a is the distance from the coordinate axis to the neutral axis. From Eq. (A.3) of Appendix A and Eq. (5.1), the following axial and shear strains, respectively, are obtained ∂u ∂ψ = − (z − a) ∂x ∂x ∂u ∂w ∂w + = −ψ γxz = ∂z ∂x ∂x εx =

(5.2)

where w is the displacement in the z-direction and it has been assumed that εy = εz = 0. The corresponding axial stress and shear stress, respectively, are ∂ψ σx = Eεx = − (z − a) E ∂x   ∂w τxz = Gγxz = G −ψ ∂x

(5.3)

where E is the Young’s modulus and G is the shear modulus, which are related to each other by G=

E 2 (1 + ν)

(5.4)

where ν is Poisson’s ratio. Notice from Eq. (5.2) that when the shear strain is zero, ∂w/∂x = ψ and σ x given in Eq. (5.3) reduces to Eq. (3.9). The shear stress given in Eq. (5.3) is independent of the position within the cross section and, therefore is constant (uniform). For a beam under static loading, it is well known that the shear distribution is parabolic. To mitigate this discrepancy, Timoshenko introduced a shear correction factor κ, sometimes referred to as a shear coefficient, so that the shear stress is represented by 

 ∂w τxz = κG −ψ . ∂x

(5.5)

In general, the shear correction factor is a function of the shape of the cross section and Poisson’s ratio and is often selected to exhibit certain limiting values at very high frequencies (Cowper 1966; Stephen 1997A; Stephen 1997B; Han et al. 1999).

5.2

Derivation of the Governing Equations and Boundary Conditions

275

From Eq. (3.3), a moment balance about the neutral axis gives

σx (z − a) dzdy = E

M=− A

∂ψ ∂x



(z − a)2 dzdy A

(5.6)

∂ψ = EI (x) ∂x where M is the bending moment,

I (x) =

(z − a)2 dzdy

(5.7)

A

and we have assumed that, in general, the limits of the integrals in Eq. (5.7) can be a function of x. The shear force acting over the cross section of the beam is given by 



V=

τxz dzdy = κG A



∂w = κA (x) G −ψ ∂x

∂w −ψ ∂x





dzdy A

(5.8)

where

A (x) =

dzdy

(5.9)

A

is the cross sectional area and we have assumed that, in general, the limits of the integrals in Eq. (5.9) can be a function of x. We shall now use these results to determine the various contributions to the total energy of a Timoshenko beam of length L. The beam under consideration is shown in Fig. 3.2 with Vo = 0.

5.2.2 Contributions to the Total Energy Kinetic Energy: Beam Element and In-Span Concentrated Mass The kinetic energy of a beam element is the sum of the translation velocity and the axial velocity of the cross section. Thus,

276

5 Timoshenko Beams

1 T= 2

,



ρ V

1 = 2

,



ρ V

1 = 2

∂w ∂t ∂w ∂t

2

 +

∂u ∂t

2 dzdydx 

2 + (z − a)

2

∂ψ ∂t

2 dzdydx

(5.10)

   2

L , ∂ψ 2 ∂w + ρI (x) ρA (x) dx ∂t ∂t 0

where we have used Eqs. (5.1), (5.7), and (5.9). The kinetic energy of a concentrated mass Mi (kg) with rotational inertia Ji (kg m2 ) located at x = Lm , 0 ≤ Lm ≤ L, is given by TMi

1 = Mi 2



∂w (Lm , t) ∂t

2

1 + Ji 2



∂ψ (Lm , t) ∂t

2

which can be written as TMi

1 = 2

L ,

 Mi

0

∂w ∂t



2 + Ji

∂ψ ∂t

2 δ (x − Lm ) dx

(5.11)

where δ (x) is the delta function. Strain Energy The strain energy is determined from Eq. (A.10) of Appendix A and Eqs. (5.2), (5.3), and (5.5). Thus,



  1 σx εx + τxy γxy dzdydx Use = 2 V

  2  

 ∂ψ 2 ∂w 1 2 dzdydx + κG −ψ (z − a) E = 2 ∂x ∂x

(5.12)

V

1 = 2

L , 0



∂ψ EI (x) ∂x

2



∂w −ψ + κGA (x) ∂x

2 dx.

External Forces: Transverse, Moment, and Axial The work done by an external force Fa (x, t) (N/m) and an external moment per unit length Ma (x, t) (N) is

L WFM =

(Fa (x, t) w (x, t) + Ma (x, t) ψ (x, t)) dx. 0

(5.13)

5.2

Derivation of the Governing Equations and Boundary Conditions

277

When the beam is subjected to a tensile axial force p (x,t) (N), Eq. (3.17) can be used directly. Thus, 1 Wp = − 2

L 0



∂w p (x, t) ∂x

2 dx.

(5.14)

Elastic Foundation, In-span Translation Spring, In-span Torsion Spring, and Spring of In-Span Single Degree-of-Freedom System When the beam is placed on a linear elastic foundation of spring constant per unit length kf (N/m2 ), the stored energy of the elastic foundation is represented by Eq. (3.20). Thus, 1 Uk f = 2

L kf w2 (x, t) dx.

(5.15)

0

When an elastic translation spring of constant ki (N/m) is attached to the beam at x = Ls , 0 ≤ Ls ≤ L, the stored energy of the spring is given by Uki =

1 2 ki w (Ls , t) 2

which can be written as

L

1 Uki = 2

ki w2 (x, t) δ (x − Ls ) dx.

(5.16)

0

When an elastic torsion spring of constant kti (Nm/rad) is attached to the beam at x = Lt , 0 ≤ Lt ≤ L, the stored energy of the spring is given by Ukti =

1 kti ψ 2 (Lt , t) 2

which can be written as

Ukti

1 = 2

L kti ψ 2 (x, t) δ (x − Lt ) dx.

(5.17)

0

When a single degree-of-freedom system is attached at x = Lo , 0 ≤ Lo ≤ L, the energy stored by the spring ko is Uk o =

1 ko (w (Lo , t) − z (t))2 2

(5.18)

where z (t) is the displacement of the mass of the single degree-of-freedom system. Equation (5.18) can written as

278

5 Timoshenko Beams

Uk o

1 = 2

L ko (w (x, t) − z (t))2 δ (x − Lo ) dx.

(5.19)

0

Single Degree-of-Freedom System The kinetic energy of the mass mo (kg) of a single degree-of-freedom system is given by   mo ∂z 2 (5.20) Tmo = 2 ∂t where z is the displacement of mass. The potential energy of the spring that is attached at x = Lo , 0 ≤ Lo ≤ L, is given by Eq. (5.18). Thus, the difference between the kinetic energy and potential energy of this system is F¯ = Tmo − Uko =

mo 2



∂z ∂t

2

1 − ko (w (Lo , t) − z (t))2 . 2

(5.21)

Attachments on the Boundaries It is assumed that at the end of the beam at x = 0 the following  elements are attached: a mass ML (kg) that has a rotational inertia JL kg m2 , a translational spring with constant kL (N/m), and a torsion spring with constant ktL (Nm/rad). At the other end of the beam at x = L it is assumed that there are  attached the following: a mass MR (kg) that has a rotational inertia JR kg m2 , a translational spring with constant kR (N/m), and a torsion spring with constant ktR (Nm/rad). Then, the difference between the kinetic energy and the potential energy of these elements is G(C1 )

,     ∂w (0, t) 2 ∂ψ (0, t) 2 1 = + JL ML 2 ∂t ∂t     ∂w (L, t) 2 ∂ψ (L, t) 2 + MR + JR ∂t ∂t  1 − kL w2 (0, t) + ktL ψ 2 (0, t) + kR w2 (L, t) + ktR ψ 2 (L, t) . 2

(5.22)

Minimization Function From Eqs. (5.10) and (5.11), the total kinetic energy is 1 TT = 2

L ,

 {ρA (x) + Mi δ (x − Lm )}

0



∂ψ + {ρI (x) + Ji δ (x − Lm )} ∂t

∂w ∂t

2 (5.23)

2 dx.

5.2

Derivation of the Governing Equations and Boundary Conditions

279

The total potential energy is obtained from the sum of Eqs. (5.12), (5.15), (5.16), (5.17), and (5.19). Thus, UT = Use + Ukf + Uki + Ukti + Uko  2  

L  ∂ψ 2 1 ∂w −ψ EI (x) = + κGA (x) 2 ∂x ∂x 0

(5.24)

+ kf w (x, t) + ki w (x, t) δ (x − Ls ) 2

2



+ kti ψ (x, t) δ (x − Lt ) + ko (w (x, t) − z (t)) δ (x − Lo ) dx. 2

2

The total external work is obtained from Eqs. (5.13) and (5.14) as WT = WFM + Wp

L =

1 (Fa (x, t) w (x, t) + Ma (x, t) ψ (x, t)) dx − 2

0

L 0



∂w p (x, t) ∂x

2 dx. (5.25)

Then, from Eqs. (5.22) to (5.25)

(C1 )

TT − UT + WT + G

L =

Gdx + G(C1 )

(5.26)

0

where   2  ∂w ∂ψ 2 1 1 + {ρI (x) + Ji δ (x − Lm )} G = {ρA (x) + Mi δ (x − Lm )} 2 ∂t 2 ∂t ,  2 2  1 ∂w ∂ψ − + κGA (x) −ψ EI (x) 2 ∂x ∂x    2 ∂w 1 2 + kf w (x, t) p (x, t) + Fa (x, t) w (x, t) + Ma (x, t) ψ (x, t) − 2 ∂x  1 − ki w2 (x, t) δ (x − Ls ) + kti ψ 2 (x, t) δ (x − Lt ) 2 1 − ko (w (x, t) − z (t))2 δ (x − Lo ) 2 (5.27) and G(C1 ) is given by Eq. (5.22).

280

5 Timoshenko Beams

5.2.3 Governing Equations To obtain the governing equations for a Timoshenko beam, we use Eq. (B.127) of Appendix B; that is, ∂Gw,x ∂Gw˙ − =0 ∂x ∂t ∂Gψ˙ ∂Gψ,x − = 0. Gψ − ∂x ∂t Gw −

(5.28)

From Eq. (5.27), it is found that Gw = Fa − kf w − ki wδ (x − Ls ) − ko (w − z (t)) δ (x − Lo )   ∂w ∂w −ψ −p Gw,x = −κGA (x) ∂x ∂x   ∂w Gψ = κGA (x) − ψ + Ma − kti ψδ (x − Lt ) ∂x Gψ,x = −EI (x)

(5.29)

∂ψ ∂x

and ∂w ∂t ∂ψ Gψ˙ = (ρI (x) + Ji δ (x − Lm )) . ∂t Gw˙ = (ρA (x) + Mi δ (x − Lm ))

(5.30)

Making the appropriate substitutions of Eqs. (5.29) and (5.30) into Eq. (5.28), we obtain the following two coupled partial differential equations governing the vibratory motion of a Timoshenko beam      ∂ ∂w ∂w ∂ −ψ + κGA (x) p (x, t) − kf w − ki wδ (x − Ls ) ∂x ∂x ∂x ∂x ∂ 2w −ko (w − z (t)) δ (x − Lo ) − (ρA (x) + Mi δ (x − Lm )) 2 = −Fa ∂t and

 κGA (x)

   ∂w ∂ ∂ψ −ψ + EI (x) − kti ψδ (x − Lt ) ∂x ∂x ∂x

∂ 2ψ − (ρI (x) + Ji δ (x − Lm )) 2 = −Ma . ∂t

(5.31)

(5.32)

To obtain the governing equation of the single degree-of-freedom system, we use ¯ where F¯ is given by Case 5 of Table B.1 of Appendix B with u = z and F = F, Eq. (5.21); thus,

5.2

Derivation of the Governing Equations and Boundary Conditions

F¯ z −

281

∂ F¯ z˙ = 0. ∂t

(5.33)

Using Eq. (5.21) in Eq. (5.33), it is found that mo

d2 z + ko z = ko w (Lo , t) . dt2

(5.34)

5.2.4 Boundary Conditions Upon comparing Eq. (5.22) with Eq. (B.95) of Appendix B, it is found that a12j = a22j = A12j = A22j = 0 j = 1, 2 a111 = ML ,

a211 = MR

a112 = JL ,

a212 = JR

A111 = kL ,

A211 = kR

A112 = ktL ,

A212 = ktR .

(5.35)

Then, using Eqs. (5.35) and (5.27) in Eqs. (B.128) of Appendix B, the following boundary conditions at each end of the beam are obtained. At x = 0 Either w (0, t) = 0 or    ∂w ∂w − κGA (x) − ψ + p (x, t) ∂x ∂x x=0

 ∂ 2w kL w + ML 2 ∂t

x=0

=0

(5.36a)

and either ψ (0, t) = 0 or  ∂ 2ψ ktL ψ + JL 2 ∂t

! ∂ψ !! − EI (x) =0 ∂x !x = 0 x=0

(5.36b)

At x = L Either w (L, t) = 0 or  kR w + MR

∂ 2w ∂t2

   ∂w ∂w − ψ + p (x, t) + κGA (x) ∂x ∂x x=L

x=L

=0

(5.37a)

and either ψ (L, t) = 0 or 

∂ 2ψ ktR ψ + JR 2 ∂t

! ∂ψ !! + EI (x) =0 ∂x !x = L x=L

(5.37b)

Boundary Condition

Clamped

Hinged (simply supported, pinned)

Hinged with torsion spring kt

Free (no axial force)

Case

1

2

3

4

y=0 ∂ψ =0 ∂η y=0 ∂ψ i (η) = so Ktα ψ ∂η

w=0 ∂ψ =0 ∂x w=0 ∂ψ = so ktα ψ EI (x) ∂x

∂y −ψ =0 ∂η

∂ψ =0 ∂η

y=0 ψ =0

w=0 ψ =0

∂ψ =0 ∂x ∂w −ψ =0 ∂x

Non dimensional form

Dimensional form

so = +1 at η = 0 (α = L) so = −1 at η = 1 (α = R) ktα L Ktα = EIo Case 1: ktα → ∞

Remarks

Table 5.1 Dimensional and non dimensional form of several boundary conditions at each end of a Timoshenko beam

282 5 Timoshenko Beams

Boundary Condition

Free with axial force

Free with torsion spring kt and translation spring k (no axial force)

Free with mass Mα with rotational inertia Jα (no axial force)

Case

5

6

7

∂2ψ ∂ψ = so jα 2 ∂η ∂τ  a (η) ∂y ∂ 2y − ψ = so mα 2 γbs R2o ∂η ∂τ i (η)

= so Kα y

∂ψ ∂2ψ = so Jα 2 EI (x) ∂x ∂t   ∂w ∂2w − ψ = so Mα 2 κGA (x) ∂x ∂t

∂ψ = so Ktα ψ ∂η

∂y ∂η

∂y −ψ a (η) ∂η



i (η)

= −S (x, t)

∂ψ =0 ∂η

∂y a (η) −ψ ∂η



Non dimensional form

∂ψ = so ktα ψ EI (x) ∂x  ∂w − ψ = so k α w κGA (x) ∂x



∂ψ =0 ∂x ∂w ∂w κGA (x) − ψ = −p (x.t) ∂x ∂x

Dimensional form

Table 5.1 (continued)

so = +1 at η = 0 (α = L) so = −1 at η = 1 (α = R) Jα Mα , mα = jα = mb mb L2 mb = ρAo L

so = +1 at η = 0 (α = L) so = −1 at η = 1 (α = R) kα L3 ktα L Kα = , Ktα = EIo EIo Case 1: ktα → ∞ & kα → ∞ Case 2: kα → ∞ & ktα → 0

Valid at either end of the beam p is tensile pL2 S= EIo

Remarks

5.2 Derivation of the Governing Equations and Boundary Conditions 283

284

5 Timoshenko Beams

It is pointed out that, as was the case with the Euler-Bernoulli beam theory, the either/or formalism for w is not required since the case of w (0, t) = 0 can be obtained from Eq. (5.36a) by taking the limit in as kL → ∞ and the case of w (L, t) = 0 can be obtained from Eq. (5.37a) by taking the limit in as kR → ∞. Similarly, the either/or formalism for ψ is not required since the case of ψ (0, t) = 0 can be obtained from Eq. (5.36b) by taking the limit in as ktL → ∞ and the case of ψ (L, t) = 0 can be obtained from Eq. (5.37b) by taking the limit in as ktR → ∞. As a final remark, it is pointed out that when an axial force is applied, the axial force always appears in the boundary condition given by Eq. (5.36a) whenever kL < ∞ and it always appears in Eq. (5.37a) whenever kR < ∞. Several special cases of Eqs. (5.36) and (5.37) are given in Table 5.1.

5.2.5 Non Dimensional Form of the Governing Equations and Boundary Conditions     If it is assumed that I (x) = Io i (x) and A (x) = Ao a (x) where Io m4 and Ao m2 are reference quantities and i (x) and a (x) are non dimensional shape functions, then Eqs. (5.31) and (5.32) can be converted to a non dimensional form by introducing the quantities given in Eq. (3.53). Those quantities are repeated below for convenience and several additional quantities are introduced. Thus, η=

x , L

y=

w , L

mb = ρAo L kg, 4o = ωo2 to2 =  ωo =

Ro =

Ko , Mo

ko rad/s, mo

Lα L

ηα = ro , L

α = m, o, s, t ro2 =

t , to

τ= γbs =

Io Ao

to2 =

ρAo L4 2 s EIo

(5.38)

2 (1 + ν) κ

and S=

pL2 , EIo

Fa L3 fˆa = , EIo

Ki =

ki L 3 , EIo

Kf =

kf L4 , EIo

Ko =

ko L3 , EIo

Mo =

mo , mb

Ma L2 EIo

m ˆa = Kti = mi =

kti L EIo

Mi , mb

(5.39) ji =

Ji . mb L2

The form of the non dimensional parameters was chosen so that the results from the Timoshenko beam can be straightforwardly reduced to those for the EulerBernoulli beam. Using Eqs. (5.38) and (5.39) in Eqs. (5.31) and (5.32), we arrive at

5.2

Derivation of the Governing Equations and Boundary Conditions

285

     ∂y ∂y ∂ 2 ∂ −ψ + γbs Ro a (η) S (η, t) − Kf γbs R2o y ∂η ∂η ∂η ∂η   −Ki γbs R2o yδ (η − ηs ) − Ko γbs R2o y − zˆ δ (η − ηo ) −γbs R2o (a (η) + mi δ (η − ηm ))

(5.40)

∂ 2y = −γbs R2o fˆa ∂τ 2

and 

   ∂y ∂ ∂ψ − ψ + γbs R2o i (η) − Kti γbs R2o ψδ (η − ηt ) ∂η ∂η ∂η  ∂ 2ψ

= −γbs R2o m ˆa − γbs R4o i (η) + ji γbs R2o δ (η − ηm ) ∂τ 2

a (η)

(5.41)

and we have used the relation δ (x) = δ (Lη) = δ (η)/|L|. The motion of the single degree of freedom system given by Eq. (3.34) becomes 1 d 2 zˆ + zˆ = y (ηo , τ ) . 4o dτ 2

(5.42)

Using Eqs. (5.38) and (5.39), the boundary conditions given by Eqs. (5.36) and (5.37) can be written in non dimensional form as follows. At η= 0 Either y (0, t) = 0 or 

 ∂ 2y KL y + mL 2 ∂τ

a (η) − γbs R2o η=0



 ∂y ∂y − ψ + S (η, t) ∂η ∂η

η=0

=0

(5.43a)

and either ψ (0, t) = 0 or  ∂ 2ψ KtL ψ + jL 2 ∂τ

! ∂ψ !! − i (η) =0 ∂η !η=0 η=0

(5.43b)

At η= 1 Either y (1, t) = 0 or  ∂ 2y KR y + m R 2 ∂τ



a (η) + γbs R2o η=1



 ∂y ∂y − ψ + S (η, t) ∂η ∂η

η=1

=0

(5.44a)

and either ψ (1, t) = 0 or  ∂ 2ψ KtR ψ + jR 2 ∂τ

! ∂ψ !! + i (η) =0 ∂η !η=1 η=1

(5.44b)

286

5 Timoshenko Beams

In Eqs. (5.43) and (5.44), kα L3 , EIo Mα mα = , mb Kα =

ktα L EIo Jα jα = mb L 2

Ktα =

(5.45) α = L, R.

Several special cases of Eqs. (5.43) and (5.44) are given in Table 5.1.

5.2.6 Reduction of the Timoshenko Equations to That of Euler-Bernoulli The Timoshenko beam equations given by Eqs. (5.40) and (5.41) can be reduced to the Euler-Bernoulli equation as follows. First, several simplifying assumptions are made so that the equations are more tractable. We remove all in-span attachments, assume that the axial force is a constant; that is, S → So , and assume that the beam has a constant cross section. Then, Eqs. (5.40) and (5.41), respectively, become  ∂ 2y

2 ∂ψ 2 2∂ y − γ R y − γ R = −γbs R2o fˆa 1 + γbs R2o So − K f bs bs o o ∂η2 ∂η ∂τ 2

(5.46)

and ∂ 2ψ ∂ 2ψ ∂y ˆ a. − ψ + γbs R2o 2 − γbs R4o 2 = −γbs R2o m ∂η ∂η ∂τ

(5.47)

To obtain the Euler-Bernoulli equation, Eqs. (5.46) and (5.47) must be combined. To do this, we first differentiate Eq. (5.47) with respect to η to obtain ∂ 3ψ ∂ 3ψ ∂m ˆa ∂ 2y ∂ψ + γbs R2o 3 − γbs R4o . − = −γbs R2o 2 ∂η ∂η ∂η ∂η ∂η∂τ 2

(5.48)

Next, we solve for ∂ψ/∂η in Eq. (5.46) and obtain  ∂ 2y

2 ∂ψ 2 2∂ y − K γ R y − γ R + γbs R2o fˆa . = 1 + γbs R2o So f bs bs o o ∂η ∂η2 ∂τ 2

(5.49)

Substituting Eq. (5.49) into Eq. (5.48), we arrive at    ∂ 4y

2 ∂ 2y ∂ 2y ∂ 2y 2 2 2∂ y − So 2 + Kf y + 2 + Kf γbs Ro Ro 2 − 2 1 + γbs Ro So ∂η4 ∂η ∂τ ∂τ ∂η  ∂ 4y ∂ 4y + γbs R4o 4 − R2o + γbs R2o + γbs R4o So ∂τ ∂η2 ∂τ 2   ∂m ˆa ∂ 2 fˆa ∂ 2 fˆa + γbs R2o R2o 2 − 2 . = fˆa − ∂η ∂τ ∂η (5.50)

5.3

Natural Frequencies and Mode Shapes of Beams with Constant Cross. . .

287

To obtain the Euler-Bernoulli equation, we return to Eq. (5.5) and recall that Eq. (5.2) reduces to Eq. (3.9) when the shear strain γxz = 0. Zero shear strain is obtained by letting G → ∞, which for the non dimensional quantities is equivalent to letting κ → ∞; that is, by setting γbs = 0. Thus, setting γbs = 0 in Eq. (5.50), we arrive at ∂ 2y ∂m ˆa ∂ 2y ∂ 4y ∂ 4y − So 2 + Kf y + 2 − R2o 2 2 = fˆa − . 4 ∂η ∂η ∂η ∂τ ∂η ∂τ

(5.51)

It is seen that Eq. (5.51) has two terms that do not appear in the comparable Euler-Bernoulli beam equation; that is, Eq. (3.60) with Ki = Ko = mi = 0. The first term is the externally applied moment m ˆ a and the second term is that which contains Ro . This latter term was introduced by Lord Rayleigh and accounts for the effect of the rotational inertia of the beam’s cross section. When this term is neglected and the applied moment is neglected, Eq. (5.51) reduces to Eq. (3.60) with Ki = Ko = mi = 0.

5.3 Natural Frequencies and Mode Shapes of Beams with Constant Cross Section, Elastic Foundation, Axial Force, and In-span Attachments 5.3.1 Introduction We shall determine the natural frequencies and mode shapes of a Timoshenko beam on an elastic foundation with constant cross section, axial force, and attachments on the interior and on the boundaries in a manner analogous to that presented in Section 3.3. After the solution to this general case has been obtained, several special cases will be examined. One of the objectives of the material presented in this section is to determine numerically the regions for which the Euler-Bernoulli beam theory will suffice. ˆ a = 0 and to To determine the natural frequencies, it is assumed that fˆa = m simplify matters, we let p (η, τ ) = po , where po is a constant; therefore, S → So . In addition, since the cross section is assumed constant, a (η) = i (η) = 1. Using Eqs. (5.43) and (5.44) as a guide, it is assumed that the boundary conditions at η = 0 are γbs R2o KL y (0, τ ) −

and those at η = 1 are

∂y (0, τ ) ∂y (0, τ ) + ψ (0, τ ) − γbs R2o So =0 ∂η ∂η ∂ψ (0, τ ) KtL ψ (0, τ ) − =0 ∂η

(5.52)

288

5 Timoshenko Beams

∂ 2 y (1, τ ) ∂τ 2 ∂y (1, τ ) ∂y (1, τ ) + − ψ (1, τ ) + γbs R2o So =0 ∂η ∂η

KR γbs R2o y (1, τ ) + mR γbs R2o

KtR ψ (1, τ ) + jR

(5.53)

∂ 2 ψ (1, τ ) ∂ψ (1, τ ) + = 0. ∂η ∂τ 2

To determine the natural frequencies, we assume solutions of the form y (η, τ ) = Y (η) e j

2τ

ψ (η, τ ) =  (η) e j zˆ (τ ) = Zo e

2τ

(5.54)

j2 τ

where 2 = ωto . Substituting Eq. (5.54) into Eqs. (5.40) to (5.42), we obtain 

 d2 Y dY −  + γbs R2o So 2 + γbs R2o 4 − Kf Y dη dη   4  M o  δ (η − ηo ) Y = 0 + γbs R2o mi 4 δ (η − ηm ) − Ki δ (η − ηs ) +  1 − 4 4o   d2  dY −  + γbs R2o 2 + γbs R4o 4  dη dη 

+ γbs R2o ji 4 δ (η − ηm ) − Kti δ (η − ηt )  = 0 (5.55) d dη



where the following expression that was determined from Eq. (5.42) was used  −1 4 Zo = Y (ηo ) 1 − 4 . o

(5.56)

Anticipating the subsequent formulation, Eq. (5.55) is written as 3  d2 Y

   d 2 2 − R k Y + γ R Ap Y (η) δ η − ηp = 0 + γ 1 + γbs R2o So bs o  bs o 2 dη dη p=1

   d2 dY + γbs R4o b  + γbs R2o + Bq Y (η) δ η − ηˆ q = 0 2 dη dη 2

γbs R2o

q=1

(5.57) where

Natural Frequencies and Mode Shapes of Beams with Constant Cross. . .

5.3

A1 = mi 4

η1 = η m

Mo 4  A2 =  1 − 4 4o

η2 = η o

A3 = −Ki

η3 = η s

B1 = ji 4

ηˆ 1 = ηm

B2 = −Kti

ηˆ 2 = ηt

289

(5.58)

and k = 4 − Kf   b = 4 − 1 γbs R4o .

(5.59)

The boundary conditions at η = 0, which are given by Eq. (5.52), become   γbs R2o KL Y (0) − 1 + γbs R2o So Y (0) +  (0) = 0 KtL  (0) −  (0) = 0

(5.60)

and those at η = 1, which are given by Eq. (5.53), become     γbs R2o KR − mR 4 Y (1) + 1 + γbs R2o So Y (1) −  (1) = 0   KtR − jR 4  (1) +  (1) = 0.

(5.61)

In Eqs. (5.60) and (5.61), the prime denotes the derivative with respect to η.

5.3.2 Solution for Very General Boundary Conditions We shall solve Eq. (5.57) subject to the boundary conditions given by Eqs. (5.60) and (5.61) using the Laplace transform on the spatial variable η. Thus, taking the Laplace transform of Eq. (5.57), we obtain    ¯ 1 (s) ¯ (s) = G 1 + γbs R2o So s2 + γbs R2o k Y¯ (s) − s   ¯ 2 (s) ¯ (s) = G sY¯ (s) + γbs R2o s2 + γbs R4o b 

(5.62)

where     ¯ 1 (s) = 1 + γbs R2o So Y (0) + 1 + γbs R2o So sY (0) −  (0) G − γbs R2o

3 

  Ap Y ηp e−sηp (5.63)

p=1

¯ 2 (s) = Y (0) + γbs R2o  (0) + sγbs R2o  (0) − γbs R2o G

2  q=1

  Bq Y ηˆ q e−sηˆ q .

290

5 Timoshenko Beams

From Eq. (5.60), it is seen that   1 + γbs R2o So Y (0) =  (0) + γbs R2o KL Y (0)  (0) = KtL  (0) .

(5.64)

Therefore, upon substituting Eq. (5.64) into Eq. (5.63), we obtain 3  

   2 2 2 ¯ Ap Y ηp e−sηp G1 (s) = γbs Ro KL + 1 + γbs Ro So s Y (0) − γbs Ro p=1

¯ 2 (s) = Y G

(0) + γbs R2o (KtL

+ s) 

(0) − γbs R2o

2 

  Bq Y ηˆ q e−sηˆ q .

(5.65)

q=1

¯ (s) and using Solving Eq. (5.62) for the transformed quantities Y¯ (s) and  Eq. (5.65), we obtain ⎧ ⎨  1 2   R S 1 + γ Y¯ (s) = s3 + γbs R2o KL s2 + γbs R4o KL b bs o o ¯ o (s) 1 + γbs R2o So ⎩ D

    + R2o 4 (1 + γbs R2o So ) − So s Y (0) + s2 + KtL s  (0) − γbs R2o

3  p=1

¯ (s) = 

⎫ 2 ⎬      2  Ap Y ηp s + R2o b e−sηp − Bq  ηˆ q se−sηˆq ⎭ q=1

⎧ ⎨

1  ¯ Do (s) 1 + γbs R2o So ⎩ 



 1 + γbs R2o So s3 + KtL 1 + γbs R2o So s2

 + γbs R2o k s + γbs R2o KtL k  (0) + [−KL s + k ] Y (0) +

3  p=1

⎫ 2 ⎬      −sη  

Ap Y ηp se p − Bq  ηˆ q 1 + γbs R2o So s2 + γbs R2o k e−sηˆq ⎭ q=1

(5.66)

where  

¯ o (s) = s2 − α 2 s2 + β 2 D

(5.67)

Natural Frequencies and Mode Shapes of Beams with Constant Cross. . .

5.3

291

and α2 =

 √  R2o  −F1 + F2 2 1 + γbs R2o So

β2 =

 √  R2o  F 1 + F2 2 2 1 + γbs Ro So

 

(5.68)

F1 = 4 + γbs k + γbs R2o So b   F2 = F12 − 4 1 + γbs R2o So γbs k b provided that for Kf = 0 and So = 0 4
0 √ F2 − F1 > 0.

(5.70)

  β 2 − α 2 = R2o F1 1 + γbs R2o So  √  β 2 + α 2 = R2o F2 1 + γbs R2o So   α 2 β 2 = −γbs R4o b k 1 + γbs R2o So .

(5.71)

It is noted that

The inverse Laplace transform of Eq. (5.66) is Y (η) = f1 (η) Y (0) + f2 (η)  (0) − H1 (η)  (η) = g1 (η) Y (0) + g2 (η)  (0) + H2 (η)

(5.72)

where H1 (η) = γbs R2o

3 

2          Ap f3p η, ηp Y ηp + Bq f4q η, ηˆ q  ηˆ q

p=1

H2 (η) =

3  p=1

q=1

2          Ap g3p η, ηp Y ηp − Bq g4q η, ηˆ q  ηˆ q

(5.73)

q=1

and the definitions of fj (η) and gj (η), j = 1, 2, and flp (η) and glp (η), l = 1, 2, are given in Appendix 5.1.

292

5 Timoshenko Beams

The two unknown constants Y (0) and (0) are determined from the boundary conditions at η = 1, which are given by Eq. (5.61). Upon substituting Eq. (5.72) into Eq. (5.61), it is found that 

d11 d12 d21 d22



Y (0)  (0)



=

P1 P2

 (5.74)

where     d11 = γbs R2o mR 4 − KR f1 (1) − 1 + γbs R2o So f1 (1) + g1 (1)     d12 = γbs R2o mR 4 − KR f2 (1) − 1 + γbs R2o So f2 (1) + g2 (1)   d21 = jR 4 − KtR g1 (1) − g 1 (1)   d22 = jR 4 − KtR g2 (1) − g 2 (1)

(5.75)

    P1 = γbs R2o mR 4 − KR H1 (1) − 1 + γbs R2o So H1 (1) − H2 (1)   P2 = − jR 4 − KtR H2 (1) + H2 (1) .

(5.76)

and

The definitions of fj (η) and g j (η) are given in Appendix 5.1 and the prime denotes the derivative with respect to η. Upon solving Eq. (5.74) for Y (0) and (0), we obtain 1 [d22 P1 − d12 P2 ] DT 1 [d11 P2 − d21 P1 ]  (0) = DT

(5.77)

DT = d11 d22 − d12 d21 .

(5.78)

Y (0) =

where

Substituting Eq. (5.77) into Eq. (5.72) and collecting terms, we arrive at ⎤ ⎡ 3 2          1 ⎣ Ap C1,p η, ηp Y ηp + Bq C2,q η, ηˆ q  ηˆ q ⎦ Y (η) = DT p=1 q=1 ⎡ ⎤ 3 2          1 ⎣  (η) = Ap C3,p η, ηp Y ηp + Bq C4,q η, ηˆ q  ηˆ q ⎦ DT p=1

q=1

(5.79)

5.3

Natural Frequencies and Mode Shapes of Beams with Constant Cross. . .

293

where

      C1,p η, ηp = d1 (η) γbs R2o γbs R2o mR 4 − KR f3p 1, ηp       1, ηp − DT γbs R2o f3p η, ηp − 1 + γbs R2o So f3p       − d1 (η) + jR 4 − KtR d2 (η) g3p 1, ηp   + d2 (η) g 3p 1, ηp

      C2,q η, ηˆ q = d1 (η) γbs R2o mR 4 − KR f4q 1, ηˆ q       1, ηˆ q − DT f4q η, ηˆ q − 1 + γbs R2o So f4q       + d1 (η) + jR 4 − KtR d2 (η) g4q 1, ηˆ q   − d2 (η) g 4q 1, ηˆ q

      C3,p η, ηp = d3 (η) γbs R2o γbs R2o mR 4 − KR f3p 1, ηp       − 1 + γbs R2o So f3p 1, ηp + DT g3p η, ηp       − d3 (η) + jR 4 − KtR d4 (η) g3p 1, ηp   + d4 (η) g 3p 1, ηp

      C4,q η, ηˆ q = d3 (η) γbs R2o mR 4 − KR f4q 1, ηˆ q       1, ηˆ q − DT g4q η, ηˆ q − 1 + γbs R2o So f4q       + d3 (η) + jR 4 − KtR d4 (η) g4q 1, ηˆ q   − d4 (η) g 4q 1, ηˆ q

(5.80)

and d1 (η) = d22 f1 (η) − d21 f2 (η) d2 (η) = −d12 f1 (η) + d11 f2 (η) d3 (η) = d22 g1 (η) − d21 g2 (η)

(5.81)

d4 (η) = −d12 g1 (η) + d11 g2 (η) . When So = 0, these results reduce to those given in (Magrab 2007).   Eq. (5.79) is in terms of five unknown constants Y ηp , p = 1, 2, 3 and  Equation  ηˆ q , q = 1, 2. To obtain the characteristic equation of the system, it is noted that Eq. (5.79) must be valid at each ηp and ηˆ q . Thus, Eq. (5.79) is evaluated for Y (η) at η = ηp , p = 1, 2, 3 and for  (η) at η = ηˆ q , q = 1, 2. Setting η to each of these values in the appropriate equations of Eq. (5.79) yields

294

5 Timoshenko Beams

Y (η1 ) DT =

3 

2          Ap C1,p η1 , ηp Y ηp + Bq C2,q η1 , ηˆ q  ηˆ q

p=1

Y (η2 ) DT =

3 

q=1 2          Ap C1,p η2 , ηp Y ηp + Bq C2,q η2 , ηˆ q  ηˆ q

p=1

Y (η3 ) DT =

3 

q=1 2          Ap C1,p η3 , ηp Y ηp + Bq C2,q η3 , ηˆ q  ηˆ q

p=1

   ηˆ 1 DT =

3 

q=1 2          Ap C3,p ηˆ 1 , ηp Y ηp + Bq C4,q ηˆ 1 , ηˆ q  ηˆ q

p=1

   ηˆ 2 DT =

3 

q=1 2          Ap C3,p ηˆ 2 , ηp Y ηp + Bq C4,q ηˆ 2 , ηˆ q  ηˆ q

p=1

q=1

which can be expressed in matrix form as [a] {W} = 0

(5.82)

where the elements of [a] and {W}, respectively, are   aij = Aj C1, j ηi , ηj − δij DT   aij = Bj−3 C2, j−3 ηi , ηˆ j−3   aij = Aj C3, j ηˆ i−3 , ηj   aij = Bj−3 C4, j−3 ηˆ i−3 , ηˆ j−3 − δij DT wi = Y (ηi )   wi =  ηˆ i−3

i, j = 1, 2, 3 i = 1, 2, 3 j = 4, 5 i = 4, 5 j = 1, 2, 3 i, j = 4, 5

(5.83)

i = 1, 2, 3 i = 4, 5

and δ ij is the Kronecker delta. The characteristic equation for a constant-cross-section Timoshenko beam on an elastic foundation, with an externally applied axial tensile force, subject to general boundary conditions, and having four different types of in-span attachments applied simultaneously, each at a different location, is given by det [a] = 0.

(5.84)

The values of  that satisfy Eq. (5.84) are the natural frequency coefficients n for the system. It is seen that the size of the characteristic determinant is directly proportional to the number of in-span attachments. This was made possible by Eq. (5.64), which is a direct result of the Laplace transform method. By being able to directly ascribe a physical interpretation to two of the four unknown constants, a system with four

5.3

Natural Frequencies and Mode Shapes of Beams with Constant Cross. . .

295

unknown constants is immediately reduced to a system with two unknown constants. This reduced system made it practical to obtain an explicit analytical solution. Since the boundary conditions were expressed in a very general form, we have, because of the analytical form of the results, a means to straightforwardly reduce the general result to numerous special cases. Thus, in a sense, the boundary conditions have been uncoupled from the number and type of in-span attachments since the form of the solution does not change as the boundary conditions change. It is noted that it is not necessary to consider that one has four different types of attachments applied at four in-span locations. If, for example, one is interested in attaching three different masses at three different locations, then the definition of each Ap is changed accordingly. Consequently, Eq. (5.84) is a very general result.

5.3.3 Special Cases Several special cases of Eq. (5.84) are now considered. No Attachments: Ap = 0 and Bq = 0 For this case, Eq. (5.84) yields DT = 0

(5.85)

where DT is given by Eq. (5.78). The values of  that satisfy Eq. (5.85) are the natural frequency coefficients n . Equation (5.85) is applicable to the general boundary conditions given by Eqs. (5.60) and (5.61) and to any of its special cases, several of which are given in Table 5.2. The form of the results in Table 5.2 have been obtained with the use of Eq. (5.71) and the identity

  γbs R2o k = β 2 − α 2 − R2o 4 1 + γbs R2o So + So . To reduce the five cases presented in Table 5.2 to those of the Euler-Bernoulli theory given in Table 3.3, it is noted that the expressions in Table 3.3 are for the case where So = Kf = 0. In Section 5.2.5, it was shown that the Timoshenko theory reduces to the Euler-Bernoulli theory as γbs → 0 and/or when Ro 0.026, which corresponds to L/h = 11, and is less than 0.01% for Ro < 0.0011, which corresponds to L/h = 262. Thus, the Timoshenko theory is increasingly more important for beams with smaller values of L/h and for higher natural frequencies. It is also of interest to compare the Rayleigh improvement to the Euler-Bernoulli theory as given by Eq. (5.51) with So = Kf = 0 to that given by Eq. (5.102). The solution to Eq. (5.51) for a beam hinged at both ends and undergoing harmonic 2 oscillations is y = sin (nπ η) ej τ . Substituting this solution into Eq. (5.51), we obtain nπ . (5.107) (R) n =  4 1 + (nπ Ro )2 Therefore, (R) n (E) n

1 =  4 1 + (nπ Ro )2

(5.108)

and the percentage difference when compared to the Euler-Bernoulli theory is (R) = 100

 2  2 9 2 (E) n(E) − (R) .  n n

(5.109)

A plot of Eq. (5.106), with n determined from Eq. (5.102), and Eq. (5.109) is given in Fig. 5.1. It is seen from this figure that the Timoshenko theory predicts increasingly lower values of the natural frequency than that predicted by the Rayleigh theory as Ro increases. This difference increases more rapidly with increasing Ro as n increases. Thus, the inclusion of the shear effects accounts for a more significant effect on the natural frequency than just including the rotational inertia effects. It is remarked that the percentage difference of the square of the natural frequency coefficients is used because the dimensional frequency is proportional to the square of the frequency coefficient. Recall Eq. (3.131). Based on these numerical results, it is seen that one should expect a value of Ro = 0.001 to recover the Euler-Bernoulli beam. This value of Ro will be used in the numerical results that follow whenever the Timoshenko beam results are compared to those of the Euler-Bernoulli beam. In presenting the numerical results in this section, we have set So = Kf = 0 and used κ = 5/6 and υ = 0.3, which from Eq. (5.38) yields γbs = 3.12. Timoshenko Beams with No In-span Attachments Representative numerical evaluations of the first four sets of boundary conditions appearing in Table 5.2 are shown in Figs. 5.2 to 5.4. In each figure, the ratio of the

5.3

Natural Frequencies and Mode Shapes of Beams with Constant Cross. . .

303

16 Δ(T) Δ(R)

14

% Difference (Δ)

12 10

n=2

8 6 4 n = 1,2 2 n=1 0

5 × 10−3

5 × 10−2

10−2 Ro

Fig. 5.1 Percentage differences of the lowest two natural frequencies between the Timoshenko theory and Euler-Bernoulli theory and between the Rayleigh theory and Euler-Bernoulli theory for a beam hinged at both ends as a function of Ro for γbs = 3.12 and Kf = So = 0 1

0.95

(Ωn/Ωn(E) )2

0.9

n=1 (E)

Ω1 = 3.142 (E)

Ω2 = 6.283 0.85

n=2

(E)

Ω3 = 9.424

0.8

0.75

0.7

n=3

5 × 10−3

5 × 10−2

10−2 Ro

Fig. 5.2 Comparison of the Timoshenko beam to that of the Euler-Bernoulli beam for the lowest three natural frequencies of a beam hinged at both ends as a function of Ro for γbs = 3.12 and Kf = So = 0

304

5 Timoshenko Beams 1 0.95 (E)

0.9

Ω1 = 4.73

0.85

Ω2 = 7.852

(Ωn/Ωn(E) )2

(E)

n=1

(E)

Ω3 = 10.99 0.8 0.75

n=2 0.7 0.65 n=3 0.6

5 × 10−3

5 × 10−2

10−2 Ro

Fig. 5.3 Comparison of the Timoshenko beam to that of the Euler-Bernoulli beam for the lowest three natural frequencies of a beam clamped at both ends as a function of Ro for γbs = 3.12 and Kf = So = 0 1

n=1 0.95

(Ωn/Ωn(E) )2

mR = 0.4 0.9

0

(E)

Ω1 = 1.472 1.875 (E)

Ω2 = 4.144 4.694 0.85

n=2

(E)

Ω3 = 7.215 7.854

mR = 0.4

0.8

mR = 0 n=3 0.75

5 × 10−3

5 × 10−2

10−2 Ro

Fig. 5.4 Comparison of the Timoshenko beam to that of the Euler-Bernoulli beam for the lowest three natural frequencies of a cantilever beam as a function of Ro for γbs = 3.12, Kf = So = 0, and mR = 0.0 and 0.4

5.3

Natural Frequencies and Mode Shapes of Beams with Constant Cross. . .

305

square of the natural frequency coefficient for the Timoshenko beam, denoted n , to that of the corresponding value obtained for the Euler-Bernoulli theory, denoted (E) (E) n , is give as a function of Ro for n = 1, 2, 3. The values for n are obtained by setting Ro = 0.001. It is also noted that for a beam of length L with a rectangular cross section of height h, Ro = 0.005 corresponds to a value of L/h = 58, Ro = 0.01 to a value of L/h = 29, and Ro = 0.05 to a value of L/h = 5.8. The results shown in these figures support the previously noted conclusions obtained for a beam hinged at both ends; that is, as the L/h ratio decreases the differences between the two theories increase and, additionally, as n increases, so does the magnitude of the differences. Furthermore, it is seen that the magnitude of these differences is a strong function of the boundary conditions. Timoshenko Beams with One In-span Attachment that Translates Only Representative numerical evaluations of Eq. (5.87) for one in-span attachment that undergoes translation only are given in Figs. 5.5 to 5.10 for three sets of boundary conditions: clamped at both ends, hinged at both ends, and a cantilever with no attachments at the free end. For each of these boundary conditions, the attachment is either a spring placed at ηs = 0.3 and 0.45 (Figs. 5.5 to 5.7) or a mass placed at ηm = 0.3 and 0.45 (Figs. 5.8 to 5.10). In addition, for each combination of these parameters two values of Ro are used: Ro = 0.03, which corresponds to 40 Ro = 0.03 35

Ro = 0.001 (Euler) ηs = 0.45

30

2

Ω1

25 ηs = 0.3 20

15

10

5 100

101

102 Ki

103

104

Fig. 5.5 Comparison of the lowest natural frequency coefficient of the Timoshenko beam to that of the Euler-Bernoulli beam for a beam hinged at both ends with an in-span spring as a function of Ki for γbs = 3.12, Kf = So = 0, and ηs = 0.3 and 0.45

306

5 Timoshenko Beams 60 Ro = 0.03 55

Ro = 0.001 (Euler)

50 ηs = 0.45

2

Ω1

45 40 35

ηs = 0.3

30 25 20 100

101

102 Ki

103

104

Fig. 5.6 Comparison of the lowest natural frequency coefficient of the Timoshenko beam to that of the Euler-Bernoulli beam for a beam clamped at both ends with an in-span spring as a function of Ki for γbs = 3.12, Kf = So = 0, and ηs = 0.3 and 0.45 9 Ro = 0.03 Ro = 0.001 (Euler) 8 ηs = 0.45

2

Ω1

7

6 ηs = 0.3

5

4

3 100

101

102 Ki

103

104

Fig. 5.7 Comparison of the lowest natural frequency coefficient of the Timoshenko beam to that of the Euler-Bernoulli beam for a cantilever beam with an in-span spring as a function of Ki for γbs = 3.12, Kf = So = 0, and ηs = 0.3 and 0.45

5.3

Natural Frequencies and Mode Shapes of Beams with Constant Cross. . .

307

10 9 8 7

2

Ω1

6

ηm = 0.3

ηm = 0.45

5 4 3 2 1

Ro = 0.03 Ro = 0.001 (Euler)

0 10−2

10−1

100 mi

101

102

Fig. 5.8 Comparison of the lowest natural frequency coefficient of Timoshenko beam to that of the Euler-Bernoulli beam for a beam hinged at both ends with an in-span mass as a function of mi for γbs = 3.12, Kf = So = 0, and ηm = 0.3 and 0.45 25

20

15 2

Ω1

ηm = 0.3 ηm = 0.45 10

5 Ro = 0.03 Ro = 0.001 (Euler) 0 10−2

10−1

100 mi

101

102

Fig. 5.9 Comparison of the lowest natural frequency coefficient of the Timoshenko beam to that of the Euler-Bernoulli beam for a beam clamped at both ends with an in-span mass as a function of mi for γbs = 3.12, Kf = So = 0, and ηm = 0.3 and 0.45

308

5 Timoshenko Beams 4

3.5

ηm = 0.3 ηm = 0.45

3

2

Ω1

2.5

2

1.5

1

Ro = 0.03 Ro = 0.001 (Euler)

0.5 10−2

10−1

100 mi

101

102

Fig. 5.10 Comparison of the lowest natural frequency coefficient of the Timoshenko beam to that of the Euler-Bernoulli beam for a cantilever beam with an in-span mass as a function of mi for γbs = 3.12, Kf = So = 0, and ηm = 0.3 and 0.45

L/h = 9.6; and Ro = 0.001, which corresponds to L/h = 289 and provides a very good approximation of the Euler-Bernoulli theory. To place in context the stiffness ratios for the various springs Ki , it is noted that a value of 3 for a cantilever beam indicates that when the spring is attached at the free end the stiffness of the beam and spring are equal. When the beam is hinged at both ends and the spring is attached at the midpoint, a value of Ki of 48 indicates that the spring stiffness and the beam stiffness are equal. Lastly, when the beam is clamped at both ends and the spring is attached at the midpoint, a value of Ki of 192 indicates that the spring stiffness and the beam stiffness are equal. It is seen from Figs. 5.5 to 5.7 that when an in-span spring is attached to a beam, the differences between the values of the lowest natural frequency of the two beam theories increase as Ki increases. This increase in the difference is true for the three sets of boundary conditions and for the two locations of the springs selected. When the in-span attachment is a mass, as shown in Figs. 5.8 to 5.10, the differences in the lowest natural frequencies between the two theories are relatively small except for the beam clamped at both ends. Also, increasing the magnitude of mi tends to decrease the differences between the two theories for all of the cases considered.

5.3

Natural Frequencies and Mode Shapes of Beams with Constant Cross. . .

309

Timoshenko Beams with Two In-span Attachments That Translate Only Representative numerical evaluations of Eq. (5.93) for two in-span attachments that undergo translation only are given in Figs. 5.11 and 5.12 for a cantilever beam with Ro = 0.03, γbs = 3.12, and mR = KR = 0. In Fig. 5.11, the lowest natural frequency coefficient 1 is obtained as a function of the location of two equally stiff springs. In Fig. 5.11a, Ki = 5 and in Fig. 5.11b Ki = 50. It is seen from these figures that the lowest natural frequency coefficient changes in a complicated manner that is dependent on the relative locations of the springs and their stiffness. It is noted that when Ki = 50, the maximum values of 1 occur when the locations of the springs are in the vicinity of the node point of the second natural frequency of the cantilever beam without any in-span attachments; that is, ηnode = 0.783. Under

4 2.8 Ω1

Ω1

2.6 2.4 2.2 2 1.8 1

3.5 3 2.5 2 1.5 1 0.8 0.6 ηp 0.4

0.8 0.6 ηp 0.4 0.2 0.2

0 0

0.4

0.6 ηr

0.8

1

0.2 0.2

0 0

(a)

0.4

0.6 ηr

0.8

1

(b)

Fig. 5.11 Lowest natural frequency coefficient of a cantilever Timoshenko beam with two in-span springs of equal stiffness whose locations vary for Ro = 0.03, γbs = 3.12, Kf = So = 0, and mR = KR = 0 (a) Ki = 5 and (b) Ki = 50

3.5 3 Ω1

Ω1

2.6 2.4 2.2 2 1.8

2.5 2 1.5 1

1.6 1

0.8

0.8 0.6 ηspring

1

0.4 0.6

0.2 0.2

0 0

(a)

0.4 ηmass

0.8

0.6 ηspring 0.4

0.2 0.2

0 0

0.4

0.6

0.8

1

ηmass

(b)

Fig. 5.12 Lowest natural frequency coefficient of a cantilever Timoshenko beam with one in-span spring and one in-span mass whose locations vary for Ro = 0.03, γbs = 3.12, Kf = So = 0, and mR = KR = 0 (a) Ki = 5 and mi = 0.1 and (b) Ki = 50 and mi = 0.5

310

5 Timoshenko Beams

these conditions, the beam is forced to assume the mode shape of the beam at its second natural frequency. In Fig. 5.12, the lowest natural frequency coefficient 1 is obtained as a function of the location of a spring and a mass. In Fig. 5.12a, Ki = 5 and mi = 0.1 and in Fig. 5.12b Ki = 50 and mi = 0.5. Again it is seen from these figures that the lowest natural frequency coefficient changes in a complicated manner that is dependent on the relative locations of the spring and the mass and their respective magnitudes.

5.4 Natural Frequencies of Beams with Variable Cross Section 5.4.1 Beams with a Continuous Taper: Rayleigh-Ritz Method In this section, the natural frequencies of cantilever Timoshenko beams with continuously variable cross sections are examined. However, we shall only solve this class of beams by using the Rayleigh-Ritz method as employed in Section 3.8.5. It is assumed that there are no in-span attachments, no externally applied forces or moments are applied, p (x, t) = po , and there are attached at the end x = L a translation spring with spring constant kR , a torsion spring with torsion spring constant ktR , and a mass MR . To employ the Rayleigh-Ritz method for the determination of the natural frequencies and mode shapes, we assume that the beam is vibrating harmonically at a frequency ω and magnitudes W (x) and  (x) and form a quantity φ, which is the difference between the maximum kinetic energy and the maximum potential energy. The maximum potential energy is obtained by replacing w (x, t) with W (x) and ψ (x, t) with  (x). The maximum kinetic energy is obtained by replacing ∂w (x, t)/∂t with ωW (x), which is the maximum translational velocity, and ∂ψ (x, t)/∂t with ω (x), which is the maximum rotational velocity. Then, from Eq. (3.280) φ = Tmax − Vmax

(5.110)

where, from Eqs. (5.22) and (5.23),

Tmax

⎧ L ⎫

⎬  ω2 ⎨  ρA (x) W 2 (x) + ρI (x)  2 (x) dx + MR W 2 (L) = ⎭ 2 ⎩

(5.111)

0

and, from Eqs. (5.22), (5.24), and (5.25),

Vmax

1 = 2

 2    

L , dW d 2 dW 2 2 + κGA (x) + kf W dx −  + po EI (x) dx dx dx 0

+

 1 kR W 2 (L) + ktR  2 (L) . 2

(5.112)

5.4

Natural Frequencies of Beams with Variable Cross Section

311

Introducing the quantities given in Eqs. (5.38) and (5.39), Eqs. (5.111) and (5.112), respectively, become

Tmax

⎤ ⎡ 1

1 4 EIo ⎣ = a (η) Y 2 (η) dη + R2o i (η)  2 (η) dη + mR Y 2 (1)⎦ (5.113) 2L 0

0

and

Vmax

⎡ 1 2   

1 EIo ⎣ 1 d 2 dY −  = i (η) dη + a dη (η) 2L dη dη γbs R2o 0

0

1



1 Y 2 (η) dη + So

+ Kf 0

0

dY dη

2 (5.114)

dη

⎤

+ KR Y 2 (1) + KtR  2 (1)⎦ . Substituting Eqs. (5.113) and (5.114) into Eq. (5.110), we obtain ⎤ ⎡ 1

1 2φL = 4 ⎣ a (η) Y 2 (η) dη + R2o i (η)  2 (η) dη + mR Y 2 (1)⎦ = EIo ⎡ −⎣

0

1

0



d i (η) dη

2

1 dη + γbs R2o

0

1 

1 Y 2 (η) dη + So

+ Kf 0

0

dY dη

1



dY − a (η) dη

2 dη

0

2 dη

-

+ KR Y (1) + KtR  (1) . 2

2

(5.115) Following the procedure of Section 3.8.5, we assume a solution of the form Y (η) =

N 

(s)

cl Yl (η)

l=1

 (η) =

N  l=1

(5.116) cl l(s) (η)

312

5 Timoshenko Beams (s)

(s)

where Yl (η) and l (η), which depend on the boundary conditions and the taper, will be determined subsequently, cl , l = 1, 2, . . ., N are unknown constants, and N is an appropriately chosen integer. It is noted that the same unknown constants cl appear for the two different quantities Y and . For justification of this form of Eq. (5.116), refer to Eq. (5.184). Substituting Eq. (5.116) into Eq. (5.115), we obtain

 = 4

N N  

cl cj I1lj −

l=1 j=1

N N  

cl cj I2lj

(5.117)

l=1 j=1

where

1 I1lj =

(s) (s) a (η) Yl (η) Yj (η) dη

1

0

+

1 I2lj =

(s)

(s)

i (η) l (η) j (η) dη

+ R2o 0

mR Yl(s) (1) Yj(s) (1) (s) (s) i (η) l (η) j (η) dη

1 + So

0

(s) (s) Yl Yj dη

1 + Kf

0

1

(s)

(s)

Yl (η) Yj (η) dη 0

 

(s) (s) (s) (s) Yj − j dη a (η) Yl − l

+

1 γbs R2o

+

(s) (s) (s) (s) KR Yl (1) Yj (1) + KtR l (1) j (1)

0

(5.118) and the prime denotes the derivative with respect to η. It is noted that I1lj = I1jl and I2lj = I2jl ; that is, they are symmetric. Using Eq. (5.117) in Eq. (3.287), yields  ∂  = cl I2ln − 4 cl I1ln = 0 ∂cn N

N

l=1

l=1

n = 1, 2, . . . , N

(5.119)

or in matrix form,   [I2 ] − 4 [I1 ] {c} = 0.

(5.120)

The elements of square matrix [I1 ] are I1lj , those of the square matrix [I2 ] are I2lj , and! those of the !column vector {c} are cl . The values of  = n that satisfy det ![I2 ] − 4 [I1 ]! = 0 are the natural frequency coefficients of the system.

5.4

Natural Frequencies of Beams with Variable Cross Section

313

We shall consider a cantilever beam with a torsion spring of constant ktR , a translation spring of constant kR , and a mass mR attached at its free end. Therefore, the non dimensional form of the boundary conditions is obtained from Eqs. (5.43) and (5.44), respectively. Thus, in the present notation, at η = 0 we have Y (0) = 0  (0) = 0

(5.121)

and at η = 1 we have     γbs R2o KR − γbs R2o 4 mR Y (1) + a (1) + γbs R2o So Y (1) − a (1)  (1) = 0 KtR  (1) + i (1)  (1) = 0. (5.122) In Eq. (5.122), the prime denotes the derivative with respect to η. To select the functions Yl(s) (η) and l(s) (η), we employ the technique of (s) (s) Section 3.8.5 and let Yl (η) and l (η) be the solution to the static Timoshenko beam equations; that is, d dη





dYl(s) (s) − l a (η) dη (s)

−Kf γbs R2o Yl

 + γbs R2o So

d2 Yl(s) dη2

(5.123)

= −γbs R2o sin ((l − 0.5) π η)

and 

   (s) dYl(s) dl (s) 2 d a (η) + γbs Ro − l i (η) = 0. dη dη dη

(5.124)

The right hand side of Eq. (5.123) has been selected to create a deflection that emulates the lth mode shape. The boundary conditions for the static case are given by Eqs. (5.121) and (5.122) with mR = 0. Tapers Following Section 3.8.2, we shall examine two different tapers: a linear tape and an exponential taper. These two tapers and their special cases are shown in Fig. 3.28, except that in this section the maximum rectangular cross section is positioned at η = 0 and the smallest cross section is located at η = 1. From Eqs. (3.237) and (3.239), a (η) = y (η) z (η) i (η) = y (η) z3 (η) .

(5.125)

When Ao is given in Eq. (3.239) and Io is given in Eq. (3.237), the functions y (η) and z (η) are as follows.

314

5 Timoshenko Beams

Linear Taper For a beam with a linear taper, y (η) = (β − 1) η + 1 z (η) = (α − 1) η + 1

(5.126)

where α = h1 /ho ≤ 1and β = b1 /bo ≤ 1. Exponential Taper For a beam with an exponential taper, y (η) = eβe η z (η) = eαe η

(5.127)

where βe = ln β ≤ 0 αe = ln α ≤ 0.

(5.128)

Numerical Results Using Eqs. (5.121) to (5.124) with mR = 0 to generate a set of Yl(s) (η) and l(s) (η), Eq. (5.120) was evaluated and compared to published results. The comparisons for several combinations of values are given in Table 5.4, where it is seen that the agreement is very good. Therefore, one can expect that the above approximate procedure will yield numerical results very close to their true values. Setting KR = mR = KtR = 0, the values of the first and second natural frequency coefficients for both a Timoshenko beam and for the Euler-Bernoulli beam have been determined. These results appear in Fig. 5.13 for a linearly tapered cantilever beam and in Fig. 5.14 for an exponentially tapered cantilever beam. It is seen from Fig. 5.13 that for a beam with a linear taper the differences in the natural frequencies between the Timoshenko theory and Euler-Bernoulli theory are functions of β and α, although for β = 0.9 and for the lowest natural frequency this difference is almost independent of α. As noted for a beam with constant cross section, the differences are much larger for the second natural frequency. Similar conclusions are drawn for the beam with an exponential taper shown in Fig. 5.14.

5.4.2 Constant Cross Section with a Step Change in Properties In this section, cantilever Timoshenko beams with a cross section that is constant for a portion of its length a and then abruptly changes to a cross section with different properties for the remainder of its length c are examined. The total length of the beam is L = a + c. It is assumed that the cantilever beam is carrying a concentrated mass ML at its free end. In addition, it is assumed that the beam does not have any

Ro

0.04

0.03

0.0289

0.04

0.0283

0.0289

Taper

Single taper Linear

Exponential

Linear

Linear

Double taper Linear

Linear

3.12

3.12

3.12

1.95

3.12

3.12

γ bs

0.8

mR = 0.6

–

0.2

0.9

0.5

mR = 20, KR = 10

–

1.0

0.6

α

–

–

Attachments at η = 1

0.2

0.9

1.0

1.0

e−1

1.0

β

6.149 6.148

3.646 3.646

1.780 1.85

0.769 0.755

4.685 4.685

3.689 3.69

21

17.985 17.987

20.581 20.574

14.395 14.43

12.166 11.950

22.879 22.874

17.886 17.88

22

38.053 38.088

53.452 53.425

40.356 40.10

36.334 32.160

56.475 56.450

43.729 43.74

23

Eq. (5.120) (Zhou and Cheung 2001)

Eq. (5.120) (Lee and Lin 1992)

Eq. (5.120) (Rossi et al. 1990)

Eq. (5.120) (Matsuda et al. 1992)

Eq. (5.120) (Tong et al.1995)

Eq. (5.120) (Rossi et al. 1990)

Source

Table 5.4 Comparison of the results from Eq. (5.120) using N = 7 and the static solution to Eqs. (5.123) and (5.124) with selected results from the literature for a tapered cantilever Timoshenko beam for So = Kf = 0

5.4 Natural Frequencies of Beams with Variable Cross Section 315

316

5 Timoshenko Beams 34

9 Ro = 0.03

Ro = 0.001 (Euler)

30 28

7

β = 0.01

26 Ω 22

Ω 12

Ro = 0.03

32

Ro = 0.001 (Euler)

8

6

24

β = 0.01

22 5

20 18

4 3

0

0.2

0.4

β = 0.9

16

β = 0.9 0.6

0.8

1

14

0

0.2

0.4

0.6

α

α

(a)

(b)

0.8

1

Fig. 5.13 Natural frequency coefficients of a linearly tapered cantilever beam as a function of α for β = 0.01 and 0.9, Ro = 0.001 and 0.03, and γbs = 3.12 (a) First natural frequency and (b) Second natural frequency. The three dimensional images of the beams at the left end are for α = 0.01 and the value of β indicated and those at the right end are for α = 1 and the value of β indicated 12

35

11

β = 0.01

9

Ro = 0.03

8

Ro = 0.001 (Euler)

β = 0.01

25

Ω 22

Ω 12

10

Ro = 0.03 Ro = 0.001 (Euler)

30

7

20

6

4

10

β = 0.9

3 2

β = 0.9

15

5

0

0.2

0.4

0.6

0.8

1

5

0

0.2

0.4

0.6

α

α

(a)

(b)

0.8

1

Fig. 5.14 Natural frequency coefficients of an exponential-tapered cantilever beam as a function of α for β = 0.01 and 0.9, Ro = 0.001 and 0.03, and γbs = 3.12 (a) First natural frequency and (b) Second natural frequency. The three dimensional images of the beams at the left end are for α = 0.05 and the value of β indicated and those at the right end are for α = 1 and the value of β indicated

in-span attachments, there is no axial force, no elastic foundation, and the applied force and applied moment are zero; thus, p = kf = Fa = Ma = ki = kti = Mi = 0. We shall follow the procedure that was used in Section 3.8.7. For the geometry shown in Fig. 3.37, the governing equations for each portion of the beam are given by Eqs. (5.31) and (5.32), which simplify to  κ l Gl A l

∂ψl ∂ 2 wl − 2 ∂xl ∂xl

 − ρ l Al

∂ 2 wl =0 ∂t2

l = 1, 2

(5.129)

5.4

Natural Frequencies of Beams with Variable Cross Section

317

and  κl Gl Al

   ∂ 2 ψl ∂wl ∂ 2 ψl − ψl + El Il I = 0 l = 1, 2. − ρ l l ∂xl ∂t2 ∂xl2

(5.130)

From Eq. (5.36a,b), the boundary conditions at x1 = 0 are w1 |x1 =0 = 0 ψ1 |x1 =0 = 0

(5.131)

and, again from Eq. (5.36a,b), those at x2 = 0 are !  ! ! ∂ 2 w2 !! ∂w2 ML 2 ! = κ2 G2 A2 − ψ2 !! ∂t x2 =0 ∂x2 x2 =0 ! ∂ψ2 !! = 0. ∂x !

(5.132)

2 x2 =0

At the common boundary between the sections of the beams with different properties, a set of continuity conditions have to be met. These continuity conditions are: the equality of the displacements; the equality of the rotations of the cross section; the sum of the moments is zero; and the sum of the shear forces is zero. Thus, using Eqs. (5.6) and (5.8), it is found that w1 |x1 =a = w2 |x2 =c ψ1 |x1 =a = − ψ2 |x2 =c ! ! ∂ψ1 !! ∂ψ2 !! E1 I1 = E I 2 2 ∂x1 !x1 =a ∂x2 !x2 =c  !  ! ! ! ∂w1 ∂w2 κ1 G1 A1 − ψ1 !! = −κ2 G2 A2 − ψ2 !! . ∂x1 ∂x2 x1 =a x2 =c

(5.133)

To put the preceding results in non dimensional form, the following parameters are introduced wl a c x1 , aL = , cL = , L = a + c m, η = L L L L 4 ρ 1 A1 L t Il x2 s2 , τ = , R2l = l = 1, 2 ξ = , to2 = L E1 I1 to Al L2 2 (1 + νl ) ρ2 A2 E1 I1 γlbs = l = 1, 2 α = , β= , m = ρ1 A1 L kg κl ρ1 A1 E2 I2 ML ML = , mb = ρ1 A1 a + ρ2 A2 c = mco kg, co = aL + αcL . mL = mb mco (5.134) yl =

318

5 Timoshenko Beams

Using Eq. (5.134) in Eqs. (5.129) and (5.130), we arrive at 2 ∂ψ1 ∂ 2 y1 2 ∂ y1 − R =0 − γ 1bs 1 ∂η ∂η2 ∂τ 2 2 ∂ 2 ψ1 ∂y1 4 ∂ ψ1 − γ R = 0 0 ≤ η ≤ aL − ψ1 + γ1bs R21 1bs 1 ∂η ∂η2 ∂τ 2

(5.135)

and 2 ∂ψ2 ∂ 2 y2 2 ∂ y2 − R =0 − αβγ 2bs 2 ∂ξ 2 ∂ξ ∂τ 2 2 ∂y2 ∂ 2 ψ2 4 ∂ ψ2 − αβγ R = 0 0 ≤ ξ ≤ cL . − ψ2 + γ2bs R22 2bs 2 ∂ξ ∂ξ 2 ∂τ 2

(5.136)

From Eq. (5.131), the boundary conditions at η = 0 are y1 |η=0 = 0

(5.137)

ψ1 |η=0 = 0 and from Eq. (5.132), those at ξ = 0 are mL co βγ2bs R22

! !  ! ∂ 2 y2 !! ∂y2 ! = − ψ 2 ! ! 2 ∂ξ ∂τ ξ =0 ξ =0 ! ! ∂ψ2 = 0. ∂ξ !

(5.138)

ξ =0

The continuity conditions given by Eq. (5.133) become y1 |η=aL = y2 |ξ =cL

β

R22 γ2bs R21 γ1bs

ψ1 |η=aL = − ψ2 |ξ =cL ! ! ∂ψ1 !! ∂ψ2 !! β = ∂η !η=aL ∂ξ !ξ =cL   !! ! ! ∂y2 ∂y1 ! = − . − ψ1 ! − ψ2 !! ! ∂η ∂ξ ξ =cL

(5.139)

η=aL

We assume solutions of the form y1 (η, τ ) = Y1 (η) e j τ ,

ψ1 (η, τ ) = 1 (η) e j

y2 (ξ , τ ) = Y2 (ξ ) e j τ ,

ψ2 (ξ , τ ) = 2 (ξ ) e j τ .

2 2

2τ 2

(5.140)

5.4

Natural Frequencies of Beams with Variable Cross Section

319

Upon substituting Eq. (5.140) into Eqs. (5.135) and (5.136), respectively, we obtain ∂ 2 Y1 ∂1 + γ1bs R21 4 Y1 = 0 − 2 ∂η ∂η ∂ 2 1 ∂Y1 + γ1bs R41 4 1 = 0 − 1 + γ1bs R21 ∂η ∂η2

(5.141) 0 ≤ η ≤ aL

and ∂ 2 Y2 ∂2 + αβγ2bs R22 4 Y2 = 0 − 2 ∂ξ ∂ξ ∂ 2 2 ∂Y2 − 2 + γ2bs R22 + αβγ2bs R42 4 2 = 0 ∂ξ ∂ξ 2

(5.142) 0 ≤ ξ ≤ cL .

The boundary conditions at η = 0, which are given by Eq. (5.137), become Y1 |η=0 = 0

(5.143)

1 |η=0 = 0 and those at ξ = 0, which are given by Eq. (5.138), become ! ! −mL co βγ2bs R22 4 Y2 ! ! ∂2 !! ∂ξ !

 ξ =0

ξ =0

=

! ! ∂Y2 − 2 !! ∂ξ ξ =0

(5.144)

= 0.

The continuity conditions given by Eq. (5.139), become Y1 |η=aL = Y2 |ξ =cL

β

R22 γ2bs R21 γ1bs

1 |η=aL = − 2 |ξ =cL ! ! ∂1 !! ∂2 !! β = ∂η !η=aL ∂ξ !ξ =cL !  ! !  ! ∂Y1 ∂Y2 ! − 1 ! − 2 !! = − . ! ∂η ∂ξ ξ =cL

(5.145)

η=aL

To obtain a solution to Eqs. (5.141) to (5.145), we take the Laplace transform of Eq. (5.141) with respect to the spatial variable η and we take the Laplace transform of Eq. (5.142) with respect to the spatial variable ξ , respectively, to obtain

320

5 Timoshenko Beams

2  ¯1 s + γ1bs R21 4 Y¯ 1 − s = Y1 (0) + sY1 (0) − 1 (0)   sY¯ 1 + γ1bs R21 s2 + γ1bs R41 4 − 1 1 = Y1 (0) + γbs R21 1 (0) + sγbs R21 1 (0)

(5.146) 0 ≤ η ≤ aL

and 2  ¯2 s + αβγ2bs R22 4 Y¯ 2 − s = Y2 (0) + sY2 (0) − 2 (0)   sY¯ 2 + γ2bs R22 s2 + αβγ2bs R42 4 − 1 2 = Y2 (0) + γ2bs R22 2 (0) + sγ2bs R22 2 (0)

(5.147) 0 ≤ ξ ≤ cL

¯ j (s), respectively, are the Laplace transforms of Yj ¯j =  where Y¯ j = Y¯ j (s) and  and  j . Using Eq. (5.143) in Eq. (5.146) and Eq. (5.144) in Eq. (5.147), respectively, we arrive at  2 ¯ 1 = Y (0) s + γ1bs R21 4 Y¯ 1 − s 1   2 2 4 4 ¯ sY1 + γ1bs R1 s + γ1bs R1  − 1 1 = γbs R21 1 (0)

0 ≤ η ≤ aL

(5.148)

and   2  ¯ 2 = s − mL co βγ2bs R2 4 Y2 (0) s + αβγ2bs R22 4 Y¯ 2 − s 2   sY¯ 2 + γ2bs R22 s2 + αβγ2bs R42 4 − 1 2 = Y2 (0) + sγ2bs R22 2 (0) 0 ≤ ξ ≤ cL . (5.149) ¯ 1 , we obtain Solving Eq. (5.148) for Y¯ 1 and 

   1  1 (0) s + s2 + R21 4 − 1 γbs R21 Y1 (0) ¯ 1 (s) D

   1  ¯ 1 (s) =  1 (0) s2 + γ1bs R21 4 − sY1 (0) γ1bs R21 ¯ 1 (s) D Y¯ 1 (s) =

(5.150)

where  

¯ 1 (s) = s2 − δ12 s2 + ε12 D and

(5.151)

5.4

Natural Frequencies of Beams with Variable Cross Section

  1 −F11 + F12 2   1 F11 + F12 ε12 = 2 F11 = (1 + γ1bs ) R21 4  2 F12 = (1 − γ1bs )2 R21 4 + 44 .

321

δ12 =

(5.152)

¯ 2 , we obtain Solving Eq. (5.149) for Y¯ 2 and  Y¯ 2 (s) =

¯ 2 (s) = 

1  3 s − mL co βγ2bs R22 4 s2 + αβR22 4 s ¯ 2 (s) D 

 −mL co β4 αβγ2bs R42 4 − 1 Y2 (0) + s2 2 (0)  1  mL co β4 s + αβ4 Y2 (0) ¯ 2 (s) D    + s3 + αβγ2bs R22 4 s 2 (0)

(5.153)

where  

¯ 2 (s) = s2 − δ22 s2 + ε22 D

(5.154)

and   1 −F21 + F22 2   1 ε22 = F21 + F22 2 F21 = (1 + γ2bs ) αβR22 4

2 F22 = (1 − γ2bs )2 αβR22 4 + 4αβ4 . δ22 =

(5.155)

The inverse Laplace transform of Eq. (5.150) is Y1 (η) = f11 (η) 1 (0) + f12 (η) Y1 (0) 1 (η) = g11 (η) 1 (0) + g12 Y1 (0)

(5.156)

where the definitions of f1i (η) and g1i (η) are given in Appendix 5.2. The inverse Laplace transform of Eq. (5.153) is Y2 (ξ ) = f21 (ξ ) Y2 (0) + f22 (ξ ) 2 (0) 2 (ξ ) = g21 (ξ ) Y2 (0) + g22 (ξ ) 2 (0) where the definitions of f2i (η) and g2i (η) are given in Appendix 5.2.

(5.157)

322

5 Timoshenko Beams

The four unknown quantities Y1 (0) , 1 (0) , Y2 (0) , and 2 (0) are determined by substituting Eqs. (5.156) and (5.157) into the continuity conditions given by Eqs. (5.145). Performing the substitutions, we obtain ⎫ ⎡ ⎤⎧ f11 (aL ) f12 (aL ) −f21 (cL ) −f22 (cL ) ⎪ 1 (0) ⎪ ⎪ ⎪ ⎢ g11 (aL ) g12 (aL ) g21 (cL ) g22 (cL ) ⎥ ⎨ Y (0) ⎬ 1 ⎢ ⎥ (5.158) ⎣ βg (aL ) βg (aL ) −g (cL ) −g (cL ) ⎦ ⎪ Y2 (0) ⎪ = 0 11 12 21 22 ⎪ ⎪ ⎭ ⎩ c41 2 (0) c42 c43 c44 where c41 = β

 R22 γ2bs  f11 (aL ) − g11 (aL ) 2 R1 γ1bs

c42 = β

 R22 γ2bs  f12 (aL ) − g12 (aL ) 2 R1 γ1bs

(5.159)

(c ) − g (c ) c43 = f21 L 21 L c44 = f22 (cL ) − g22 (cL ) , etc., are given in Appendix 5.2. The prime on the functions and the quantities f11 f1 l and g1 l denotes the derivative with respect to η and the prime on the functions f2 l and g2 l denotes the derivative with respect to ξ . The natural frequency coefficients are obtained by finding those values of  = n for which the determinant of the coefficients of Eq. (5.158) equals zero.

5.4.3 Numerical Results To obtain representative numerical results, it is assumed that both portions of the beam are the same material; that is, ρ1 = ρ2 and E1 = E2 , and they have rectangular cross sections. We then consider the three special cases shown in Fig. 3.38 and given by Eqs. (3.303) to (3.305). If αo = h2 /h1 , then these three cases can be expressed as: Case 1: b2 /b1 = h2 /h1 α = αo2 β=

1 α2

(5.160)

Case 2: b2 /b1 = 1 α = αo β=

1 α3

(5.161)

5.4

Natural Frequencies of Beams with Variable Cross Section

323

Case 3: h2 /h1 = 1 b2 b1 1 β= α α=

(5.162)

Furthermore, it is noted that only R1 or R2 is an independent quantity since, for a rectangular cross section, h2 h1 h2 R2 = √ = √ = αo R1 . L 12 L 12 h1

(5.163)

The evaluation of the determinant of the coefficients of Eq. (5.158) for the lowest natural frequency coefficients are shown in Figs. 5.15 to 5.17. Included in these figures are the results for an Euler-Bernoulli beam, which have been given in Figs. 3.39 to 3.41. It is seen from these results that there is a difference between the EulerBernoulli and Timoshenko theories for mL = 0 of 2 to 3% and that this difference diminishes as the value of mL increases. Thus, for stepped beams, it is adequate, in most cases, to use the Euler-Bernoulli beam theory for the determination of the lowest natural frequency. 5.5 5 4.5 mL = 0

4 3.5

mL = 0.2

Ω 21

3 2.5 2

mL = 1

1.5 β = 1/α, α = b2 / b1, R2 = 0.05 β = 1/α2, α = (h2/h1)2, R2 = 0.032 β = 1/α3, α = h2/h1, R2 = 0.02 Euler 0.6 0.8 1

1 0.5 0

0

0.2

0.4 aL

Fig. 5.15 Lowest natural frequency of a stepped cantilever Timoshenko beam as a function of aL for α = 0.4, R1 = 0.001 (Euler-Bernoulli beam) and 0.05, γbs = 3.12, mL = 0, 0.2, and 1.0, and for the three combinations of α, β, and R2 indicated; R2 is determined from Eq. (5.163)

324

5 Timoshenko Beams 4.5 mL = 0

4 3.5 3

mL = 0.2

Ω 21

2.5 2

mL = 1

1.5 β = 1/α, α = b2 / b1, R2 = 0.05 1

β = 1/α2, α = (h2/h1)2, R2 = 0.039 β = 1/α3, α = h2/h1, R2 = 0.03

0.5

Euler 0

0

0.2

0.4

0.6

0.8

1

aL

Fig. 5.16 Lowest natural frequency of a stepped cantilever Timoshenko beam as a function of aL for α = 0.6, R1 = 0.001 (Euler-Bernoulli beam) and 0.05, γbs = 3.12, mL = 0, 0.2, and 1.0, and for the three combinations of α, β, and R2 indicated; R2 is determined from Eq. (5.163) 4.5 4 mL = 0 3.5 3

mL = 0.2

Ω 21

2.5 2 mL = 1 1.5 β = 1/α, α = b2/b1, R2 = 0.05 1

β = 1/α2, α = (h2/h1)2, R2 = 0.045 β = 1/α3, α = h2/h1, R2 = 0.04

0.5 0

Euler 0

0.2

0.4

0.6

0.8

1

aL

Fig. 5.17 Lowest natural frequency of a stepped cantilever Timoshenko beam as a function of aL for α = 0.8, R1 = 0.001 (Euler-Bernoulli beam) and 0.05, γbs = 3.12, mL = 0, 0.2, and 1.0, and for the three combinations of α, β, and R2 indicated; R2 is determined from Eq. (5.163)

5.5

Beams Connected by a Continuous Elastic Spring

325

5.5 Beams Connected by a Continuous Elastic Spring As indicated in Section 3.9.2, elastically connected beams have been used as a relatively simple approximation to determine the natural frequencies of double-wall carbon nanotubes. In this section, we shall determine the natural frequencies for two Timoshenko beams with no in-span attachments, no externally applied forces, and no axial force; that is, with p = ki = ko = Mi = kti = Ji = Fa = Ma = 0. In addition, it is assumed that the beams are of the same length and they are each of constant cross section so that i (η) = a (η) = 1. The displacements of each beam are denoted as wj and the rotations of the cross sections due to bending are denoted as ψj , j = 1, 2. Then, for beam 1, Eqs. (5.31) and (5.32) simplify to  ∂ 2 w1 ∂w1 − ψ1 − kc (w1 − w2 ) − ρ1 A1 2 = 0 ∂x ∂t   2 ∂ ψ1 ∂ 2 ψ1 ∂w1 κ1 G1 A1 − ρ I =0 − ψ1 + E1 I1 2 2 ∂x ∂x2 ∂t2

κ1 G1 A1

∂ ∂x



(5.164)

and for beam 2 

 ∂w2 ∂ 2 w2 − ψ2 − kc (w2 − w1 ) − ρ2 A2 2 = 0 ∂x ∂t   ∂ 2 ψ2 ∂ 2 ψ2 ∂w2 κ2 G2 A2 − ρ2 I2 2 = 0 − ψ2 + E2 I2 2 ∂x ∂x ∂t

κ2 G2 A2

∂ ∂x

(5.165)

where we have replaced the term accounting for the elastic foundation –kf w with the term −kc (w1 − w2 ), where kc represents the elastic coupling of the continuous spring connecting the two beams. This sign convention assumes that w1 > w2 . Using the definitions presented in Eqs. (5.38), (5.39), and (5.134), Eqs. (5.164) and (5.165), respectively, can be rewritten as ∂ ∂η



 ∂y1 ∂ 2 y1 − ψ1 − Kc γ1bs R21 (y1 − y2 ) − γ1bs R21 2 = 0 ∂η ∂τ 2 ∂y1 ∂ 2 ψ1 4 ∂ ψ1 − γ R =0 − ψ1 + γbs R21 1bs 1 ∂η ∂η2 ∂τ 2

(5.166)

and ∂ ∂η



 ∂ 2 y2 ∂y2 − ψ2 − Kc γ2bs βR22 (y2 − y1 ) − αβγ2bs R22 2 = 0 ∂η ∂τ 2 ∂y2 ∂ 2 ψ2 4 ∂ ψ2 − ψ2 + γ2bs R22 − αβγ R =0 2bs 2 ∂η ∂η2 ∂τ 2

where

(5.167)

326

5 Timoshenko Beams

yl =

wl , L

γlbs =

ρ2 A2 , α= ρ1 A1

2 (1 + νl ) , κl

E1 I1 β= , E2 I2

R2l =

Il Al L2

kc L4 Kc = , E1 I1

to2

l = 1, 2 ρ1 A1 L4 = . E1 I 1

(5.168)

To determine the natural frequencies, we assume solutions of the form yl (η, τ ) = Yl (η) e j

2τ

ψl (η, τ ) = l (η) e j

2τ

l = 1, 2

(5.169)

where 2 = ωto . Substituting Eq. (5.169) into Eqs. (5.166) and (5.167), respectively, we obtain ∂ ∂η



 ∂Y1 − 1 − Kc γ1bs R21 (Y1 − Y2 ) + γ1bs R21 4 Y1 = 0 ∂η ∂Y1 ∂ 2 1 + γ1bs R41 4 1 = 0 − 1 + γ1bs R21 ∂η ∂η2

(5.170)

and ∂ ∂η



 ∂Y2 − 2 − Kc γ2bs βR22 (Y2 − Y1 ) + αβγ2bs R22 4 Y2 = 0 ∂η ∂Y2 ∂ 2 2 − 2 + γ2bs R22 + αβγ2bs R42 4 2 = 0. ∂η ∂η2

(5.171)

To obtain a solution to Eqs. (5.170) and (5.171), it is assumed that Yl (η) = Wln (η) l (η) = ln (η)

l = 1, 2

(5.172)

where W1n and 1n are the nth mode shapes corresponding to the solution of ∂ ∂η



 ∂W1n ˆ 4n W1n = 0 − 1n + γ1bs R21  ∂η

∂W1n ∂ 2 1n ˆ 4n 1n = 0 + γ1bs R41  − 1n + γ1bs R21 ∂η ∂η2

(5.173)

for a set of prescribed boundary conditions and W2n and 2n are the nth mode shapes corresponding to the solutions of

5.5

Beams Connected by a Continuous Elastic Spring

∂ ∂η



327

 ∂W2n ˜ 4n W2n = 0 − 2n + αβγ2bs R22  ∂η

∂W2n ∂ 2 2n ˜ 4n 2n = 0 + αβγ2bs R42  − 2n + γ2bs R22 ∂η ∂η2

(5.174)

and for a set of prescribed boundary conditions. Explicit forms of Wln and ln can be obtained from Table 5.3 for four sets of boundary conditions by setting Kf = So = 0; their form for other boundary conditions can be obtained from Eqs. (5.72) ˆ 4n and  ˜ 4n are obtained for each and (5.74) with H1 = H2 = 0. The values of  beam from the characteristic equations appearing in Table 5.2 for the corresponding boundary conditions appearing in Table 5.3; for other boundary conditions, one uses Eq. (5.78) with Kf = So = 0. Substituting Eq. (5.172) into Eqs. (5.170) and (5.171) and using Eqs. (5.173) and (5.174), the following systems of equations are obtained

 ˆ 4n W1n (η) + Kc W2n (η) = 0 4 − Kc − 

 ˆ 4n 1n (η) = 0 γ1bs R41 4 − 

(5.175)

and

 ˜ 4n W2n (η) + Kc W1n (η) = 0 α 4 − Kc α − 

 ˜ 4n 2n (η) = 0. αβγ1bs R42 4 − 

(5.176)

ˆ 4n , It is seen that the second equation of Eq. (5.175) is satisfied if either 4 =  which is the solution for the beam without the elastic coupling, or 1n = 0, which is a trivial solution. Neither solution is of interest; therefore, each is ignored. Similar reasoning is applied to the second equation of Eq. (5.176). It is noted from the ˆ 4n corresponds to the case where discussion in Section 3.9.2 that the solution 4 =  the displacements of the two beams are in phase; that is, there is no extension or compression of the elastic spring connecting them. The first equation of Eq. (5.175) and the first equation of Eq. (5.176) are rewritten in matrix form as 

ˆ 4n 4 − Kc −  Kc ˜ 4n Kc α4 − Kc − α 



W1n (η) W2n (η)

 =0

(5.177)

which has a non trivial solution when  = n is a solution to

ˆ 4n 4n − Kc − 



 ˜ 4n − Kc2 = 0. α4n − Kc − α 

(5.178)

328

5 Timoshenko Beams

At this point, it is assumed that both beams have the same material, the same ˜ 4n . Then, ˆ 4n =  geometry and the same boundary conditions; thus, α = β = 1 and  Eq. (5.178) simplifies to ˆ 4n + 2Kc 4n = 

(5.179)

which is the Timoshenko beam analog of the result obtained for the Euler-Bernoulli beam; that is, Eq. (3.339) when λ = β = 1. Equation (5.179) indicates that when the elastically connected beams are the same and the boundary conditions on each beam are the same, the natural frequencies can be determined from only Eq. (5.166) with y2 = 0 and with Kc replaced by 2Kc . If the beams are hinged at both ends, then the natural frequency coefficients are given by Eq. (5.100) with Kf replaced by 2Kc and So = 0. This result agrees with (Yoon et al. 2005) after their results are converted to the current notation and their determinant is evaluated algebraically.

5.6 Forced Excitation 5.6.1 Boundary Conditions and the Generation of Orthogonal Functions As was done in Section 3.10, the response of a Timoshenko beam to forced excitation will be determined by using separation of variables and the application of orthogonal functions. From the discussion in Section B.2.2 of Appendix B, the boundary conditions given by Eqs. (5.43) and (5.44) allows one to generate orthogonal functions provided that G given by Eq. (5.27) and G(C1 ) given by Eq. (5.22) are symmetric quadratics. The expression for G(C1 ) is a symmetric quadratic. The expression for G becomes a symmetric quadratic when Fa = Ma = 0. In addition, as discussed at the end of Section 3.2.3, orthogonal functions cannot be created when a single degree-of-freedom system is attached; therefore, we also set ko = 0. The general orthogonality condition for a system described by two dependent variables is given by Eq. (B.117) of Appendix B with N = 2. To apply these equations to the Timoshenko beam, several sets of equations have to be compared. First, (n) (n) it is noted that U1 = Yn (η) and U2 = n (η), where Yn (η) and n (η) are given by either Eq. (5.86), Eq. (5.88), Eq. (5.91), Eq. (5.94), or Eq. (5.97) and their corresponding characteristic equations as the case may be. Upon comparing Eq. (5.22) with the second equation of Eq. (B.95) for N = 2, it is found that the constants appearing in this equation are those given in Eq. (5.35). Comparing Eq. (5.27) to the first equation of Eq. (B.95), it is found that p11 = ρA (x) + Mi δ (x − Lm ) , p22 = ρI (x) + Ji δ (x − Lm ), and p12 = p21 = 0. Then, in the current notation, the orthogonality condition given by Eq. (B.117) can be written as

5.6

Forced Excitation

329

1 Bnm =

(a (η) Yn (η) Ym (η) + mi δ (η − ηm ) Yn (η) Ym (η)) dη 0

1  + R2o i (η) n (η) m (η) + ji δ (η − ηm ) n (η) m (η) dη

(5.180)

0

+ ML Yn (0) Ym (0) + JL n (0) m (0) + MR Yn (1) Ym (1) + JR n (1) m (1) = δnm Nn where δ nm is the kronecker delta and Nn =

1

   [a (η) + mi δ (η − ηm )] Yn2 (η) + i (η) R2o + ji δ (η − ηm ) n2 (η) dη

0

+ ML Yn2 (0) + JL n2 (0) + MR Yn2 (1) + JR n2 (1) . (5.181)

5.6.2 General Solution We shall obtain the response of a Timoshenko beam to an externally applied excitation for the general boundary conditions given by Eqs. (5.43) and (5.44). It is assumed that the axial tensile force is constant so that S (η, τ ) = So and that no single degree-of-freedom system is attached to the beam. We are interested in obtaining the solution to the governing equations given by Eqs. (5.40) and (5.41), respectively, which now become    ∂y ∂ 2y ∂ −ψ + γbs R2o So 2 − Kf γbs R2o y a (η) ∂η ∂η ∂η −Ki γbs R2o yδ (η − ηs ) − γbs R2o (a (η) + mi δ (η − ηm ))

∂ 2y = −γbs R2o fˆa ∂τ 2 (5.182)

and 

   ∂ ∂y ∂ψ − ψ + γbs R2o i (η) − Kti γbs R2o ψδ (η − ηt ) ∂η ∂η ∂η

 ∂ 2ψ − γbs R4o i (η) + ji γbs R2o δ (η − ηm ) = −γbs R2o m ˆ a. ∂τ 2

a (η)

(5.183)

330

5 Timoshenko Beams

To solve Eqs. (5.182) and (5.183), we assume a solution of the form y (η, τ ) =

∞ 

Yn (η) ϕn (τ )

n=1

ψ (η, τ ) =

∞ 

(5.184) n (η) ϕn (τ )

n=1

where ϕ n are to be determined and Yn and  n , respectively, are solutions to    ∂ ∂ 2 Yn ∂Yn + γbs R2o So 2 − Kf γbs R2o Yn − n a (η) ∂η ∂η ∂η −Ki γbs R2o δ (η

− ηs ) Yn + γbs R2o 4n (a (η) + mi δ (η

(5.185)

− ηm )) Yn = 0

and    ∂n ∂ ∂Yn − n + γbs R2o i (η) − Kti γbs R2o δ (η − ηt ) n ∂η ∂η ∂η 

+ 4n γbs R4o i (η) + ji γbs R2o δ (η − ηm ) n = 0. 

a (η)

(5.186)

The functions Yn and  n satisfy the following boundary conditions at η = 0     γbs R2o KL − mL 4n Yn (0) − a (0) Yn (0) − n (0) − γbs R2o So Yn (0) = 0   KtL − jL 4n n (0) − i (0) n (0) = 0 (5.187) and the following boundary conditions at η = 1     γbs R2o KR − mR 4n Yn (1) + a (1) Yn (1) − n (1) + γbs R2o So Yn (1) = 0   KtR − jR 4n n (1) + i (1) n (1) = 0. (5.188) In Eqs. (5.187) and (5.188), the prime denotes the derivative with respect to η. The following operations are now performed. Equation (5.184) is substituted into Eq. (5.182), and then Eq. (5.185) is used to arrive at ∞  n=1



 ∂ 2 ϕn (τ ) 4 + n ϕn (τ ) Yn (η) = fˆa (η, τ ). (a (η) + mi δ (η − ηm )) ∂τ 2

Substituting Eq. (5.184) into Eq. (5.183) and using Eq. (5.186), we arrive at

(5.189)

5.6

Forced Excitation ∞ 

331

R2o i (η) + ji δ (η − ηm )

n=1

   ∂ 2 ϕ (τ ) n 4 +  ϕ ˆ a (η, τ ) . n (η) = m (τ ) n n ∂τ 2 (5.190)

Equation (5.184) is substituted into the boundary conditions given by Eq. (5.43), and then Eq. (5.187) is used to obtain at η = 0 ∞ 

 mL

n=1 ∞ 

 jL

n=1

 ∂ 2 ϕn 4 + n ϕn Yn (0) = 0 ∂τ 2

 ∂ 2 ϕn 4 + n ϕn n (0) = 0. ∂τ 2

(5.191)

Equation (5.184) is substituted into the boundary conditions given by Eq. (5.44) and then Eq. (5.188) is used to obtain at η = 1 ∞ 

 mR

n=1 ∞ 

 jR

n=1

 ∂ 2 ϕn 4 + n ϕn Yn (1) = 0 ∂τ 2

 ∂ 2 ϕn 4 + n ϕn n (1) = 0. ∂τ 2

(5.192)

Next, Eq. (5.189) is multiplied by Yl (η) and integrated with respect to η from 0 to 1 to obtain ∞  2  ∂ ϕn (τ )

∂τ 2

n=1

1 =

 1 + 4n ϕn (τ )

(a (η) + mi δ (η − ηm )) Yn (η) Yl (η) dη 0

(5.193)

fˆa (η, τ ) Yl (η) dη

0

and Eq. (5.190) is multiplied by l (η) and integrated with respect to η from 0 to 1 to obtain ∞  2  ∂ ϕn (τ )

∂τ 2

n=1

+ 4n ϕn (τ )

 1  R2o i (η) + ji δ (η − ηm ) n (η) l (η) dη 0

1 =

m ˆ a (η, τ ) l (η) dη. 0

(5.194)

332

5 Timoshenko Beams

For the boundary conditions, Eqs. (5.191) and (5.192), we perform the following four sets of operations: (1) the first equation of Eq. (5.191) is multiplied by Yl (0); (2) the second equation of Eq. (5.191) is multiplied by l (0); (3) the first equation of Eq. (5.192) is multiplied by Yl (1); and (4) the second equation of Eq. (5.192) is multiplied by l (1). These operations result in Eq. (5.191) becoming ∞ 

 mL

n=1 ∞ 

 jL

n=1

 ∂ 2 ϕn 4 +  ϕ n n Yn (0) Yl (0) = 0 ∂τ 2

 ∂ 2 ϕn 4 + n ϕn n (0) l (0) = 0 ∂τ 2

(5.195)

and Eq. (5.192) becoming ∞ 

 mR

n=1 ∞  n=1

 jR

 ∂ 2 ϕn 4 + n ϕn Yn (1) Yl (1) = 0 ∂τ 2

 ∂ 2 ϕn 4 + n ϕn n (1) l (1) = 0. ∂τ 2

(5.196)

Upon adding Eqs. (5.193) to (5.196) and collecting terms, we arrive at ∞ 

 Bnl

n=1

 1

1 ∂ 2 ϕn (τ ) 4 ˆ + n ϕn (τ ) = fa (η, τ ) Yl (η) dη + m ˆ a (η, τ ) l (η) dη ∂τ 2 0

0

(5.197) where Bnl is given by Eq. (5.180); that is, Bnl = δnl Nn , where Nn is given by Eq. (5.181). Therefore, Eq. (5.197) reduces to ∂ 2 ϕn (τ ) + 4n ϕn (τ ) = gn (τ ) ∂τ 2

(5.198)

where 1 gn (τ ) = Nn

1

1 fˆa (η, τ ) Yn (η) dη + Nn

0

1 m ˆ a (η, τ ) n (η) dη.

(5.199)

0

The solution to Eq. (5.198) is given by Eq. (C.6) of Appendix C. Assuming that the initial conditions are zero and ζ = 0 in Eq. (C.6), we obtain 1 ϕn (τ ) = 2 n

τ 0

 sin 2n ξ gn (τ − ξ ) dξ .

(5.200)

5.6

Forced Excitation

333

Substituting Eqs. (5.199) and (5.200) into Eq. (5.184), we arrive at ⎛ 1

τ

∞

  Yn (η) 2 ⎝ fˆa (η, τ − ξ ) Yn (η) dη sin  ξ y (η, τ ) = n Nn 2n n=1

0

1 +

0

m ˆ a (η, τ − ξ ) n (η) dη⎠ dξ

0

∞  n (η)

τ

ψ (η, τ ) =

n=1

Nn 2n

1 +

⎞

⎛  1 sin 2n ξ ⎝ fˆa (η, τ − ξ ) Yn (η) dη

0

0

(5.201)

⎞

m ˆ a (η, τ − ξ ) n (η) dη⎠ dξ .

0

5.6.3 Impulse Response The impulse response shall be determined for a Timoshenko beam with a constant cross section; that is, for a (η) = i (η) = 1. It is assumed that m ˆ a = 0 and that an impulse of magnitude Fo is applied at η = η1 ; thus, fˆa (η, τ ) = Fˆ o δ (η − η1 ) δ (τ )

(5.202)

where, from Eq. (5.39) Fˆ o =

Fo L3 . EIo

Substituting Eq. (5.202) into Eq. (5.201) yields y (η, τ ) = Fˆ o

∞  Yn (η) Yn (η1 ) n=1

ψ (η, τ ) = Fˆ o

Nn 2n

 sin 2n τ

∞  n (η) n (η1 ) n=1

Nn 2n

 sin 2n τ .

(5.203)

A cantilever beam with a mass attached to its free end is chosen as the system to illustrate Eq. (5.203). The natural frequency coefficients are determined from the characteristic equation given by Case 4 of Table 5.2 and the corresponding modes shapes Yn and  n are given by Case 4 of Table 5.3. In addition, Eq. (5.181) simplifies to

1  Nn = Yn2 (η) + R2o n2 (η) dη + MR Yn2 (1) . 0

(5.204)

334

5 Timoshenko Beams η1 = 0.6

1

y(1, τ)

0.5 0 −0.5 −1

0

1

2

3

4

5

3

4

5

η1 = 1 2

y(1, τ)

1 0 −1 −2

0

1

2 τ

Fig. 5.18 Displacement response of the free end of a cantilever Timoshenko beam when an impulse is applied at η1 = 0.6 and 1.0 for mR = 0 and Ro = 0.03. The Euler-Bernoulli theory response is shown with the dotted line and was obtained by setting Ro = 0.001 η1 = 0.6

y(1, τ)

0.5

0

−0.5

0

1

2

3

4

5

3

4

5

η1 = 1 1

y(1, τ)

0.5 0 −0.5 −1

0

1

2 τ

Fig. 5.19 Displacement response of the free end of a cantilever Timoshenko beam when an impulse is applied at η1 = 0.6 and 1.0 for mR = 0.2 and Ro = 0.03. The Euler-Bernoulli theory response is shown with the dotted line and was obtained by setting Ro = 0.001

Appendix 5.1 Definitions of the Solution Functions fl and gl and Their Derivatives

335

The displacements of the free end of the beam are shown in Figs. 5.18 and 5.19 for Fˆ o = 1, γbs = 3.12, Ro = 0.03, mR = 0 and 0.2, and η1 = 0.5 and 1.0. In addition, the displacements for an Euler-Bernoulli beam are also shown for this set of parameters. For both beam theories, 13 mode shapes have been used. The results show that the differences in the response for the two beam theories is in the detailed structure of the waveform and is due to the differences in the higher frequency content of the response. The maximum and minimum amplitudes are very close, although in certain cases they occur at slightly different times.

Appendix 5.1 Definitions of the Solution Functions fl and gl and Their Derivatives The quantities fj (η) and flk (η, ξ ) are given by   1  1 + γbs R2o So Qαβ (η, α, β) f1 (η) =  1 + γbs R2o So + γbs R2o KL Rαβ (η, α, β)

 

+ R2o 4 1 + γbs R2o So − So Sαβ (η, α, β) + γbs R4o KL b Tαβ (η, α, β)



  1  Rαβ (η, α, β) + KtL Sαβ (η, α, β) f2 (η) =  2 1 + γbs Ro So        1  Rαβ η − ηp , α, β u η − ηp f3p η, ηp =  2 1 + γbs Ro So     + R2o b Tαβ η − ηp , α, β u η − ηp

p = 1, 2, 3

      1  Sαβ η − ηˆ q , α, β u η − ηˆ q f4q η, ηˆ q =  2 1 + γbs Ro So and the quantities gj (η) and glk (η, ξ ) are given by

q = 1, 2

(5.205)

336

5 Timoshenko Beams

  1  −KL Sαβ (η, α, β) + k Tαβ (η, α, β) 1 + γbs R2o So   1  1 + γbs R2o So Qαβ (η, α, β) g2 (η) =  2 1 + γbs Ro So 

+ 1 + γbs R2o So KtL Rαβ (η, α, β)  + γbs R2o k Sαβ (η, α, β) + γbs R2o KtL k Tαβ (η, α, β) g1 (η) = 

    1  Sαβ η − ηp , α, β u η − ηp p = 1, 2, 3 2 1 + γbs Ro So         1  1 + γbs R2o So Rαβ η − ηˆ q , α, β u η − ηˆ q g4q η, ηˆ q =  2 1 + γbs Ro So     q = 1, 2 + γbs R2o k Tαβ η − ηˆ q , α, β u η − ηˆ q (5.206)

  g3p η, ηp = 

where Qαβ (η, α, β) , . . . , are given by Eq. in Appendix  (C.22)    C with  α and  β , η , η , η ˆ η = g η = f η ˆ = given by Eq. (5.68). It is noted that f 3p p p 3p p p 4p q q   g4p ηˆ q , ηˆ q = 0. The derivatives of the quantities given in Eqs. (5.205) and (5.206), respectively, are

f1 (η) = 

  1  1 + γbs R2o So Vαβ (η, α, β) 1 + γbs R2o So

+ γbs R2o KL Qαβ (η, α, β)

 

+ R2o 4 1 + γbs R2o So − So Rαβ (η, α, β)  + γbs R4o KL b Sαβ (η, α, β) f2 (η) = 

  1  Qαβ (η, α, β) + KtL Rαβ (η, α, β) 2 1 + γbs Ro So

     1  Qαβ η − ηp , α, β u η − ηp 2 1 + γbs Ro So     p = 1, 2, 3 + R2o b Sαβ η − ηp , α, β u η − ηp

  η, η =  f3p p

  η, ηˆ f4q q = 

and

    1  Rαβ η − ηˆ q , α, β u η − ηˆ q 1 + γbs R2o So

q = 1, 2

(5.207)

Appendix 5.2 Definitions of Solution Functions fli and gli and Their Derivatives

337

  1  −KL Rαβ (η, α, β) + k Sαβ (η, α, β) g 1 (η) =  1 + γbs R2o So   1  1 + γbs R2o So Vαβ (η, α, β) + γbs R2o k Rαβ (η, α, β) g 2 (η) =  2 1 + γbs Ro So

  + KtL 1 + γbs R2o So Qαβ (η, α, β) + γbs R2o KtL k Sαβ (η, α, β)       1  Rαβ η − ηp , α, β u η − ηp p = 1, 2, 3 g 3p η, ηp =  2 1 + γbs Ro So         1  1 + γbs R2o So Qαβ η − ηˆ q , α, β u η − ηˆ q g 4q η, ηˆ q =  2 1 + γbs Ro So     q = 1, 2 + γbs R2o k Sαβ η − ηˆ q , α, β u η − ηˆ q (5.208) where we have used Eq. (C.23) of Appendix C. There should also be terms containing the derivatives of u (η); however, since f j and g j will be evaluated only at η = 1 these terms will equal zero. Therefore, they have been omitted.

Appendix 5.2 Definitions of Solution Functions fli and gli and Their Derivatives The quantities fli are given by f11 (η) = Sδε (η, δ1 , ε1 )

  f12 (η) = Rδε (η, δ1 , ε1 ) + R21 4 − 1 γbs R21 Tδε (η, δ1 , ε1 )

f21 (ξ ) = Qδε (ξ , δ2 , ε2 ) − mL co βγ2bs R22 4 Rδε (ξ , δ2 , ε2 ) + αβR22 4 Sδε (ξ , δ2 , ε2 )

 − mL co β4 αβγ2bs R42 4 − 1 Tδε (ξ , δ2 , ε2 )

(5.209)

f22 (ξ ) = Rδε (ξ , δ2 , ε2 ) and the quantities gli are given by g11 (η) = Rδε (η, δ1 , ε1 ) + γ1bs R21 4 Tδε (η, δ1 , ε1 )   g12 (η) = −Sδε (η, δ1 , ε1 ) γ1bs R21 g21 (ξ ) = mL co β4 Sδε (ξ , δ2 , ε2 ) + αβ4 Tδε (ξ , δ2 , ε2 ) g22 (ξ ) = Qδε (ξ , δ2 , ε2 ) + αβγ2bs R22 4 Sδε (ξ , δ2 , ε2 )

(5.210)

338

5 Timoshenko Beams

where Qδε (η, δ, ε) , . . ., are given by Eq. (C.22) with δ = δl and ε = εl , l = 1, 2, as the case may be. The derivatives of the quantities given in Eqs. (5.209) and (5.210), respectively, are (η) = R (η, δ , ε ) f11 δε 1 1

  (η) = Q (η, δ , ε ) + R2 4 − 1 f12 γbs R21 Sδε (η, δ1 , ε1 ) δε 1 1 1 (ξ ) = V (ξ , δ , ε ) − m c βγ R2 4 Q (ξ , δ , ε ) f21 δε 2 2 L o 2bs 2 δε 2 2

+ αβR22 4 Rδε (ξ , δ2 , ε2 ) 

− mL co β4 αβγ2bs R42 4 − 1 Sδε (ξ , δ2 , ε2 )

(5.211)

(ξ ) = Q (ξ , δ , ε ) f22 δε 2 2

and g 11 (η) = Qδε (η, δ1 , ε1 ) + γ1bs R21 4 Sδε (η, δ1 , ε1 )   g 12 (η) = −Rδε (η, δ1 , ε1 ) γ1bs R21 g 21 (ξ ) = mL co β4 Rδε (ξ , δ2 , ε2 ) + αβ4 Sδε (ξ , δ2 , ε2 )

(5.212)

g 22 (ξ ) = Vδε (ξ , δ2 , ε2 ) + αβγ2bs R22 4 Rδε (ξ , δ2 , ε2 ) where the prime denotes the derivative of the function with respect to its argument, either η or ξ as the case may be. In arriving at Eqs. (5.211) and (5.212), we have used Eq. (C.23).

References Cowper GR (1966) The shear coefficients in Timoshenko’s beam theory. Trans ASME J Appl Mech 33:335–340 Ginsberg JH, Pham H (1995) Forced harmonic response of a continuous system displaying eigenvalue steering phenomena. ASME J Vib Acoust 117:439–444 Han SM, Benaroya H, Wei T (1999) Dynamics of transversely vibrating beams using four engineering theories. J Sound Vib 225(5):935–988 Huang TC (1961) The effect of rotary inertia and of shear deformation on the frequency and normal modes equations of uniform beams with simple end conditions. AMSE J Appl Mech 28(4): 579–584 Lee SY, Lin SM (1992) Exact solutions for non uniform Timoshenko beams with attachments. AIAA J 30(12):2930–2934 Magrab EB (2007) Natural frequencies and mode shapes of Timoshenko beams with attachments. J Vib Control 13(7):905–934 Matsuda H, Morita C, Sakiyama C (1992) A method for vibration analysis of a tapered Timoshenko beam with constraint at any points and carrying a heavy tip mass. J Sound Vib 158(20):331–339

References

339

Rossi RE, Laura PAA, Gutierrez RH (1990) A note on transverse vibrations of a Timoshenko beam of non-uniform thickness clamped at one end and carrying a concentrated mass at the other. J Sound Vib 143(3):491–502 Stephen NG (1997A) Mindlin plate theory: best shear coefficient and higher spectra validity. J Sound Vib 202:539–553 Stephen NG (1997B) On ‘A check on the accuracy of Timoshenko’s beam theory’. J Sound Vib 257:809–812 Tong X, Tabarrok B, Yeh KY (1995) Vibration analysis of Timoshenko beams with nonhomogeneity and varying cross-section. J Sound Vib 186(5):821–835 Yoon J, Ru CQ, Mioduchowski A (2005) Terahertz vibration of short carbon nanotubes modeled as Timoshenko beams. ASME J Appl Mech 72:10–17 Zhou D, Cheung YK (2001) Vibrations of tapered Timoshenko beams in terms of static Timoshenko beam functions. ASME J Appl Mech 68:596–602

Chapter 6

Thin Plates

The natural frequencies and mode shapes of thin rectangular and circular plates are obtained for a wide variety of boundary conditions. Plates that have constant inplane forces applied and that are resting on elastic foundations are also considered. The responses of circular plates to externally applied forces are determined and the extensional vibrations of circular plates, which have been used in MEMS RF devices, are examined.

6.1 Introduction Plates are ubiquitous and important structural members. Their excitation can occur in any number of ways: from wind loads on windows in high-rise buildings; from turbulent flow over a ship’s hull; from the landing and takeoff of jet planes on aircraft carriers; from vehicular loads on bridge surfaces; and when supporting rotating machinery. They can become acoustic radiators and amplifiers of noise in such systems as car and truck panels and in submarine hulls. In recent years, the extensional vibrations of piezoelectric plates have been used in MEMS RF devices. In this chapter, we shall develop the equations of motion for rectangular plates and then convert these results to polar coordinates when circular plates are considered. It will be seen that ‘closed form’ analytical solutions can be obtained for rectangular plates only under a very limited set of boundary conditions. Consequently, we shall only use the Rayleigh-Ritz procedure when examining rectangular plates. For circular plates, we shall be able to obtain analytical solutions for a wide range of boundary conditions. An excellent summary of the results of published articles concerning the natural frequencies and mode shapes of all types of plates can be found in (Leissa 1969).

E.B. Magrab, Vibrations of Elastic Systems, Solid Mechanics and Its Applications 184, C Springer Science+Business Media B.V. 2012 DOI 10.1007/978-94-007-2672-7_6, 

341

342

6 Thin Plates

6.2 Derivation of Governing Equation and Boundary Conditions: Rectangular Plates 6.2.1 Introduction Consider the rectangular plate of constant thickness shown in Fig. 6.1, which has a length a in the x-direction, a length b in the y-direction, and a thickness h in the z-direction, where h 0  Kb wˆ r +

  ∂w mb α ∂ 2 wˆ r ˆ r r + Vˆ rˆ rˆ , θ , τ − N 2 ∂τ 2 ∂r    ∂w ˆr ˆ rˆ rˆ , θ , τ +M Ktb ∂ rˆ

rˆ =α rˆ =α

=0 (6.57) =0

When the plate is a solid plate α = 0 and Eq. (6.57) is not applicable; it is replaced by the requirement that the displacement at the center of the plate remains finite. At rˆ = 1    ∂ wˆ r ma ∂ 2 wˆ r + Nˆ r Ka wˆ r + − Vˆ rˆ rˆ , θ , τ 2 2 ∂τ ∂ rˆ    ∂w ˆr ˆ rˆ rˆ , θ , τ −M Kta ∂ rˆ

rˆ =1 rˆ =1

=0 (6.58) =0

6.4 Natural Frequencies and Mode Shapes of Circular Plates for Very General Boundary Conditions 6.4.1 Introduction We shall determine the natural frequencies and mode shapes for annular and solid circular plates on an elastic foundation and subjected to a constant tensile in-plane force. It is assumed that the boundary conditions are those given by Eqs. (6.57) and (6.58). After the solution for these boundary conditions has been obtained, several of its special cases will be examined. The governing equation is given by Eq. (6.53), which for convenience is repeated below ˆ r − Nˆ r ∇rˆ2θ wˆ r + Kf wˆ r + ∇rˆ4θ w

  ∂ 2 wˆ r = fa rˆ , θ , τ . 2 ∂τ

(6.59)

To determine the natural frequencies, we set fa = 0 and assume a solution of the form   2 wˆ r = W rˆ , θ ej τ

(6.60)

where 2 = ωtp . Substituting Eq. (6.60) into Eq. (6.59), we arrive at ∇rˆ4θ W − Nˆ r ∇rˆ2θ W − λ4 W = 0

(6.61)

λ4 = 4 − Kf .

(6.62)

where

6.4

Natural Frequencies and Mode Shapes of Circular Plates for Very General. . .

357

Substituting Eq. (6.60) into Eqs. (6.57) and (6.58), respectively, gives the following boundary conditions. At rˆ = α > 0

mb α 4  Kb −  W (α, θ ) − Nˆ r W (α, θ ) + Vˆ rˆ (α, θ ) = 0 2 ˆ rˆ (α, θ ) = 0 Ktb W (α, θ ) + M

(6.63)

When the plate is a solid plate α = 0 and Eq. (6.63) is not applicable; is replaced by the requirement that the displacement at the center of the plate remains finite. At rˆ = 1

ma 4   W (1, θ ) + Nˆ r W (1, θ ) − Vˆ rˆ (1, θ ) = 0 Ka − 2 ˆ rˆ (1, θ ) = 0 Kta W (1, θ ) − M

(6.64)

In Eqs. (6.63) and (6.64), the prime denotes the derivative with respect to rˆ and   ∂W ∂2 W ∂2 W +ν + 2 2 rˆ ∂ rˆ ∂ rˆ 2 rˆ ∂θ   3   ∂ W ∂2 W ∂ 2  . − ∇rˆ θ W + (1 − ν) 2 Vˆ rˆ rˆ , θ = ∂ rˆ rˆ ∂ rˆ ∂θ 2 rˆ 3 ∂θ 2

  ˆ rˆ rˆ , θ = − M



(6.65)

A solution to Eq. (6.61) can be obtained by assuming that W(ˆr, θ ) =

∞ 

  ˆ n rˆ cos nθ W

(6.66)

n=0

and by assuming a similar solution where sin (nθ ) replaces cos (nθ ), n = 1, 2 . . . . However, since the plate is continuous from 0 ≤ θ ≤ 2π , Eq. (6.66) is sufficient for the determination of the natural frequencies and mode shapes. Substituting Eq. (6.66) into Eq. (6.61) gives 4 ˆ ˆ n − Nˆ r ∇ 2 W ˆ ∇rˆ4 W rˆ n − λ Wn = 0

(6.67)

where ∇rˆ2 =

d2 d n2 + − 2. 2 dˆr rˆ dˆr rˆ

(6.68)

When the solution is independent of θ , n = 0 and the solution is referred to as the axisymmetric solution. The boundary conditions given by Eqs. (6.63) and (6.64), respectively, become as follows.

358

6 Thin Plates

At rˆ = α > 0

mb α 4  ˆ n (α) − Nˆ r W ˆ n (α) + Vˆ rˆ (α) = 0 Kb −  W 2 ˆ n (α) + M ˆ rˆ (α) = 0 Ktb W

(6.69)

When the plate is a solid plate α = 0 and Eq. (6.69) is not applicable; it is replaced by the requirement that the displacement at the center of the plate remains finite. At rˆ = 1

Ka −

ma 4  ˆ n (1) + Nˆ r W ˆ n (1) − Vˆ rˆ (1) = 0  W 2 ˆ n (1) − M ˆ rˆ (1) = 0 Kta W

(6.70)

In Eqs. (6.69) and (6.70), the prime denotes the derivative with respect to rˆ and 



 ˆn dW n2 ˆn − 2W rˆ dˆr rˆ   ˆn   n2 dW d 2  1 ˆn . ˆ n − (1 − ν) − W ∇rˆ W Vˆ rˆ rˆ = dˆr rˆ 2 dˆr rˆ

  ˆ rˆ rˆ = − M

ˆn d2 W +ν dˆr2

(6.71)

Equation (6.67) can be factored as follows

  ˆn =0 ∇rˆ2 + ε 2 ∇rˆ2 − δ 2 W

(6.72)

where       1 1 −Nˆ r + Nˆ r2 + 4λ4 = −Nˆ r + Nˆ r2 + 4 4 − Kf 2 2       1 1 δ2 = Nˆ r + Nˆ r2 + 4λ4 = Nˆ r + Nˆ r2 + 4 4 − Kf . 2 2

ε2 =

(6.73)

      ˆ δ rˆ + W ˆ ε rˆ , then, from Eq. (6.72), ˆ n rˆ = W If it is assumed that W

    ˆ δ − δ2W ˆ δ + ∇ 2 − δ2 ∇ 2 W ˆ ε + ε2 W ˆ ε = 0. ∇rˆ2 + ε 2 ∇rˆ2 W rˆ rˆ Thus, we seek the solution to the following two equations   2 ˆε ˆε d2 W dW n 2 ˆε =0 W + − ε − dˆr2 rˆ dˆr rˆ 2  2  ˆδ ˆδ n d2 W dW 2 ˆδ =0 W − + + δ rˆ dˆr dˆr2 rˆ 2

(6.74)

6.4

Natural Frequencies and Mode Shapes of Circular Plates for Very General. . .

359

which are Bessel’s equations. Adding the well-known solutions to Eq. (6.74), we arrive at           ˆ n rˆ = An Jn εˆr + Bn Yn εˆr + Cn In δˆr + Dn Kn δˆr W

0 < rˆ ≤ 1

(6.75)

where Jn and Yn , respectively, are the Bessel functions of the first and second kind of order n and In and Kn , respectively, are the modified Bessel functions of the first and second kind of order n. Equation (6.75) is used for an annular plate. Since Yn and Kn approach infinity as rˆ → 0, in order for the solution to remain finite at rˆ = 0, we set Bn = Dn = 0 and Eq. (6.75) reduces to       ˆ n rˆ = An Jn εˆr + Cn In δˆr W

0 ≤ rˆ ≤ 1.

(6.76)

Equation (6.76) is used for a solid plate and only the boundary conditions given by Eq. (6.70) are applicable. It is noted that when Nˆ r = 0, ε = δ = λ and when Nˆ r = Kf = 0, ε = δ = . In view of Eq. (6.74), Eq. (6.71) can be simplified as follows. For the portion of the solution involving Jn and Yn , it is found that   2 ˆ   d W n n ˆ n + (1 − ν) ˆn ˆ rˆ rˆ = ε2 W − 2W M rˆ dˆr rˆ   ˆn ˆn   dW n2 dW 1 2 ˆn − (1 − ν) 2 − W Vˆ rˆ rˆ = − ε dˆr rˆ dˆr rˆ

(6.77)

and for the portion of the solution involving In and Kn , it is found that   ˆn   dW n2 2 ˆ n + (1 − ν) ˆn ˆ rˆ rˆ = − δ W − 2W M rˆ dˆr rˆ   ˆn ˆn   dW n2 dW 1 2 ˆ − (1 − ν) 2 − Wn . Vˆ rˆ rˆ = δ dˆr rˆ dˆr rˆ

(6.78)

It is noted that Eqs. (6.77) and (6.78) are also used when Nˆ r = 0, Kf = 0, or Nˆ r = Kf = 0.

6.4.2 Natural Frequencies and Mode Shapes of Annular and Solid Circular Plates The natural frequencies and mode shape are determined for an annular plate on and elastic foundation and subject to an in-plane radial tensile force by substituting Eq. (6.75) into the boundary conditions given by Eqs. (6.69) and (6.70). Performing these substitutions, results in the following system of equations

360

6 Thin Plates

⎤⎧ ⎫ h11 (εα) h12 (εα) h13 (δα) h14 (δα) ⎪ ⎪ ⎪ An ⎪ ⎢ h21 (εα) h22 (εα) h23 (δα) h24 (δα) ⎥ ⎨ Bn ⎬ ⎥ ⎢ =0 [H ()] {A} = ⎣ h31 (ε) h32 (ε) h33 (δ) h34 (δ) ⎦ ⎪ Cn ⎪ ⎪ ⎭ ⎩ ⎪ h41 (ε) h42 (ε) h43 (δ) h44 (δ) Dn ⎡

(6.79)

where   h11 (εα) = a1n Jn (εα) − b1n ε 2 (Jn−1 (εα) − Jn+1 (εα))   h12 (εα) = a1n Yn (εα) − b1n ε 2 (Yn−1 (εα) − Yn+1 (εα))   h13 (δα) = a1n In (δα) − c1n δ 2 (In−1 (δα) + In+1 (δα))   h14 (δα) = a1n Kn (δα) + c1n δ 2 (Kn−1 (δα) + Kn+1 (δα))   h21 (εα) = a2n ε 2 (Jn−1 (εα) − Jn+1 (εα)) + b2n Jn (εα)   h22 (εα) = a2n ε 2 (Yn−1 (εα) − Yn+1 (εα)) + b2n Yn (εα)   h23 (δα) = a2n δ 2 (In−1 (δα) + In+1 (δα)) + c2n In (δα)   h24 (δα) = − a2n δ 2 (Kn−1 (δα) + Kn+1 (δα)) + c2n Kn (δα)   h31 (ε) = a3n Jn (ε) + b3n ε 2 (Jn−1 (ε) − Jn+1 (ε))   h32 (ε) = a3n Yn (ε) + b3n ε 2 (Yn−1 (ε) − Yn+1 (ε))   h33 (δ) = a3n In (δ) + c3n δ 2 (In−1 (δ) + In+1 (δ))   h34 (δ) = a3n Kn (δ) − c3n δ 2 (Kn−1 (δ) + Kn+1 (δ))   h41 (ε) = a4n ε 2 (Jn−1 (ε) − Jn+1 (ε)) + b4n Jn (ε)   h42 (ε) = a4n ε 2 (Yn−1 (ε) − Yn+1 (ε)) + b4n Yn (ε)   h43 (δ) = a4n δ 2 (In−1 (δ) + In+1 (δ)) + c4n In (δ)   h44 (δ) = − a4n δ 2 (Kn−1 (δ) + Kn+1 (δ)) + c4n Kn (δ)

(6.80)

and ajn , bjn , and cjn , j = 1, 2, 3, 4, are given in Table 6.1. The natural frequencies are those values of  = nm for which det [H (nm )] = 0; that is, for a given value of n, n = 0, 1, 2, . . ., there are nm natural frequency

Table 6.1 Constants appearing in Eq. (6.80) j

ajn

1

Kb −

2

Ktb +

3 4

bjn mb α 4 (1 − ν) 2  + n 2 α3

1−ν α ma 4  − (1 − ν) n2 Ka − 2 Kta − (1 − ν)

cjn

Nˆ r + ε 2 + ε2 −

(1 − ν) 2 n α2

(1 − ν) n2 α2

Nˆ r − δ 2 + −δ2 −

(1 − ν) 2 n α2

(1 − ν) n2 α2

Nˆ r + ε 2 + (1 − ν) n2

Nˆ r − δ 2 + (1 − ν) n2

−ε2 + (1 − ν) n2

δ 2 + (1 − ν) n2

6.4

Natural Frequencies and Mode Shapes of Circular Plates for Very General. . .

361

coefficients, m = 0, 1, 2, . . . . The value of n indicates the number of nodal diameters and the value of m indicates the number of nodal circles including the boundary. When the plate is a solid plate, Eq. (6.79) reduces to 

h (ε) [Hs ()] {A} = 31 h41 (ε)

h33 (δ) h43 (δ)



An Cn

 = 0.

(6.81)

The natural frequencies are those values of  for which det [Hs (nm )] = 0. The mode shapes for an annular plate are obtained from Eqs. (6.66), (6.75), and (6.79) as     Wnm rˆ , θ = Rˆ nm rˆ cos nθ

0 < rˆ ≤ 1

n = 0, 1, 2, . . .

m = 1, 2, . . . (6.82)

where           Rˆ nm rˆ = Jn εnm rˆ + E1nm Yn εnm rˆ + E2nm In δnm rˆ + E3 nm Kn δnm rˆ

(6.83)

and ε nm and δ nm are given Eq. (6.73) with  replaced by nm . In Eq. (6.83), Ejnm , j = 1, 2, 3, are determined from ⎡

h22 (εnm α) ⎣ h32 (εnm ) h42 (εnm )

h23 (δnm α) h33 (δnm ) h43 (δnm )

⎫ ⎫ ⎧ ⎤⎧ h24 (δnm α) ⎨ E1nm ⎬ ⎨ −h21 (εnm α) ⎬ −h31 (εnm ) . (6.84) h34 (δnm ) ⎦ E2nm = ⎩ ⎭ ⎭ ⎩ E3 nm h44 (δnm ) −h41 (εnm )

In arriving at Eqs. (6.83) and (6.84), we have arbitrarily set Anm to unity. The mode shapes for a solid plate are obtained from Eqs. (6.66), (6.76), and (6.81) as     Wnm rˆ , θ = Rˆ nm rˆ cos nθ

0 ≤ rˆ ≤ 1

n = 0, 1, 2, . . .

m = 1, 2, . . . (6.85)

where       Rˆ nm rˆ = Jn εnm rˆ + E2nm In δnm rˆ h31 (εnm ) E2nm = − h33 (δnm )

(6.86)

and we have again arbitrarily set Anm to unity. Equation (6.79) is a very general result: it includes the effects of a tensile inplane force per unit length acting in the radial direction, an elastic foundation, and various attachments on the inner and outer boundaries. This equation can be reduced to several common boundary conditions such as clamped, hinged, and free by taking the limiting process that was used in Chapters 3 and 5 for beams. For example, if the edge rˆ = α were clamped, then Kb → ∞ and Ktb → ∞. Therefore the first row of Eq. (6.79) is divided by Kb and then Kb → ∞. After this limit has been taken, it is found from Eq. (6.80) and Table 6.1 that a1n → 1 and b1n = 0 and c1n = 0. Similarly, the second row of Eq. (6.79) is divided by Ktb and then Ktb → ∞. After

362

6 Thin Plates

this limit has been taken, it is found from Eq. (6.80) and Table 6.1 that a2n → 1 and that b2n = 0 and c2n = 0. The values of ajn , bjn , and cjn for a clamped boundary, a boundary hinged with a torsion spring, and a free edge with a mass, respectively, are summarized in Tables 6.2 to 6.4. In Table 6.3, the case of a plate hinged at either if its edges or both of its edges is obtained by setting either Ktb = 0 or Kta = 0 or both of them to zero as the case may be. In Table 6.4, the case of an annular plate free at either if its edges or both of its edges is obtained by setting either mb = 0 or ma = 0 or both of them to zero as the case may be. In Table 6.5, a plate that is hinged with a torsion spring at its outer boundary and has a rigid solid circular disk of radius b and mass Md at its inner boundary is given. For this case, the mass ratio md = Md / mp α 2 and Ktb → ∞; that is, the slope in the radial direction at the common boundary of the disk and the inner edge of the annular plate is zero. It is also noted that this configuration is different from the case of a plate with a concentrated mass along its inner boundary. In the case of a concentrated mass, the moment at the inner boundary is zero, whereas for the case of the disk, the slope at the inner boundary is zero. Table 6.2 Constants appearing in Eq. (6.80) for the case when the inner and outer edges of an annular circular plate are clamped

Table 6.3 Constants appearing in Eq. (6.80) for the case when the inner and outer edges of an annular circular plate are hinged with an elastic torsion spring

j

ajn

bjn

cjn

1 2 3 4

1 1 1 1

0 0 0 0

0 0 0 0

j

ajn

bjn

cjn

1

1

0

0

2

1−ν Ktb + α

(1 − ν) n2 ε2 − α2

−δ2 −

3

1

0

0

Kta − (1 − ν)

−ε 2

4

+ (1 − ν) n2

(1 − ν) n2 α2

δ 2 + (1 − ν) n2

Table 6.4 Constants appearing in Eq. (6.80) for the case when the inner and outer edges of an annular circular plate are free and there is a mass along each edge j

ajn

1

−

2 3 4

mb α 4 (1 − ν) 2  + n 2 α3

1−ν α ma 4 −  − (1 − ν) n2 2 −(1 − ν)

bjn

cjn

Nˆ r + ε2 + ε2 −

(1 − ν) 2 n α2

(1 − ν) n2 α2

Nˆ r − δ 2 + −δ2 −

(1 − ν) 2 n α2

(1 − ν) n2 α2

Nˆ r + ε2 + (1 − ν) n2

Nˆ r − δ 2 + (1 − ν) n2

−ε2 + (1 − ν) n2

δ 2 + (1 − ν) n2

6.4

Natural Frequencies and Mode Shapes of Circular Plates for Very General. . .

363

Table 6.5 Constants appearing in Eq. (6.80) for the case when there is a rigid disk of radius α and mass ratio md attached to the inner edge of an annular circular plate and the outer edge of the plate is hinged with an elastic torsion spring j

ajn

1

−

2 3

1 1

0 0

0 0

4

Kta − (1 − ν)

−ε2 + (1 − ν) n2

δ 2 + (1 − ν) n2

md α 4 (1 − ν) 2  + n 2 α3

bjn Nˆ r + ε 2 +

cjn (1 − ν) 2 n α2

Nˆ r − δ 2 +

(1 − ν) 2 n α2

Tables 6.2 to 6.5 are used as follows. If a plate with a clamped inner edge and a free outer edge is being considered, then the values obtained from the first two rows of Table 6.2 for a1n , b1n , c1n , a2n , b2n , and c2n are used in Eq. (6.80) and the values obtained from last two rows of Table 6.4 for a3n , b3n , c3n , a4n , b4n , and c4n are used for the remaining quantities in Eq. (6.80). For solid plates, only the last two rows in Tables 6.2 to 6.4 are used.

6.4.3 Numerical Results Representative natural frequency coefficients and corresponding mode shapes have been obtained from the evaluation of Eqs. (6.79) and (6.82) for an annular plate and Eqs. (6.81) and (6.85) for a solid plate for various combinations of boundary conditions, in-plane forces, and masses attached to a boundary. From Eqs. (6.82) and (6.85), it is seen that the value of n determines the number of nodal diameters: for n = 0 there are no nodal diameters and the displacement at a given value of rˆ is the same at every value of θ . This is the axisymmetric case. At each value of n, there is an infinite number of natural frequencies identified with the subscript m, m = 0, 1, 2, . . . . Associated with each value of m is the corresponding number of nodal circles, including the boundaries. The natural frequency coefficients and mode shapes for a solid plate for Kf = Nr = 0 and for clamped, hinged, and free boundary conditions, respectively, are shown in Figs. 6.2 to 6.4. Included in these figures are the locations of the radii of the nodal circles. For the free plate shown in Fig. 6.4, ν = 0.3. In Fig. 6.5, the effects of an in-plane force on a clamped solid circular plate are shown along with the magnitude of the lowest value of Nr at which buckling occurs, Nr,buckle . As had been found with beams, a tensile in-plane force increases the natural frequency and a compressive in-plane force decreases it. For the case of annular plates with Kf = Nr = 0, the natural frequency coefficients for several combinations of boundary conditions have been determined for six values of α, 0.01 ≤ α ≤ 0.5, and for n = 0, 1, and 2 and m = 1, 2, and 3. In Table 6.6, we have tabulated the natural frequency coefficients for an annual plate that is clamped on its inner and outer boundaries; in Table 6.7 the values are for a

2

2

2

Ω02 = 39.771

Ω03 = 89.104

RNC: 1

RNC: 0.379, 1

RNC: 0.255, 0.583, 1

2

Ω12 = 60.829

Ω13 = 120.08

RNC: 1

RNC: 0.49, 1

RNC: 0.35, 0.639, 1

2

Ω22 = 84.583

Ω23 = 153.82

RNC: 1

RNC: 0.557, 1

RNC: 0.414, 0.678, 1

Ω01 = 10.216

Ω11 = 21.26

Ω21 = 34.877

2

2

2

2

Fig. 6.2 Values of 2nm for a clamped solid circular plate, the corresponding mode shapes, and the radii of the nodal circles of each mode shape including the boundary for Nr = Kf = 0: RNC = radii of the nodal circles and the dashed lines indicate the locations of the nodal lines and circles

2

2

2

Ω02 = 29.72

Ω03 = 74.156

RNC: 1

RNC: 0.442, 1

RNC: 0.279, 0.641, 1

2

Ω12 = 48.479

Ω13 = 102.77

RNC: 1

RNC: 0.551, 1

RNC: 0.378, 0.692, 1

2

Ω22 = 70.117

Ω23 = 134.3

RNC: 1

RNC: 0.613, 1

RNC: 0.443, 0.726, 1

Ω01 = 4.9351

Ω11 = 13.898

Ω21 = 25.613

2

2

2

2

Fig. 6.3 Values of 2nm for a hinged solid circular plate, the corresponding mode shapes, and the radii of the nodal circles of each mode shape including the boundary for Nr = Kf = 0: RNC = radii of the nodal circles and the dashed lines indicate the locations of the nodal lines and circles

2

2

2

Ω01 = 9.0031

Ω02 = 38.443

RNC: 0.68

RNC: 0.391, 0.841

2

Ω03 = 87.75

2

2

Ω11 = 20.475

Ω12 = 59.812

RNC: 0.781

RNC: 0.497, 0.87

2

Ω13 = 118.96

RNC: 0.351, 0.645, 0.907 2

2

Ω20 = 5.3583

RNC: 0.257, 0.591, 0.893

Ω21 = 32.56

Ω22 = 84.366

RNC: 0.822

RNC: 0.56, 0.888

Fig. 6.4 Values of 2nm for a free solid circular plate, the corresponding mode shapes, and the radii of the nodal circles of each mode shape including the boundary for Nr = Kf = 0 and ν = 0.3: RNC = radii of the nodal circles and the dashed lines indicate the locations of the nodal lines and circles 80 70

Nr,buckle = −14.68 2

Ω12 60 50 2 Ωnm

2

Ω02

40 30

2

Ω11

20

2

Ω01

10 0 −20

−10

0

10

20

30

40

Nr

Fig. 6.5 Values of 2nm for a clamped solid circular plate for Kf = 0 as a function of the magnitude of an in-plane force: Nr > 0 indicates a tensile force and Nr < 0 indicates a compressive force

366

6 Thin Plates

Table 6.6 Natural frequency coefficients 2nm for an annular circular plate clamped at rˆ = α and clamped at rˆ = 1 and for Nr = Kf = 0 n=0

n=1

n=2

α

m=1

m=2

m=3

m=1

m=2

m=3

m=1

m=2

m=3

0.01 0.1 0.2 0.3 0.4 0.5

22.924 27.281 34.609 45.346 61.872 89.251

62.604 75.366 95.741 125.36 170.9 246.34

122.63 148.21 188.15 246.16 335.37 483.22

23.698 28.916 36.103 46.644 62.996 90.23

65.456 78.635 98.278 127.38 172.56 247.74

127.39 152.54 191.23 248.5 337.25 484.78

34.895 36.617 41.82 51.139 66.672 93.321

84.645 90.448 106.52 133.67 177.63 251.97

153.97 167.14 200.86 255.67 342.94 489.49

Table 6.7 Natural frequency coefficients 2nm for an annular circular plate clamped at rˆ = α and free at rˆ = 1 for ν = 0.3 and for Nr = Kf = 0 n=0 α

m=1

0.01 3.766 0.1 4.2374 0.2 5.1811 0.3 6.6604 0.4 9.0206 0.5 13.024

n=1 m=2

m=3

21.088 61.286 25.262 73.901 32.291 94.084 42.614 123.47 58.549 168.69 85.033 243.69

n=2

m=1

m=2

m=3

2.1096 3.4781 4.813 6.5523 9.1155 13.29

22.743 64.422 27.673 77.487 34.526 96.954 44.631 125.82 60.378 170.69 86.706 245.44

m=1

m=2

m=3

5.361 5.6227 6.4471 7.9565 10.465 14.704

35.277 36.941 41.959 50.947 65.946 91.738

84.428 90.183 106.13 133.08 176.76 250.68

plate with the inner boundary clamped and the outer boundary free and ν = 0.3; in Table 6.8, the values are for a plate with the inner boundary free and the outer boundary clamped and ν = 0.3; in Table 6.9, the values are for a plate with the inner and outer boundaries free and ν = 0.3; and in Table 6.10, the values are for a plate with both boundaries hinged. It is noted upon comparing the values of 2nm for α = 0.01 in Table 6.8 to the values of 2nm in Fig. 6.2 that they are very closely equal to one another. A similar conclusion is reached when comparing the values of 2nm for α = 0.01 in Table 6.9 to the values of 2nm in Fig. 6.4. One can conclude from these comparisons that the Table 6.8 Natural frequency coefficients 2nm for an annular circular plate free at rˆ = α and clamped at rˆ = 1 for ν = 0.3 and for Nr = Kf = 0 n=0

n=1

n=2

α

m=1

m=2

m=3

m=1

m=2

m=3

m=1

m=2

m=3

0.01 0.1 0.2 0.3 0.4 0.5

10.214 10.159 10.408 11.424 13.603 17.715

39.755 39.521 43.02 51.745 67.159 93.847

89.051 90.445 105.01 132.41 177.02 252.19

21.26 21.195 20.551 19.54 19.594 22.015

60.829 60.061 56.897 59.759 72.215 97.376

120.08 117.08 117.07 138.66 181.01 255.05

34.873 34.535 33.735 32.594 31.535 32.116

84.569 83.478 80.836 79.061 86.013 107.49

153.78 151.34 146.38 156.47 192.71 263.56

6.4

Natural Frequencies and Mode Shapes of Circular Plates for Very General. . .

367

Table 6.9 Natural frequency coefficients 2nm for an annular circular plate free at rˆ = α and free at rˆ = 1 for ν = 0.3 and for Nr = Kf = 0 n=0

n=1

n=2

α

m=1

m=2

m=3

m=1

m=2

m=3

m=0

m=1

m=2

0.01 0.1 0.2 0.3 0.4 0.5

9.0002 8.7745 8.4442 8.3535 8.6138 9.3135

38.428 38.236 41.704 50.353 65.682 92.308

87.698 89.026 103.36 130.48 174.7 249.39

20.475 20.406 19.695 18.292 17.243 17.198

59.811 59.072 55.98 58.784 71.156 96.266

118.96 116.00 115.84 137.11 179.07 252.61

5.3578 5.3034 5.1461 4.906 4.6066 4.2711

35.256 34.931 34.155 32.973 31.599 31.115

84.353 83.27 80.656 78.932 85.95 107.51

Table 6.10 Natural frequency coefficients 2nm for an annular circular plate hinged at rˆ = α and hinged at rˆ = 1 and for Nr = Kf = 0 n=0

n=1

n=2

α

m=1

m=2

m=3

m=1

m=2

m=3

m=1

m=2

m=3

0.01 0.1 0.2 0.3 0.4 0.5

14.688 14.485 16.78 21.079 28.123 40.043

49.002 51.781 63.371 81.737 110.56 158.64

102.77 112.99 140.6 182.53 247.7 356.09

15.195 16.776 19.222 23.317 30.109 41.797

51.152 56.507 67.106 84.635 112.88 160.57

107.00 119.33 144.89 185.64 250.12 358.05

25.616 25.936 27.24 30.273 36.156 47.089

70.129 71.687 78.552 93.417 119.89 166.35

134.33 138.86 157.91 195.04 257.38 363.95

limiting condition of α → 0 requires the inner edge to be a free edge when the boundary conditions at the outer edge of the annular and solid plates are the same. In Fig. 6.6, the values of 2nm for an annular plate clamped at its inner boundary and free at its outer boundary for Kf = Nr = 0 and with a mass Ma attached to the outer boundary are given for α = 0.1, 0.25, and 0.5 and ν = 0.3. It is seen that the natural frequency coefficient decreases rapidly as the magnitude of ma increases from 0 to about 0.9, after which the rate of decrease is much slower with increasing values of ma . An annular plate with a mass Mb on its inner boundary can also be used to determine the axisymmetric natural frequencies of a solid plate carrying a concentrated mass Mi at its center. For a solid plate that is carrying a concentrated mass Mi , we  define its mass ratio as mi = Mi /mp , where from Eq. (6.55), mb = Mb / mp α 2 . If the masses in each case are equal; that is, Mb = Mi , then in the boundary condition for the annular plate mb = mi /α 2 , where α  1. The natural frequencies of a clamped solid circular plate with a concentrated mass at its center is shown in Fig. 6.7 as a function of mi for n = 0 (axisymmetric case). These results were obtained with α = 0.0001 and they are in good agreement with published results (Roberson 1951). The natural frequency coefficients 201 of a solid circular plate with a free edge restrained by a translation spring with constant Ka and torsion spring with constant Kta are shown in Fig. 6.8 as a function of Ka and Kta . These results show that for

368

6 Thin Plates 14 α = 0.1 α = 0.25

12

α = 0.5 2

Ω11

10

8

2

2 Ωnm

Ω01

6

2

Ω01 2

Ω11

4

2

0

2

Ω01 0

2

Ω11 0.5

1

1.5 ma

2

2.5

3

Fig. 6.6 Values of 2nm for an annular plate clamped at its inner boundary and free at its outer boundary as a function of a mass ma attached to the outer boundary for α = 0.1, 0.25, and 0.5 and for Nr = Kf = 0 11 2

Ω01 2 Ω02 /5

10

2 Ω03 /10

9

2 Ω0m

8 7 6 5 4 3

0

0.2

0.4

0.6

0.8

1

mi

Fig. 6.7 Lowest three natural frequency coefficients of a clamped solid circular plate with a concentrated mass at its center as a function of mi for n = 0 and for Nr = Kf = 0

6.4

Natural Frequencies and Mode Shapes of Circular Plates for Very General. . .

369

2 Ωclamped

10 8

2 Ωhinged

2 Ω 01

6 4 2 0 4 3

4

2

2

1 log10(Ka)

0

0 −1 −2

log10(Kta)

Fig. 6.8 Lowest natural frequency coefficient of a free solid circular plate as a function of the stiffness of a translation spring and a torsion spring attached to the free edge for ν = 0.3 and for Nr = Kf = 0 Table 6.11 Natural frequency coefficients 2nm for an annular circular plate carrying a solid rigid disk with a mass ratio md and a radius rˆ = α and clamped at rˆ = 1 for Nr = Kf = 0. n=0 md = 0.1

md = 0.5

md = 1

md = 2

α

m=1

m=2

m=1

m=2

m=1

m=2

m=1

m=2

0.05 0.1 0.2 0.3 0.4 0.5

10.332 10.671 11.991 14.304 18.07 24.347

40.642 42.959 51.051 64.188 84.913 119.2

10.305 10.567 11.589 13.355 16.149 20.617

40.471 42.378 49.206 60.44 78.254 107.84

10.272 10.44 11.134 12.386 14.414 17.69

40.26 41.699 47.369 57.367 73.824 101.77

10.205 10.199 10.356 10.933 12.135 14.295

39.851 40.483 44.748 53.836 69.588 96.794

n=1 md = 0.1

md = 0.5

md = 1

md = 2

α

m=1

m=2

m=1

m=2

m=1

m=2

m=1

m=2

0.05 0.1 0.2 0.3 0.4 0.5

23.561 23.182 20.655 19.651 21.445 26.57

64.277 60.961 58.709 68.197 87.482 121.04

23.559 23.149 20.221 18.504 19.253 22.551

64.255 60.518 56.418 63.95 80.397 109.33

23.558 23.107 19.7 17.29 17.239 19.372

64.226 59.959 54.016 60.409 75.663 103.08

23.554 23.024 18.73 15.399 14.556 15.668

64.169 58.828 50.409 56.311 71.143 97.958

370

6 Thin Plates

small values of Ka the system behaves as a single degree-of-freedom system with the plate acting as a rigid mass mounted on the translation spring; the torsion spring has very little effect on the natural frequency. As the stiffness of the translation spring increases the system approaches a hinged plate for small values of Kta and approaches that of a clamped plate as both Ka and Kta become very large. The final numerical evaluation is for an annular plate that is clamped along its  outer boundary and has a rigid solid circular disk of mass ratio md = Md / mp α 2 and non dimensional radius α attached to the plate inner boundary at rˆ = α. The values of the frequency coefficients for Kf = Nr = 0 and for a range of values for md are given in Table 6.11 for n = 0 and 1, m = 1 and 2, and for several values of α.

6.5 Natural Frequencies and Mode Shapes of Rectangular and Square Plates: Rayleigh-Ritz Method 6.5.1 Introduction The natural frequencies and mode shapes for rectangular and square plates with and without an interior mass Mi located at (ηo , ξo ) shall be determined. It is assumed that kf = ki = Nx = Ny = 0 and, since only the natural frequencies and mode shapes will be determined, Fˆ a = 0. The governing equation given by Eq. (6.38) simplifies to 4 wˆ + (1 + mi αδ (η − ηo ) δ (ξ − ξo )) ∇ηξ

∂ 2 wˆ = 0. ∂τ 2

(6.87)

The plate is undergoing harmonic oscillations of the form ˆ (η, ξ ) ej wˆ (η, ξ , τ ) = W

2τ

(6.88)

where 2 = ωtp . Substituting Eq. (6.88) into Eq. (6.87), we arrive at 4 ˆ ˆ = 0. W − 4 (1 + mi αδ (η − ηo ) δ (ξ − ξo )) W ∇ηξ

(6.89)

When mi = 0, separable solutions to Eq. (6.89) are those given by either ˆ (η, ξ ) = W

∞ 

Yn (ξ ) sin εη

(6.90a)

Xn (η) sin δξ

(6.90b)

n=1

or ˆ (η, ξ ) = W

∞  n=1

6.5

Natural Frequencies and Mode Shapes of Rectangular and Square Plates. . .

371

or ˆ (η, ξ ) = W

∞ 

sin εη sin δξ

(6.90c)

n=1

and a similar set of solutions with sine replaced by cosine. Examining the boundary conditions, it is seen that the sine function only satisfies the hinged boundary condition provided that ε = mπ and δ = mπ/α, where m = 1, 2, . . . and from Eq. (6.37) α = b/a. The same is true for the cosine function, although the expressions for ε and δ are different. Thus, a straightforward analytical solution can be obtained only if two opposite edges of the plate are hinged. To overcome this restriction on the boundary conditions, we shall forego seeking an analytical solution to Eq. (6.89) for a restricted set of boundary conditions and instead use the Rayleigh-Ritz method; this will allow us to consider a much wider range of combinations of boundary conditions.

6.5.2 Natural Frequencies and Mode Shapes of Rectangular and Square Plates The starting point for determining the natural frequencies and mode shapes for rectangular and square plates using the Rayleigh-Ritz method is to apply the procedure that was used in Section 3.8.5 for beams of variable cross section. Thus, the quantity to be minimized for a system undergoing harmonic excitation at frequency ω and magnitude W (x, y) is the difference between the maximum kinetic energy and the maximum potential energy; that is, =

D (Tmax − Vmax ) 2

(6.91)

where, from Eqs. (6.12) and (6.13) and the definitions given by Eq. (6.37), the maximum kinetic energy is

α 1 Tmax = 

4 0

ˆ 2 (η, ξ ) dηdξ . (1 + mi αδ (η − ηo ) δ (ξ − ξo )) W

(6.92)

0

The maximum potential energy is obtained from Eq. (6.11) and the definitions given by Eq. (6.37) to arrive at

Vmax

⎡  2 ⎤

α 1  2 ˆ 2  2 ˆ 2 2W 2W ˆ d2 W ˆ ˆ d d d d W W ⎣ ⎦ dηdξ . = + + 2ν 2 + 2 (1 − ν) dξ dη dη2 dξ 2 dη dξ 2 0 0

(6.93)

372

6 Thin Plates

ˆ (η, ξ ) can be expressed as (Young 1950) It is assumed that W ˆ (η, ξ ) = W

N N   n

Anm Xn (η) Ym (ξ )

(6.94)

m

where the trail functions Xn and Ym are chosen as the mode shapes for a beam of constant cross section as given in the rightmost column of Table 3.3 for the first five sets of boundary conditions; that is, for p = 1, , 2, . . . , 5. For specificity, these functions have been reproduced in Table 6.12. The functions given in Table 6.12 satisfy the clamped and hinged boundary conditions for a plate exactly, but only approximately satisfy those for the free boundary condition. Substituting Eq. (6.94) into Eqs. (6.92) and (6.93) and these results in turn into Eq. (6.91) yields ⎧ N N N N   D ⎨ 4  Anm Apq I1np I2mq + mi αXn (ηo ) Xp (ηo ) Ym (ξo ) Yq (ξo )  = 2⎩ n

−

m

p

N  N  N N   n

− 2ν

m

p

q

  Anm Apq I5np I2mq + I1np I6mq + 2 (1 − ν) I3np I4mq

q

N  N  N N   n

m

p

⎫ ⎬ Anm Apq I7np I8mq

q

⎭ (6.95)

(p)

(p)

Table 6.12 Trial functions Xn = Xn and Yn = Yn , p = 1, 2, . . . , 5, for several sets of boundary conditions. The values of n , where 4n = ρAo a4/EIo , are given in the fourth column of Table 3.5 for the corresponding value of p Boundary conditions Case η = 0 (p) ξ =0

η=1 ξ =α

Xn (η)

Yn (ξ )

1

Hinged

sin (n η)

sin (n ξ/α) −

Hinged

(p)

2

Clamped

Clamped

S (n ) T (n η) + S (n η) − T (n )

3

Clamped

Free

−

4

Free

Free

5

Clamped

Hinged

S (n ) T (n ξ/α) + S (n ξ/α) T (n )

T (n ) T (n ) T (n η) + S (n η) − T (n ξ/α) + S (n ξ/α) Q (n ) Q (n )

n = 1: 1 n = 2: 1 − 2η n > 2: S (n ) − Q (n η) + R (n η) R (n ) −

(p)

S (n ) T (n η) + S (n η) T (n )

n = 1: 1 n = 2: 1 − 2ξ/α n > 2: S (n ) Q (n ξ/α) + R (n ξ/α) − R (n ) −

S (n ) T (n ξ/α) + S (n ξ/α) T (n )

6.5

Natural Frequencies and Mode Shapes of Rectangular and Square Plates. . .

373

where

1 I1np =

α Xn (η) Xp (η) dη

I2mq =

0

1 I3np =

0

Xn (η) Xp (η) dη

α I4mq =

0

1 I5np =

I7np =

Ym (ξ ) Yq (ξ ) dξ

0

Xn (η) Xp (η) dη

α I6mq =

0

1

Ym (ξ ) Yq (ξ ) dξ

(6.96) Ym (ξ ) Yq (ξ ) dξ

0

Xn (η) Xp (η) dη

0

α I8mq =

Ym (ξ ) Yq (ξ ) dξ

0

and the prime denotes the derivative with respect to the variable of integration. It is noted that I7np and I8mq are not symmetric, whereas all the other integrals are symmetric. The derivatives of the trial functions appearing in Eqs. (6.96) as given in Table 6.12 are obtained by using Eq. (C.20) of Appendix C. The necessary condition for  to be a minimum value is attained when ∂ =0 ∂Alk

l, k = 1, 2, . . . , N.

(6.97)

Thus, substituting Eq. (6.95) into Eq. (6.97) and performing the indicated operation results in the following system of equations N N   n

m

Anm Cnmlk − 

4

N  N  n

Anm Bnmlk = 0

l, k = 1, 2 . . . , N

(6.98)

m

where Bnmlk = I1nl I2mk + mi αXn (ηo ) Xl (ηo ) Ym (ξo ) Yk (ξo ) (6.99) Cnmlk = I5nl I2mk + I1nl I6mk + 2 (1 − ν) I3nl I4mk + ν (I7ln I8km + I7nl I8mk ) . In arriving at Eqs. (6.98) and (6.99), the symmetric properties of Ijnp , j = 1, 2, . . . , 6 have been used. Equation (6.98) is a system of N2 equations in terms of N2 unknown constants Anm , n, m = 1, 2, . . . , N, that can be expressed in matrix notation as [C] {A} − 4 [B] {A} = 0

(6.100)

where the elements of [C], [B], and {A} are given in Appendix 6.1. It is seen that Eq. (6.100) is a standard eigenvalue formulation that can be solved by readily available procedures yielding the N2 natural frequency coefficients j ,

374

6 Thin Plates (j)

j = 1, 2, . . . , N 2 , and the corresponding N2 components of the modal vector Anm . Then, from Eq. (6.94), the mode shape corresponding to j is given by ˆ j (η, ξ ) = W

N N   n

(j) Anm Xn (η) Ym (ξ ).

(6.101)

m

6.5.3 Numerical Results Equation (6.100) is solved for a wide variety of boundary conditions. For the case where mi = 0, many of the results from these various combinations of boundary conditions have been compared to those appearing in the literature and excellent agreement was obtained for all cases; that is, in virtually all cases, the results presented here agree to better than 0.25%. The results in the literature for the cases where one pair of opposing edges were simply supported were obtained using an analytical solution. The results obtained from Eq. (6.100) for a plate hinged on all four edges are exact. Nine combinations of boundary conditions have been considered and the natural frequency coefficients and mode shapes for each set of boundary conditions are shown in Figs. 6.9 to 6.17. For each set of boundary conditions, the lowest nine natural frequencies and the corresponding mode shapes are presented. A value of ν = 0.3 has been used in all cases involving at least one free edge and a value of N = 5 has been found to provide excellent agreement even for the ninth lowest natural frequency coefficient. For all but two boundary combinations, α = 1.5. For the cases of a cantilever plate and for a plate with two adjacent edges clamped and the remaining two edges free, α = 1; that is, the plate is square. In the figures, the broken lines indicate the location of the nodal lines. In order to speak more concisely about the various configurations examined, the following short hand notation is introduced to indicate the boundary conditions: (edge x = 0)-(edge y = 0)-(edge x = 1)-(edge y = α). Thus, if h = hinged edge, c = clamped edge, and f = free edge, then c-c-f-f indicates a plate in which the two adjacent edges along x = 0 and y = 0 are clamped and the remaining edges along x = 1 and y = α are free. In Figs. 6.9 to 6.13, the lowest nine natural frequencies and the corresponding mode shapes are presented for rectangular plates with two opposite edges hinged. In Fig. 6.14, the nine lowest natural frequency coefficients and corresponding mode shapes are given for a rectangular plate clamped on all four of its edges. In Figs. 6.15 to 6.17, the nine lowest natural frequency coefficients and corresponding mode shapes are given for plates with different combinations of clamped and free boundaries. The results in Fig. 6.17 are for a cantilever plate (c-f-f-f). From Figs. 6.9 to 6.17 it is clear that it is not possible to predict a priori which natural frequency in the ordering from lowest to highest will correspond to a specific set of nodal lines in the x and y directions. This unpredictability is also dependent on the value of α.

6.5

Natural Frequencies and Mode Shapes of Rectangular and Square Plates. . .

Fig. 6.9 Lowest 9 natural frequency coefficients and corresponding mode shapes for a rectangular h-h-h-h plate with α = 1.5: the dashed lines indicate the location of the nodal lines

Fig. 6.10 Lowest 9 natural frequency coefficients and corresponding mode shapes for a rectangular h-c-h-c plate with α = 1.5: the dashed lines indicate the location of the nodal lines

375

Ω12 = 14.256

Ω22 = 27.416

Ω32 = 43.865

h

h

h

h

h

h

h

h

h

h

h

h

Ω42 = 49.348 h

Ω52 = 57.024 h

Ω62 = 78.957 h

h

h

h

h

h

h

h

h

h

Ω72 = 80.053 h

Ω82 = 93.213 h

Ω92 = 106.37 h

h

h

h

h

h

h

h

h

h

Ω12 = 17.381 c

Ω22 = 35.379 c

Ω32 = 45.451 c

h

h

h

h c

h c

h c

2 Ω4 = 62.13 c

Ω52 = 62.424 c

Ω62 = 88.95 c

h

h

h

h

h

h

c

c

c

2 Ω7 = 94.259 c

Ω82 = 97.583 c

Ω92 = 110.33 c

h

h

h

h c

h c

h c

376 Fig. 6.11 Lowest 9 natural frequency coefficients and corresponding mode shapes for a rectangular h-c-h-h plate with α = 1.5: the dashed lines indicate the location of the nodal lines

6 Thin Plates Ω12 = 15.582

Ω22 = 31.086

Ω32 = 44.577

h

h

h

h

h

h

h

h

c

c

c

2 Ω4 = 55.425 h

Ω52 = 59.507 h

Ω62 = 83.691 h

h

h

h

h

h

h

c

c

c

Ω72 = 88.506

Ω82 = 93.705 h

Ω92 = 108.21 h

h

h

h h

Fig. 6.12 Lowest 9 natural frequency coefficients and corresponding mode shapes for a rectangular h-c-h-f plate with α = 1.5 and ν = 0.3: the dashed lines indicate the location of the nodal lines

h

h

h

h

c

c

c

Ω12 = 11.038

Ω22 = 20.354

Ω32 = 37.987

f

f

f

h

h

h

h

h

h

c

c

c

Ω42 = 40.492

Ω52 = 49.855

Ω62 = 64.213

f

f

f

h

h

h

h

h

h

c

c

c

2 Ω7 = 68.118 f

Ω82 = 89.781

Ω92 = 94.749

f

f

h

h

h c

h c

h

h c

6.5

Natural Frequencies and Mode Shapes of Rectangular and Square Plates. . .

Fig. 6.13 Lowest 9 natural frequency coefficients and corresponding mode shapes for a rectangular h-f-h-f plate with α = 1.5 and ν = 0.3: the dashed lines indicate the location of the nodal lines

Ω12 = 9.7703

Ω22 = 13.054

Ω32 = 22.973

f

f

f

h

h

h

h

h

h

f

f

f

Ω42 = 39.338

Ω52 = 40.454

Ω62 = 42.998

f

f

f

h

Fig. 6.14 Lowest 9 natural frequency coefficients and corresponding mode shapes for a rectangular c-c-c-c plate with α = 1.5: the dashed lines indicate the location of the nodal lines

377

h

h

h

h

h

f

f

f

2 Ω7 = 54.387 f

Ω82 = 66.235

Ω92 = 73.939

f

f

h

h

h

h

h

h

f

f

f

Ω12 = 27.026

Ω22 = 41.759

Ω32 = 66.025

c

c

c

c

c

c

c

c

c

c

c

c

2 Ω4 = 66.625

Ω52 = 79.994

Ω62 = 101.01

c

c

c

c

c

c

c

c

c

c

c

c

Ω72 = 103.4 c

Ω82 = 125.5 c

Ω92 = 136.63 c

c

c

c

c c

c c

c c

378

6 Thin Plates

Fig. 6.15 Lowest 9 natural frequency coefficients and corresponding mode shapes for a rectangular c-f-c-f plate with α = 1.5 and ν = 0.3: the dashed lines indicate the location of the nodal lines

Ω12 = 22.327

Ω22 = 24.325

Ω32 = 31.707

f

f

f

c

c

c

c

c

f

f

f

Ω42 = 46.932

Ω52 = 61.624

Ω62 = 64.413

f

f

f

c

c

c

c

c

c

f

f

f

2 Ω7 = 71.093 f

Ω82 = 73.597 f

Ω92 = 90.59 f

c

c

c

c

c

f

Fig. 6.16 Lowest 9 natural frequency coefficients and corresponding mode shapes for a square c-c-f-f plate with ν = 0.3: the dashed lines indicate the location of the nodal lines

c

c

f

f

Ω12 = 6.9443

Ω22 = 24.043

Ω32 = 26.7

f

f

f

c

f

c

f

c

f

c

c

c

Ω42 = 47.796

Ω52 = 63.053

Ω62 = 65.869

f

f

f

c

f

c

f

c

f

c

c

c

2 Ω7 = 85.968 f

Ω82 = 88.799 f

Ω92 = 122.01 f

f

c c

c

f c

c

N/A c

f

6.5

Natural Frequencies and Mode Shapes of Rectangular and Square Plates. . .

Fig. 6.17 Lowest 9 natural frequency coefficients and corresponding mode shapes for a square c-f-f-f plate, called a cantilever plate, with ν = 0.3: the dashed lines indicate the location of the nodal lines

379

Ω12 = 3.4919

Ω22 = 8.5318

Ω32 = 21.429

f

f

f

c

f

c

f

c

f

f

f

f

Ω42 = 27.339

Ω52 = 31.163

Ω62 = 54.437

f

f

f

c

f

c

f

c

f

f

f

f

Ω72 = 61.618 f

Ω82 = 64.495 f

Ω92 = 71.54 f

f

c

f

c

f

f

c

f

f

The final set of results, which is shown in Fig. 6.18, gives the lowest natural frequency coefficient for a rectangular plate carrying a concentrated mass mi = 0.5 as a function of its position for α = 1.5 and ν = 0.3. The plate has two adjacent edges clamped and the remaining two edges free. As expected, the minimum lowest natural frequency occurs when the mass is placed at the intersection of the two free edges. For this set of parameters, it is seen that when the mass is placed within the approximate regions defined by 0 ≤ ηo ≤ 0.2 and 0 ≤ ξo ≤ α and 0 ≤ ηo ≤ 1 and 0 ≤ ξo ≤ 0.2 the natural frequency is minimally affected by the mass.

6.5.4 Comparison with Thin Beams It is instructive to compare the natural frequency coefficients of a plate that has two opposite edges free to that obtained for a thin beam in Chapter 3. The boundary conditions that are on the remaining two edges of the plate correspond to the boundary conditions that are prescribed for the beam. To be able to compare the natural frequencies, it is noted from Eqs.  (3.53),  (6.37), and (6.10) that for a beam of rectangular cross section tp2 = tb2 1 − ν 2 , where the subscript p indicates the plate and the subscript b the beam. In arriving at this relation, we have set L = a where L is the length of the beam. Consequently, the natural frequency of the plate   and the beam are equal when 2p = 2b 1 − ν 2 . It is expected that as α → 0 this relationship between the plate and beam should become true. In Table 6.13, we have

380

6 Thin Plates

Fig. 6.18 Lowest natural frequency coefficient of a rectangular plate that is clamped on two adjacent edges and free on the remaining two edges as a function of the position of a concentrated mass ratio mi for mi = 0.5, ν = 0.3, and α = 1.5

5

Ω1

2

4 3 2 1 0

0 0.2

0.5

0.4 ηo

0.6 1

Table 6.13 Comparison of the lowest three natural frequency coefficients of a narrow plate with those of an Euler-Bernoulli beam as a function of the boundary conditions and α for ν = 0.3

ξo

1 0.8 1.5

2p,n

2b,n

c-f-c-f

c-c

n

α = 0.1

α = 0.05

α = 0.02

1 2 3

21.919 60.383 118.26

21.879 60.283 117.92

21.865 60.250 117.81

h-f-h-f

  1 − ν2

21.343 58.832 115.33 h-h

n

α = 0.1

α = 0.05

α = 0.02

1 2 3

9.502 38.069 85.85

9.499 38.010 85.578

9.497 37.992 85.492

c-f-f-f

9.415 37.660 84.735 c-f

n

α = 0.1

α = 0.05

α = 0.02

1 2 3

3.426 21.423 59.939

3.422 21.404 59.842

3.421 21.398 59.811

3.354 21.020 58.855

6.5

Natural Frequencies and Mode Shapes of Rectangular and Square Plates. . .

381

tabulated the results for the lowest three natural frequency coefficients for a beam that is hinged at both ends, clamped at both ends, and clamped at one end and free at the other end for α = 0.1, 0.05, and 0.02. In arriving at these results, values of N = 5 and ν = 0.3 were used. An examination of the results in Table 6.13 indicates that when α = 0.02,   2 2 2 p and b 1 − ν differ by less than 1% for most of the cases and does not exceed 2.4% for all α. While the natural frequency coefficients presented in Table 6.13 are in good agreement with those of an Euler-Bernoulli beam for the lowest three natural frequencies, they are not the complete picture. To gain a clearer understanding of what is happening as α decreases, consider the mode shapes shown for a c-f-c-f plate for α = 0.05 and α = 0.1, respectively, in Figs. 6.19 and 6.20.

2

2

Ω1 = 21.88 f

c

2

Ω3 = 117.9

Ω2 = 60.28 c

f

c

c

f

c

f

f

f

Ω4 = 195

Ω5 = 270.9

Ω6 = 291.1

2

f

c

2

c

f

c

2

c

f

c

f

f

f

Ω7 = 546.2

Ω8 = 815.8

Ω9 = 1099

2

f

c

2

c

f

c

c

2

c

f

c

c

f

f

f

c

Fig. 6.19 Lowest 9 natural frequency coefficients and corresponding mode shapes for a c-f-c-f plate for α = 0.05 and ν = 0.3. The horizontal dashed lines and the interior vertical lines indicate the location of the nodal lines 2

2

c

2

Ω3 = 118.3 f

Ω2 = 60.38 f

Ω1 = 21.92 f c

c

c

c

c

f

f

f

Ω4 = 137.6

Ω5 = 195.7 f

Ω6 = 279 f

2

2

f c

c

c

2

c

c

c

f

f

f

Ω7 = 292.8

Ω8 = 421.9

Ω9 = 575.5

2

2

f c

f c

f

2

c

f c

f

c

c f

Fig. 6.20 Lowest 9 natural frequency coefficients and corresponding mode shapes for a c-f-c-f plate for α = 0.1 and ν = 0.3. The horizontal dashed lines and the interior vertical lines indicate the location of the nodal lines

382

6 Thin Plates

It is seen that even for these relatively narrow plates, several additional plate modes appear: when α = 0.05 one plate mode appears between the fourth and fifth beamlike modes and when α = 0.1 one plate mode appears between the third and fourth beam-like modes and another between the fourth and fifth beam-like modes. For the other two types of boundary conditions, it has been found that similar behavior occurs, except that the locations of the plate modes interspersed between the beam-like modes are different.

6.6 Forced Excitation of Circular Plates 6.6.1 General Solution to the Forced Excitation of Circular Plates We shall obtain the general solution to the forced excitation of an annular circular plate resting on an elastic foundation and subjected to a constant in-plane tensile radial force. The equation of motion is given by Eq. (6.53). It is assumed that the boundary conditions are those given by Eqs. (6.57) and (6.58) and that the initial conditions are zero. Based on Eq. (6.82) for an annular plate and Eq. (6.85) for a solid plate, we assume a solution to Eq. (6.53) of the form ∞ ∞ ∞        wˆ rˆ , θ , τ = Rˆ 0m rˆ ϕ0m (τ ) + Rˆ nm (ˆr) cos (nθ ) ϕc,nm (τ ) m=1 ∞ ∞  

+

n=1 m=1

  Rˆ nm rˆ sin (nθ ) ϕs,nm (τ )

(6.102)

n=1 m=1

  where Rˆ nm rˆ , which is given by Eq. (6.83) for an annular plate or by Eq. (6.86) for a solid plate, is a solution to ∇rˆ4 Rˆ nm − Nˆ r ∇rˆ2 Rˆ nm − λ4nm Rˆ nm = 0 m = 1, 2, . . . , n = 0, 1, 2, . . .

(6.103)

and λ4nm = 4nm − Kf .

(6.104)

The modal amplitudes ϕ0m (τ ) , ϕc,nm (τ ) , and ϕs,nm (τ ) are unknown quantities that are to be determined. To reduce the algebraic complexity in what follows, we shall only use the double summation involving cos (nθ ). The remaining two summations will be dealt with after the results using this expression have been obtained.   The function Rˆ nm rˆ for an annular plate, which is given by Eq. (6.83), satisfies the following boundary conditions.

6.6

Forced Excitation of Circular Plates

383

At rˆ = α > 0

mb α 4  nm Rˆ nm (α) − Nˆ r Rˆ nm (α) + Vˆ rˆ,nm (α) = 0 Kb − 2 ˆ rˆ ,nm (α) = 0 Ktb Rˆ nm (α) + M

(6.105)

ma 4  Ka − Rˆ nm (1) + Nˆ r Rˆ nm (1) − Vˆ rˆ ,nm (1) = 0  2 nm ˆ rˆ ,nm (1) = 0 Ktb Rˆ nm (1) − M

(6.106)

At rˆ = 1

where the prime denoted the derivative with respect to rˆ and      dRˆ nm d2 Rˆ nm n2 ˆ rˆ ,nm rˆ = − M +ν − 2 Rˆ nm dˆr2 rˆ dˆr rˆ   

  n2 dRˆ nm d 1 2 Vˆ rˆ ,nm rˆ = − Rˆ nm . ∇rˆ Rˆ nm − (1 − ν) 2 dˆr rˆ dˆr rˆ

(6.107)

  For a solid plate, Rˆ nm rˆ is given by Eq. (6.86) and satisfies only Eq. (6.106). Substituting only the portion of the expression involving cos (nθ ) in Eq. (6.102) into Eq. (6.53) and using Eq. (6.103) yields ∞  2 ∞   ∂ ϕc,nm (τ )

∂τ 2

n=1 m=1

    + 4nm ϕc,nm (τ ) Rˆ nm rˆ cos (nθ ) = fa rˆ , θ , τ .

(6.108)

Upon substituting only the portion of the expression involving cos (nθ ) in Eq. (6.102) into the boundary conditions given by Eqs. (6.57) and (6.58), respectively, the boundary conditions become At rˆ = α > 0 ∞  ∞    Kb Rˆ nm (α) − Nˆ r Rˆ nm (α) + Vˆ rˆ ,nm (α) ϕc,nm (τ ) n=1 m=1

mb α ∂ 2 ϕc,nm (τ ) Rˆ nm (α) cos (nθ ) = 0 2 ∂τ 2 ∞  ∞    ˆ rˆ,nm (α) ϕc,nm (τ ) cos (nθ ) = 0 Ktb Rˆ nm (α) + M +

n=1 m=1

(6.109)

384

6 Thin Plates

At rˆ = 1 ∞  ∞  

 Ka Rˆ nm (1) + Nˆ r Rˆ nm (1) − Vˆ rˆ ,nm (1) ϕc,nm (τ )

n=1 m=1

ma ∂ 2 ϕc,nm (τ ) Rˆ nm (1) cos (nθ ) = 0 2 ∂τ 2 ∞  ∞    ˆ rˆ ,nm (1) ϕc,nm (τ ) cos (nθ ) = 0 Kta Rˆ nm (1) − M

+

(6.110)

n=1 m=1

Using Eq. (6.105) in Eq. (6.109) and Eq. (6.106) in Eq. (6.110), it is found that the second equation of Eq. (6.109) is satisfied and the second equation of Eq. (6.110) is satisfied and that the first equation of each of these equations, respectively, becomes  ∞  ∞  mb α ∂ 2 ϕc,nm (τ ) + 4nm ϕc,nm (τ ) Rˆ nm (α) cos (nθ ) = 0 2 ∂τ 2

(6.111)

 ∞ ∞   ma ∂ 2 ϕc,nm (τ ) + 4nm ϕc,nm (τ ) Rˆ nm (1) cos (nθ ) = 0. 2 ∂τ 2

(6.112)

n=1 m=1

and

n=1 m=1

The following operations are now performed on Eqs. (6.108), (6.111) and   (6.112). First, Eq. (6.108) is multiplied by Rˆ pi rˆ cos (pθ ) rˆ dˆrdθ and integrated over the area of the plate to arrive at ∞  ∞  2  ∂ ϕc,nm (τ )

∂τ 2

n=1 m=1

1 ×

+ 4nm ϕc,nm (τ )

    Rˆ nm rˆ Rˆ pi rˆ rˆ dˆr

α

(6.113)

2π cos (nθ ) cos (pθ ) dθ = gc,pi (τ ) 0

where

1 2π gc,pi (τ ) = α

    fa rˆ , θ , τ Rˆ pi rˆ cos (pθ ) rˆ dˆrdθ .

(6.114)

0

However,

2π cos (nθ ) cos (pθ ) dθ = π δnp 0

n, p = 1, 2, . . .

(6.115)

6.6

Forced Excitation of Circular Plates

385

and, therefore, Eq. (6.113) becomes ∞  m=1



∂ 2 ϕc,nm (τ ) π + 4nm ϕc,nm (τ ) ∂τ 2

1

    Rˆ nm rˆ Rˆ ni rˆ rˆ dˆr = gc,ni (τ ) . (6.116)

α

Next, Eq. (6.111) is multiplied by Rˆ pi (α) cos (pθ ) dθ and integrated with respect to θ , 0 ≤ θ ≤ 2π , to arrive at  ∞  mb απ ∂ 2 ϕc,nm (τ ) + 4nm ϕc,nm (τ ) Rˆ nm (α) Rˆ ni (α) = 0 2 ∂τ 2

(6.117)

m=1

where we have used Eq. (6.115). Similarly, Eq. (6.112) is multiplied by Rˆ pi (1) cos (pθ ) dθ and integrated with respect to θ , 0 ≤ θ ≤ 2π , to arrive at  ∞  mb π ∂ 2 ϕc,nm (τ ) + 4nm ϕc,nm (τ ) Rˆ nm (1) Rˆ ni (1) = 0 2 ∂τ 2

(6.118)

m=1

and we have again used Eq. (6.115). For the last operation, Eqs. (6.113), (6.117) and (6.118) are added to obtain ∞  m=1

 π Anmi

∂ 2 ϕc,nm (τ ) + 4nm ϕc,nm (τ ) = gc,ni (τ ) ∂τ 2

(6.119)

where

1 Anmi = α

    mb mb α Rˆ nm rˆ Rˆ ni rˆ rˆ dˆr + Rˆ nm (α) Rˆ ni (α) + Rˆ nm (1) Rˆ ni (1) . (6.120) 2 2

To evaluate Eq. (6.120), it is noted that when Fa = 0, Eq. (6.24) is a symmetric quadratic. In addition, Eqs. (6.18) and (6.19) are also symmetric quadratics. Therefore, when these results are transformed to polar coordinates they remain symmetric quadratics. Consequently, for the   boundary conditions given by Eqs. (6.57) and (6.58) and their special cases, Rˆ ni rˆ is an orthogonal function. From Eq. (6.29) and the discussion following Eq. (6.49), it is seen that a11 = Mb , a21 = Ma , (i) and a12 = a22 = 0. Therefore, from Eqs.   (B.90) to (B.92) with U (x) → ˆRni rˆ , the orthogonality condition for Rˆ ni rˆ in terms of the non dimensional parameters is

1 α

    ma αmb Rˆ ni rˆ Rˆ nj rˆ rˆ dˆr + Rˆ ni (α) Rˆ nj (α) + Rˆ ni (1) Rˆ nj (1) = Nˆ ni δij (6.121) 2 2

386

6 Thin Plates

where

Nˆ ni =

1 α

  αmb 2 ma 2 Rˆ ni (α) + Rˆ (1) . Rˆ 2ni rˆ rˆ dˆr + 2 2 ni

(6.122)

Thus, from Eq. (6.121), Eq. (6.120) becomes Anmi = δmi Nˆ ni

(6.123)

gc,nm (τ ) ∂ 2 ϕc,nm (τ ) + 4nm ϕc,nm (τ ) = . ∂τ 2 π Nˆ nm

(6.124)

and Eq. (6.119) simplifies to

To obtain a solution to Eqs. (6.53), (6.57), and (6.58) using the double summation expression involving sin (nθ ) in Eq. (6.102), we employ the same procedure that was just used to obtain Eq. (6.124) and arrive at gs,nm (τ ) ∂ 2 ϕs,nm (τ ) + 4nm ϕs,nm (τ ) = ∂τ 2 π Nˆ nm

(6.125)

where

1 2π gs,nm (τ ) = α

    fa rˆ , θ , τ Rˆ nm rˆ sin (nθ ) rˆ dˆrdθ .

(6.126)

0

For the special case of axisymmetric excitation; that is, when n = 0, we use the single summation expression in Eq. (6.102) that is independent of θ and the preceding procedure to arrive at g0m (τ ) ∂ 2 ϕ0m (τ ) + 4nm ϕ0m (τ ) = ∂τ 2 Nˆ 0m

(6.127)

where

1 g0m (τ ) = α

    fa rˆ , τ Rˆ 0m rˆ rˆ dˆr.

(6.128)

6.6

Forced Excitation of Circular Plates

387

The solutions to Eqs. (6.124), (6.125), and (6.127) are given by Eq. (C.6) of Appendix C. Thus, for zero initial conditions, 1 ϕ0m (τ ) = 2  Nˆ 0m 0i

ϕc,nm (τ ) =

ϕs,nm (τ ) =

τ



g0m (τ − t) sin 20m t dt

0

τ

1 π 2nm Nˆ nm

π 2nm Nˆ nm

(6.129)

0

τ

1



gc,nm (τ − t) sin 2nm t dt 

gs,nm (τ − t) sin 2nm t dt.

0

Substituting Eqs. (6.114), (6.126) and (6.128) into Eq. (6.129) and the results in turn into Eq. (6.102), we obtain the displacement response as   τ 1 ∞ ˆ 

       R0m rˆ 2 ˆ f t r ˆ , τ − t R r ˆ sin  wˆ rˆ , θ , τ = a 0m 0m rˆ dˆr dt 2 Nˆ m=1 0m 0m 0 α   ∞  ∞  Rˆ nm rˆ + cos (nθ ) × π 2nm Nˆ nm n=1 m=1

τ 1 2π

+

0 α 0 ∞ ∞   n=1 m=1

  Rˆ nm rˆ sin (nθ ) × π 2nm Nˆ nm

τ 1 2π 0 α

     fa rˆ , θ , τ − t Rˆ nm rˆ cos (nθ ) sin 2nm t rˆ dˆrdθ dt



    fa rˆ , θ , τ − t Rˆ nm rˆ sin (nθ ) sin 2nm t rˆ dˆrdθ dt.

0

(6.130)

6.6.2 Impulse Response of a Solid Circular Plate We shall now determine the response of a clamped solid circular plate to an impulse force applied at the center of the plate when Kf = Nˆ r = 0. This type of excitation is independent of θ and, therefore, is axisymmetric and requires only the solution for n = 0. Thus, Eq. (6.130) simplifies to   τ 1 ∞ ˆ 

       R0m rˆ ˆ 0m rˆ sin 20m t rˆ dˆrdt f r ˆ , τ − t R wˆ rˆ , τ = a 2 Nˆ m=1 0m 0m 0

0

(6.131)

388

6 Thin Plates

where, from the last two rows of Table 6.2 and Eq. (6.80) it is found that Eq. (6.86) can be written as  J0 (0m )      I0 0m rˆ R0m rˆ = J0 0m rˆ − I0 (0m )

(6.132)

and from Eq. (6.81) 0m are solutions to J0 (0m ) I1 (0m ) + I0 (0m ) J1 (0m ) = 0.

(6.133)

In obtaining Eqs. (6.132) and (6.133), we have used Eq. (6.73) to find that δ0m = ε0m = 0m . A non dimensional impulse force applied at the center of a solid plate can be expressed as   1   fa rˆ , τ = δ rˆ δ (τ ) 2π rˆ

(6.134)

where the definition of the delta function in polar coordinates for the axisymmetric case has been used. Substituting Eq. (6.134) into Eq. (6.131) yields ∞    wˆ rˆ , τ = m=1



  Rˆ 0m (0) ˆ 0m rˆ sin 20m τ . R 2π 2 Nˆ 0m

(6.135)

0m

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

w(0, τ)

w(0, τ)

The evaluation of Eq. (6.135) for the response of the center of the plate is shown in Fig. 6.21. In Fig. 6.21a, the response was obtained by summing the modes corresponding to the lowest 15 natural frequencies and in Fig. 6.21b the response was obtained by summing the modes corresponding to the lowest 3 natural frequencies. As was the case for an Euler-Bernoulli cantilever beam shown in Fig. 3.49, only

0

0

−0.1

−0.1

−0.2

−0.2

−0.3

−0.3

−0.4

0

0.5

1

1.5

2 τ

(a)

2.5

3

3.5

4

−0.4

0

0.5

1

1.5

2 τ

2.5

3

3.5

4

(b)

Fig. 6.21 Response of the center of a solid circular plate to an impulse applied to its center (a) Response using 15 modes and (b) Response using 3 modes

6.7

Circular Plate with Concentrated Mass Revisited

389

a few modes are required to capture the main features of the response. It is noted that the non dimensional period τp of the mode associated with the lowest natural frequency is τp = 2π/201 = 0.615.

6.7 Circular Plate with Concentrated Mass Revisited We shall now use orthogonal functions to determine the natural frequencies of a clamped solid circular plate carrying a concentrated mass Mo at its center when Kf = Nˆ r = 0. This configuration is only described by axisymmetric motion since, for n > 0, the center of the plate in always on a nodal line and, therefore, the concentrated mass does not interact with these modes. To include the inertia force created by Mo , it is noted that when the mass is positioned at the center of the plate its inertial force can be expressed as Mo δ (r) ∂ 2 w 2π r ∂t2

(6.136)

where we have again used the definition of the delta function for the axisymmetric case in polar coordinates. Then the governing equation for axisymmetric motion is obtained by modifying Eq. (6.59) as   ˆr mo δ(ˆr) ∂ 2 w ˆr + 1 + =0 ∇a4 w 2 rˆ ∂τ 2

(6.137)

  where we have use the relation δ (r) = δ rˆ /|a| and Mo ρπha2 d2 d ∇a2 = 2 + . rˆ dˆr dˆr

mo =

(6.138)

To solve Eq. (6.137) for the natural frequencies, we assume a solution of the form ∞      2 Am Rˆ 0m rˆ ej τ w ˆ r rˆ , τ =

(6.139)

m=1

  where Rˆ 0m rˆ is given by Eq. (6.132) and is a solution to ∇a4 Rˆ 0m − 40m Rˆ 0m = 0

(6.140)

390

6 Thin Plates

and 2 = ωtp . The quantity 0m , m = 1, 2, . . . , are solutions to Eq. (6.133). Upon substituting Eq. (6.139) into Eq. (6.137) and using Eq. (6.140), we obtain ∞  m=1

∞     mo 4    δ(ˆr) Am 40m − 4 Rˆ 0m rˆ − Am Rˆ 0m rˆ = 0. 2 rˆ

(6.141)

m=1

  Equation (6.141) is multiplied by Rˆ 0i rˆ rˆ dˆr and integrated over rˆ , 0 ≤ rˆ ≤ 1, to arrive at , ∞ mo  4 ˆ 4 Am Rˆ 0m (0) Rˆ 0i (0) = 0 i = 1, 2, . . . (6.142) Ai 0i N0i −  Ai Nˆ 0i + 2 m=1

where we have used Eqs. (6.121) and (6.123), and from Eq. (6.122) Nˆ 0i =

1

  Rˆ 20i rˆ rˆ dˆr.

(6.143)

0

In matrix notation, Eq. (6.142) can be written as   [a] − 4 [b] {A} = 0

(6.144)

where the elements of the matrices are aij = 40i Nˆ 0i δij mo Rˆ 0i (0) Rˆ 0j (0) bij = Nˆ 0i δij + 2 T  {A} = A1 A2 · · ·

(6.145)

and the superscript T indicates the transpose of the vector. A numerical evaluation of Eq. (6.144) yields the results shown previously in Fig. 6.7. However, Eq. (6.144) converges very slowly and 13 terms in the expansion are required for the third natural frequency to be within 0.5% of those determined by Eq. (6.81) with α = 0.0001. Recall the discussion in Section 6.4.3 regarding Fig. 6.7. The differences for the first and second natural frequency coefficients are 0.25% or less.

6.8 Extensional Vibrations of Plates 6.8.1 Introduction The extensional vibrations of plates are those for which there are only in-plane displacements; that is, there are no transverse displacements. These types of vibrations are proving useful in creating high-Q filters at the micrometer scale for RF devices

6.8

Extensional Vibrations of Plates

391

(Wang et al. 2004A; Wang et al. 2004B; Lin et al. 2005). We shall determine the equations of motion and the boundary conditions for a rectangular geometry and then convert these equations to those in polar coordinates in order to consider circular plates (Love, 1947, pp. 497–498). The configuration that we are ultimately interested in is a free solid circular plate.

6.8.2 Contributions to the Total Energy We start by defining the following stress resultants for a plate of constant thickness h. Using Eqs. (6.1) and (6.2)

h/2 Tx = −h/2

h/2 Ty =

σy dz = −h/2

h/2 Txy = −h/2



 Eh  Eh σx dz = εx + υεy = 2 1−υ 1 − υ2



 Eh  Eh εy + υεx = 2 1−υ 1 − υ2

Eh Eh τxy dz = γxv = 2 (1 + υ) 2 (1 + υ)



∂v ∂u +υ ∂x ∂y ∂v ∂u +υ ∂y ∂x



 (6.146)

 ∂u ∂v + . ∂y ∂x

Strain Energy From Eqs. (6.1) and (6.2) and Eq. (A.11) of Appendix A, the strain energy is U=

Eh  2 1 − υ2





Eh  =  2 1 − υ2 1−υ + 2



εx2 + εy2 + 2υεx εy + A

" A

∂u ∂x

∂u ∂v + ∂y ∂x



2 +

∂v ∂y

 1−υ 2 γxy dA 2 

2 + 2υ

∂u ∂x



∂v ∂y

 (6.147)

2 # dA.

Kinetic Energy The kinetic energy is ρh T= 2

" A

∂u ∂t

2

 +

∂v ∂t

2 # dA.

(6.148)

392

6 Thin Plates

Minimization Function The function to be minimized is

T −U =

GdA

(6.149)

A

where "

 # ∂v 2 + ∂t "   2      # ∂v ∂u ∂u 2 Eh 1 − υ ∂u ∂v 2 ∂v  + + 2υ −  + + . ∂x ∂y ∂x ∂y 2 ∂y ∂x 2 1 − υ2

ρh G= 2

∂u ∂t

2



(6.150)

6.8.3 Governing Equations and Boundary Conditions Based on the form of G given by Eq. (6.150), the governing equations are determined from the Euler-Lagrange equation given by Case 3 of Table B.1 of Appendix B as applied to the case for N = 2, where u1 = u, and u2 = v. In addition, since G does not contain terms involving u and v, only their derivatives, the governing equations can be determined from ∂Gu,y ∂Gu,x ∂Gu˙ + + =0 ∂x ∂y ∂t ∂Gv,y ∂Gv,x ∂Gv˙ + + = 0. ∂x ∂y ∂t

(6.151)

From Eq. (6.150), it is seen that  Eh ∂v ∂u  Gu,x = −  +υ 2 ∂x ∂y 1−υ  Eh ∂u ∂v Gu,y = − + = Gv,x 2 (1 + υ) ∂y ∂x  Eh ∂u ∂v  Gv,y =  +υ 2 ∂y ∂x 1−υ Gu˙ = ρh

∂u , ∂t

Gv˙ = ρh

∂v . ∂t

(6.152)

6.8

Extensional Vibrations of Plates

393

Substituting Eq. (6.152) into Eq. (6.151) yields the following governing equations   ρ 1 − υ 2 ∂ 2u ∂ 2u 1 − υ ∂ 2u 1 + υ ∂ 2v + + = ∂x2 2 ∂y2 2 ∂x∂y E ∂t2   ρ 1 − υ 2 ∂ 2v ∂ 2v 1 − υ ∂ 2v 1 + υ ∂ 2u + + . = ∂y2 2 ∂x2 2 ∂x∂y E ∂t2

(6.153)

The boundary conditions are given by Case 3 of Table B.2 of Appendix B. It is assumed that there are no attachments to the boundaries and it is noted that G is not a function of derivatives higher than the first; therefore, these boundary conditions can be written as follows. Along the edges x = x1 and x = x2 they are either

u=0

or

 ∂v ∂u Eh   +υ = Tx = 0 2 ∂x ∂y 1−υ

either

v=0

or

 ∂u ∂v Eh + = Txy = 0. 2 (1 + υ) ∂y ∂x

(6.154a)

(6.154b)

and

The boundary conditions along the edges y = y1 and y = y2 are either

v=0

or

 ∂u ∂v Eh   +υ = Ty = 0 2 ∂y ∂x 1−υ

either

u=0

or

 Eh ∂u ∂v + = Txy = 0. 2 (1 + υ) ∂y ∂x

(6.155a)

(6.155b)

and

To solve Eq. (6.153), the following two quantities are introduced, a dilation  = ε x + εy =

∂u ∂v + ∂x ∂y

(6.156)

and a rotation 1  = 2



 ∂v ∂u − . ∂x ∂y

(6.157)

Using Eqs. (6.156) and (6.157), the first equation of Eq. (6.153) becomes   ρ 1 − υ2 ∂ 2u ∂ ∂ − (1 − υ) = ∂x ∂y E ∂t2

(6.158)

394

6 Thin Plates

and the second equation of Eq. (6.153) becomes   ρ 1 − υ2 ∂ 2v ∂ ∂ . + (1 − υ) = ∂y ∂x E ∂t2

(6.159)

If Eq. (6.158) is differentiated with respect to x and Eq. (6.159) is differentiated with respect to y and the resulting equations are added, then we obtain 2  ∇xy

  ρ 1 − υ 2 ∂ 2 = E ∂t2

(6.160)

2 is given in Eq. (6.28). If Eq. (6.158) is now differentiated with respect to where ∇xy y and Eq. (6.159) is differentiated with respect to x and the resulting equations are subtracted, then we obtain

2  = ∇xy

2ρ (1 + υ) ∂ 2  . E ∂t2

(6.161)

Free Solid Circular Plates The preceding results are converted to polar coordinates using Eq. (6.46) and noting that u → ur and v → uθ . Then the coupled equations given by Eqs. (6.158) and (6.159), respectively, become   ρ 1 − υ 2 ∂ 2 ur ∂ ∂ − (1 − υ) = ∂r r∂θ E ∂t2   ρ 1 − υ 2 ∂ 2 uθ ∂ ∂ + (1 − υ) = r∂θ ∂r E ∂t2

(6.162)

and the uncoupled equations given by Eqs. (6.160) and (6.161), respectively, become 2  ∇rθ

  ρ 1 − υ2 ∂ 2 = E ∂t2

2ρ (1 + υ) ∂ 2  2 ∇rθ  = E ∂t2

(6.163)

2 is given in Eq. (6.48) and the dilatation and rotation, respectively, given where ∇rθ by Eqs. (6.156) and (6.157) become (Love, 1947, p. 56)

ur ∂uθ ∂ur + + ∂r r r∂θ   uθ ∂ur 1 ∂uθ + − .  = 2 ∂r r r∂θ

 =

(6.164)

6.8

Extensional Vibrations of Plates

395

We are only interested in a free solid circular plate; therefore, the general boundary conditions given by Eqs. (6.154) and (6.155) for a rectangular plate simplify as follows. Equations (6.155a, b) are not applicable and Eqs. (6.154a, b), which apply only to the free edge at r = a, become Tr Trθ

   Eh ∂uθ ur ∂ur  =  =0 +υ + ∂r r∂θ r 1 − υ2 r=a  Eh ∂uθ uθ ∂ur = =0 + − 2 (1 + υ) r∂θ ∂r r r=a

(6.165)

where we have used Eq. (A.12) of Appendix A and Eq. (6.146) with εx → εe , εy → εθ , and γxy → γrθ .

6.8.4 Natural Frequencies and Mode Shapes of a Circular Plate To determine the natural frequencies and mode shapes of a free solid circular plate undergoing extensional oscillations, we start by assuming a separable solution of the form  = Dn (r) cos (nθ ) ejωt  = Wn (r) sin (nθ ) ejωt ur = Un (r) cos (nθ ) ejωt

(6.166)

uθ = Vn (r) sin (nθ ) ejωt . Another set of solutions can be assumed by replacing in Eq. (6.166) cos (nθ ) with sin (nθ ) and sin (nθ ) with cos (nθ ). Substituting Eq. (6.166) into Eqs. (6.163) and (6.162), respectively, we obtain  κ12 n2 − 2 Dn = 0 a2 r   κ22 d 2 Wn dWn n2 + + − 2 Wn = 0 rdr dr2 a2 r d 2 Dn dDn + + 2 dr rdr



(6.167)

and κ12 nWn dDn + (1 − υ) Un = − dr r a2 κ12 nDn dWn Vn = − (1 − υ) r dr a2

(6.168)

396

6 Thin Plates

where   ρ 1 − υ 2 a2 ω2 = E 2 κ 2. κ22 = (1 − υ) 1 κ12

(6.169)

The solutions to Eq. (6.167) that remain finite at r = 0 are Dn (r) = A n Jn (κ1 r/a)

(6.170)

Wn (r) = B n Jn (κ2 r/a)

where Jn (x) is the Bessel function of the first kind of order n and A n and B n are unknown constants. Substituting Eq. (6.170) into Eq. (6.168), it is found that d nBn Jn (κ1 r/a) + Jn (κ2 r/a) dr r (6.171) d nAn Vn = − Jn (κ1 r/a) − Bn Jn (κ2 r/a) r dr where An = −A n a2 κ12 and Bn = 2B n a2 κ22 . The characteristic equation is obtained by substituting Eq. (6.171) into the last two equations of Eq. (6.166) and the result, in turn, into Eq. (6.165). These operations result in Un = An

,

g11 (κ1 , n, υ)

g12 (κ2 , n, υ)

g21 (κ1 , n, υ)

g22 (κ2 , n, υ)

-"

An Bn

# =0

(6.172)

where   g11 (κ1 , n, υ) = − (1 − υ) κ1 Jn (κ1 ) + (1 − υ) n2 − κ12 Jn (κ1 )   g12 (κ2 , n, υ) = n (1 − υ) κ2 Jn (κ2 ) − Jn (κ2 )   g21 (κ1 , n, υ) = 2n −κ1 Jn (κ1 ) + Jn (κ1 )   g22 (κ2 , n, υ) = 2κ2 Jn (κ2 ) + κ22 − 2n2 Jn (κ2 )

(6.173)

and the prime denotes the derivative with respect to its argument. In arriving at Eq. (6.173), Eq. (6.167) has been used. The characteristic equation from which the natural frequency coefficients κ1 = κ1nm are determined is obtained by setting the determinant of the coefficients of Eq. (6.172) to zero; thus, g11 (κ1nm , n, υ) g22 (κ2nm , n, υ) − g12 (κ2nm , n, υ) g21 (κ1nm , n, υ) = 0.

(6.174)

6.8

Extensional Vibrations of Plates

397

The corresponding mode shapes are determined from Eqs. (6.166), (6.171), and (6.172) as  nCnm ur,nm (η, θ ) = κ1nm Jn (κ1nm η) + Jn (κ2nm η) cos (nθ ) η  n Jn (κ1nm η) + κ2nm Cnm Jn (κ2nm η) sin (nθ ) uθ,nm (η, θ ) = η

(6.175)

√ 2κ1nm / (1 − υ), η = r/a, the prime denotes the derivative with where κ2nm = respect to its argument, and Cnm = −

g11 (κ1nm , n, υ) . g12 (κ2nm , n, υ)

(6.176)

When n = 0, Eq. (6.174) simplifies to [κ1 J0 (κ1 ) − (1 − υ) J1 (κ1 )] [κ2 J0 (κ2 ) − 2J1 (κ2 )] = 0.

(6.177)

The values of κ 10m that satisfy κ1 J0 (κ10m ) − (1 − υ) J1 (κ10m ) = 0

(6.178)

correspond to the case of in-plane radial motions only (Love, 1947). In this case, the shape of the plate remains circular.

6.8.5 Numerical Results Equations (6.174) and (6.178) have been evaluated to determine the values of κ 1nm as a function of Poisson’s ratio, υ (Onoe 1956). The results have been plotted in Fig. 6.22, where it is seen that κ 1nm is somewhat dependent on υ . In practice, there are two modes of vibration that are of interest: the purely radial mode (n = 0) and the so-called “wine glass” mode (n = 2). The natural frequency coefficient for the radial mode is κ 10m and its values are obtained from Eq. (6.178). The natural frequency coefficient for the “wine glass” mode is κ 12m and its values are obtained from Eq. (6.174). The corresponding mode shapes are shown in Fig. 6.23. The “wine glass” mode is useful in that the four node points are locations where the plate can be supported while minimally affecting the natural frequency. It is noted that for, say υ = 0.25, κ121 /κ101 = 1.4357/2.0172 = 0.712. Thus, for this value of Poisson’s ratio, the lowest value of the natural frequency associated with the “wine glass” mode is about 29% lower than that associated with the purely radial mode.

398

6 Thin Plates 6 κ102

5.5 5 4.5

κ1 nm

4 κ112 3.5 3

κ122

2.5 κ101

2

κ111

1.5 1

κ121 0

0.1

0.2

0.3

0.4

0.5

ν

Fig. 6.22 Lowest two natural frequency coefficients κ 1nm as a function υ for n = 0, 1, and 2

(a)

(b)

Fig. 6.23 (a) Mode shape for the radial extensional mode corresponding to κ 101 and (b) Mode shape for the “wine glass” extensional mode corresponding to κ 121 . Circle with dashed line is the original shape

References

399

Appendix 6.1 Elements of Matrices in Eq. (6.100) The elements of [B] and {A} are ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

B1111 . . . B111 N .. . B11N1 . . . B11NN

B1211 · · · B1N11 B2111 B2211 . . . B121 N .. . B12N1 . . . B12NN

· · · B2N11 · · · BN111 BN211 · · · BNN11 . . . ··· BNN1N .. . ··· BNNN1 . . . ··· BNNNN

⎫ ⎤⎧ A11 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ . ⎥⎪ . ⎪ . ⎪ ⎪ ⎪ ⎥⎪ ⎪ ⎪ ⎪ ⎥⎪ ⎥ ⎨ A1. N ⎬ ⎥ ⎥ ⎪ .. ⎪ . ⎪ ⎥⎪ ⎪ ⎪ AN1 ⎪ ⎥⎪ ⎪ . ⎪ ⎦⎪ ⎪ ⎪ .. ⎪ ⎪ ⎪ ⎭ ⎩ ANN

(6.179) The elements of Cnmlk are obtained by replacing Bnmlk with Cnmlk in Eq. (6.179).

References Leissa AW (1969) Vibration of plates. NASA SP-160 Lin Y-W, Li S-S, Ren Z, Nguyen CT-C (2005) Vibrating micromechanical resonators with solid dielectric capacitive-transducer ‘gaps’. Proceedings, Joint IEEE International Frequency Control/Precision Time & Time Interval Symposium, Vancouver, Canada, pp 128–134 Love AEH (1947) A treatise on the mathematical theory of elasticity, 4th edn. Dover, New York, NY, pp 497–498 Onoe M (1956) Contour vibrations of isotropic circular plates. J Acoust Soc Am 28:1158–1162 Roberson RE (1951) Transverse vibrations of a free circular plate carrying concentrated mass. ASME J Appl Mech 18(3):280–282 Wang J, Butler JE, Feygelson T, Nguyen CT-C (2004A) 1.51-GHz nanocrystalline diamond micromechanical disk resonator with material-mismatched isolating support. Proceedings 17th International IEEE MEMS Conference, Maastricht, The Netherlands, pp 641–644 Wang J, Ren Z, Nguyen CT-C (2004B) 1.156-GHz self-aligned vibrating micromechanical disk resonator. IEEE Trans Ultrason Ferroelectri Freq Control 51(12):1607–1628 Young D (1950) Vibration of rectangular plates by the Ritz method. ASME J Appl Mech 17(4):448–453

Chapter 7

Cylindrical Shells and Carbon Nanotube Approximations

The Flügge and Donnell theories for thin cylindrical shells are used to obtain the approximate natural frequencies and mode shapes of single-wall and double-wall carbon nanotubes.

7.1 Introduction Vibrations of carbon nanotubes are important in a number of nano-mechanical devices such as oscillators, charge detectors, clocks, field emission devices, and sensors (Gibson et al. 2007). Electron microscope observations of vibrating carbon nanotubes have been used indirectly and nondestructively to determine the effective elastic modulus and other aspects of mechanical behavior of carbon nanotubes. There are two approaches to modeling carbon nanotubes: atomistic and continuum. Atomistic approaches include classical molecular dynamics simulation and tight binding molecular dynamics simulation methods (Qian et al. 2002). These atomistic methods tend to be very computationally intensive. We shall confine our investigations to the continuum models, which use equivalent material properties to represent the material constants in these models. The reason for having to use equivalent material properties is that a nanotube is essentially a structure comprised of carbon atoms in several different configurations. Such properties as density, thickness, and tensile modulus do not directly apply. We shall continue the discussion of the choice of equivalent values when numerical results are presented in Section 7.4.2. Two shell theories that can be used to approximate single-wall and double-wall carbon nanotubes are introduced: Flügge’s theory and Donnell’s theory. We shall obtain the governing equations and boundary condition for these two shell theories and then use them to estimate the natural frequencies of carbon nanotubes.

E.B. Magrab, Vibrations of Elastic Systems, Solid Mechanics and Its Applications 184, C Springer Science+Business Media B.V. 2012 DOI 10.1007/978-94-007-2672-7_7, 

401

402

7 Cylindrical Shells and Carbon Nanotube Approximations

7.2 Derivation of Governing Equations and Boundary Conditions: Flügge’s Theory 7.2.1 Introduction Consider the element of a circular cylindrical shell of constant thickness h and middle surface radius a as shown in Fig. 7.1. If the total displacement normal to the shell surface is denoted W (x, θ , t), the total in-plane displacement in the x-direction is denoted Ux (x, θ , ξ , t), and the total in-plane displacement in the θ -direction is denoted Uθ (x, θ , ξ , t), then referring to Fig. 7.1 it is seen that these quantities are given by Ux (x, θ , ξ , t) = ux (x, θ , t) + ξβx (x, θ , t) Uθ (x, θ , ξ , t) = uθ (x, θ , t) + ξβθ (x, θ , t) W (x, θ , ξ , t) = w (x, θ , t)

(7.1)

where ux and uθ , respectively, represent the stretching of the middle surface of the shell in the x and θ directions and β x and β θ are the rotations of the normal to the

Nx

dθ

Nxθ Mx

Mxθ

dξ

ξ

Middle surface

a

σx τθx

Nθx

ux

Mθ Nθ

βθ

Vx W

Mθx

σθ

dsx = dx

uθ

Fig. 7.1 Differential element of a cylindrical shell and the force resultants and moment resultants

h

dAθ = (a + ξ)dθdξ

τθx βx dAx = dxdξ

Uθ

dsθ = adθ Txθ

Ux

qn

7.2

Derivation of Governing Equations and Boundary Conditions: Flügge’s Theory

403

middle surface as shown in the figure; that is, they are the rotations due to the transverse shear. It seen in Eq. (7.1) that it has been assumed that there is no change in thickness of the shell; that is, W is independent of ξ . Equation (7.1) is a consequence of Kirchhoff’s hypothesis, which assumes that the shear strains through the thickness of the shell γxξ = γθξ = 0 and that the strain eξ = 0. From Eq. (7.1), it is seen that βx =

∂Ux , ∂ξ

βθ =

∂Uθ . ∂ξ

It can be shown (Leissa 1973) that in order for γxξ = γθξ = 0 ∂w βx = − , ∂x  1 ∂w uθ − . βθ = a ∂θ

(7.2)

The linear strain displacement relations for a cylindrical shell are given by (Leissa 1973) ex =

∂Ux ∂x

  1 ∂Uθ eθ = +W a (1 + ξ/a) ∂θ ∂Ux ∂Uθ 1 + . γxθ = a (1 + ξ/a) ∂θ ∂x

(7.3)

Upon substituting Eqs. (7.1) and (7.2) into Eq. (7.3) and introducing the notation u¯ x =

ux , a

u¯ θ =

uθ , a

w¯ =

w , a

η=

x a

(7.4)

we obtain ex = εx0 + ξ χx 

1 eθ = εθ0 + ξ χθ (1 + ξ/a) 

1 0 + ξ (1 + ξ/ (2a)) τs γxθ = εxθ (1 + ξ/a) where

(7.5)

404

7 Cylindrical Shells and Carbon Nanotube Approximations

∂ u¯ x , ∂η 1 ∂ 2 w¯ χx = − , a ∂η2

∂ u¯ θ +w ¯ ∂θ  1 ∂ u¯ θ χθ = − a ∂θ  ∂ u¯ x 2 ∂ u¯ θ ∂ u¯ θ = + , τs = − ∂θ ∂η a ∂η

εx0 =

0 εxθ

εθ0 =

∂ 2 w¯ ∂θ 2



¯ ∂ 2w ∂η∂θ

(7.6)  .

0 represent the normal and shear strains in the middle The quantities εx0 , εθ0 , and εxθ surface of the shell, as can be confirmed by setting ξ = 0 in Eq. (7.5). The quantities χ x and χ θ represent the curvature of the middle surface of the shell during deformation. The quantity τs represents the change in twist of the middle surface of the shell. In the theory of thin shells, it is usually assumed1 that σξ = 0. If we let y correspond to θ , then from Eqs. (A.1) and (A.6) of Appendix A, the stress-strain relations are

E  (ex + νeθ ) 1 − ν2 E  (eθ + νex ) σθ =  1 − ν2 σx = 

(7.7)

τxθ = Gγxθ where ν is Poisson’s ratio and E is Young’s modulus. Upon substituting Eq. (7.5) into Eq. (7.7), we obtain   0 νε νξ χ E θ θ  εx0 + ξ χx + + σx =  (1 + ξ/a) (1 + ξ/a) 1 − ν2   εθ0 E ξ χθ 0  νεx + ξ νχx + σθ =  + (1 + ξ/a) (1 + ξ/a) 1 − ν2   0 εxθ ξ (1 + ξ/ (2a)) τxθ = G + τs . (1 + ξ/a) (1 + ξ/a)

(7.8)

In anticipation of what is to follow, we introduce the following definitions for force resultants per unit length on the middle surface of the shell and the moments per unit length on the middle surface of the shell. Referring to Fig. 7.1, the force resultants per unit length are defined as

1 This assumption leads to an inconsistency in the theory, since, having already assumed that eξ = 0, it is straightforward to show that σξ = ν (σx + σθ ).

7.2

Derivation of Governing Equations and Boundary Conditions: Flügge’s Theory



Nx Nxθ



h/2 = −h/2

σx τxθ



dAθ = dsθ

h/2 −h/2

σx τxθ

  ξ 1+ dξ a

405

(7.9)

and

Nθ Nθx



h/2 = −h/2

σθ τθx



dAx = dsx

h/2 −h/2

 σθ dξ . τθx

(7.10)

The moments per unit length are defined as



Mx Mxθ

h/2 = −h/2

σx τxθ



dAθ ξ = dsθ

h/2 −h/2

σx τxθ

  ξ ξ dξ 1+ a

(7.11)

and

Mθ Mθx



h/2 = −h/2

 

h/2 dAx σθ σθ ξ ξ dξ . = τθx τθx dsx

(7.12)

−h/2

As shown in Fig. 7.1, we have used the following relations: dsθ = adθ, dsx = dx, dAθ = (a + ξ ) dθ dξ , and dAx = dxdξ . Upon substituting Eq. (7.8) into Eq. (7.9), we arrive at E  Nx =  1 − ν2 = E˜

εx0

h/2  εx0 (1 + ξ/a) + χx ξ (1 + ξ/a) + νεθ0 + νξ χθ dξ −h/2

+ ha aχx + νεθ0



= E˜ N¯ x Nxθ

(7.13)

h/2  0 εxθ =G + ξ (1 + ξ/ (2a)) τs dξ −h/2



= Gh

0 εxθ

   ha a ha a 0 ˜ + τs = EKo εxθ + τs 2 2

˜ o N¯ xθ = EK where Eh  N/m, E˜ =  1 − ν2

Ko =

1−ν , 2

ha =

h2 5 and for both sets of boundary conditions and for both values of a/h. Consequently, the Timoshenko beam theory can be used as a first approximation to the lowest n = 1 mode of a thin cylindrical shell described by Flügge’s theory. In Figs. 7.4 and 7.5, respectively, several mode shapes are shown for a cantilever shell and a shell clamped at both ends using Flügge’s theory. These results are in qualitative agreement with (Sakhaee-Pour et al. 2009; Li and Chou 2004). The mode shapes were obtained by solving Eq. (7.81) for the natural frequency coefficients l and the modal coefficients Anm,l , Bnm,l , and Cnm,l . These coefficients were then used in Eqs. (7.75) and (7.71) to determine the mode shapes. A value of M = 7 was used

428 Fig. 7.4 Mode shapes for the lowest three natural frequencies nm for a cantilever shell using Flügge’s theory (a) n = 1 (b) n = 2 and (c) n = 3

7 Cylindrical Shells and Carbon Nanotube Approximations Top view

Δ11 = 0.0104

Front view

Free end

Top view

Δ12 = 0.0573

Front view

Free end

Top view

Δ13 = 0.138

Front view

Free end

(a)

to obtain these results. It is seen that the axial motion has a significant effect on the mode shapes of a cantilever cylindrical shell in that it causes a relatively substantial distortion of the free end in the axial direction. The effects of Poisson’s ratio on the natural frequency coefficients using Flügge’s theory are shown in Figs. 7.6 and 7.7. It is seen that Poisson’s ratio influences the natural frequency coefficient for n = 1 and has less effect for n ≥ 2 and L/a > 8. Application to Carbon Nanotubes The equations from which the natural frequency coefficients l are determined are a function of the ratio of the shell thickness to the radius of its middle surface h/a, Poisson’s ratio ν, the circumferential wave number n, and the shell length to radius

7.4

Natural Frequencies of Clamped and Cantilever Shells: Single-Wall. . .

Fig. 7.4 (continued)

Top view

Δ21 = 0.0261

429 Front view

Free end

Top view

Δ22 = 0.0335

Front view

Free end

Top view

Δ23 = 0.0613

Front view

Free end

(b)

ratio L/a. To convert the result to a frequency in Hz for a specific carbon nanotube, one needs an estimate of the equivalent thickness h, the equivalent Young’s modulus E, the equivalent Poisson’s ratio ν, and the equivalent density ρ. There have been numerous attempts at estimating these values by various methods (Huang et al. 2006; Gupta and Batra 2008; Lee and Oh 2008; Thostenson et al. 2001; Popov et al. 2000). These quantities have a range of values that depend on the method used and the model of the orientation of the carbon atoms in the nanotube: armchair or zigzag. From the references cited, the thickness range is 0.066 ≤ h ≤ 6.8 nm with the most frequently appearing value being 0.34 nm. The range of the Young’s modulus is 0.6 ≤ E ≤ 1.4 TPa with a few outliers of 0.2, 3.4 and 5.5 TPa. In numerical works, a value of 1 TPa is frequently assumed. The range of Poisson’s ratio is

430 Fig. 7.4 (continued)

7 Cylindrical Shells and Carbon Nanotube Approximations Top view

Δ31 = 0.0731

Front view

Free end

Top view

Δ32 = 0.0746

Front view

Free end

Top view

Δ33 = 0.0801

Front view

Free end

(c)

0.14 ≤ ν ≤ 0.43. A value of ν = 0.2 to 0.3 is often assumed. For density, a value of ρ = 2300 kg/m3 is most often assumed. With regard to the approximate range of the geometric parameters, 4 ≤ h/a ≤ 30 and 10 ≤ L/a ≤ 200 are what have been considered experimentally. To obtain an estimate of a typical value of the natural frequency of a carbon nanotube in Hz, the following parameters are selected from the ranges given above and from Figs. 7.2 and 7.3:  = 0.05, E = 1 TPa, a = 1.5 nm, ρ = 2300 kg/m3 , and ν = 0.25. From Eq. (7.28), it is found that to = 6.97 × 10−14 s. Then, from the definition of , f1 = /(2π to ) = 114.3 gHz.

7.4

Natural Frequencies of Clamped and Cantilever Shells: Single-Wall. . . Δ11 = 0.0558

Top view

431

Top view

Δ12 = 0.134

Top view

Δ22 = 0.0605

(a)

Δ21 = 0.0333

Top view

(b)

Fig. 7.5 Mode shapes for the lowest two natural frequencies nm for a shell clamped at each end using Flügge’s theory (a) n = 1 and (b) n = 2

1

1

n=3

0.1

0.1 n=2

Δn1

Δn1

n=3 n=2

ν = 0.15 ν = 0.4

0.01

ν = 0.15 ν = 0.4

0.01

n=1

0.001

1

2

3

0.001

n=1

L /a

4 5 6 7 8910 L /a

(a)

(b)

4 5 6 7 8910

20

30 40 50

1

2

3

20

30 40 50

Fig. 7.6 Natural frequency coefficients n1 for a cantilever shell according to the Flügge theory as a function of L/a for v = 0.15 and 0.4 and for n = 1, 2, 3 (a) a/h = 15 and (b) a/h = 50

432

7 Cylindrical Shells and Carbon Nanotube Approximations

1

1

n=3

0.1

0.1 n=3

Δn1

Δn1

n=2

n=2

ν = 0.15

0.01

n=1

ν = 0.4

0.001

ν = 0.15

0.01

n=1

ν = 0.4

0.001 1

2

3

4 5 6 7 8 910 L /a

20

30 40 50

1

2

3

4 5 6 7 8 9 10 L/a

20

30 40 50

(b)

(a)

Fig. 7.7 Natural frequency coefficients n1 for a shell clamped at both ends according to the Flügge theory as a function of L/a for v = 0.15 and 0.40 and for n = 1, 2, 3 (a) a/h = 15 and (b) a/h = 50

7.5 Natural Frequencies of Hinged Shells: Double-Wall Carbon Nanotube Approximation We shall represent a double-wall carbon nanotube by two co-axial cylindrical shells of the same length, same thickness, and same physical properties and assume that they are coupled by the van der Waals interaction force as shown in Fig. 7.8. When the two shells are positioned such that their respective ends lie in the same plane, the van der Waals interaction force (Ru 2001; Natsuki and Morinobu 2006) between them can be represented by a spring-like force that is proportional to the net displacement of the radial displacements of the two shells. We denote the displacement of the outer shell w(1) and assume that the radius of its middle surface is a1 ; likewise, we denote the displacement of the inner shell w(2) and  assume  that the radius of its middle surface is a2 . In addition, it is assumed that co N/m3 is a constant that

w(1)

co w(2)

h

a2 a1

Fig. 7.8 Geometry of two coaxial nanotubes coupled by the van der Waals force

h

7.5

Natural Frequencies of Hinged Shells: Double-Wall Carbon Nanotube. . .

433

represents the equivalent stiffness of the van der Waals force over the surface of the shell at their static equilibrium separation distance. Values for co will be discussed subsequently. If it is assumed that w(1) > w(2) , then the pressure on the interior surface of the outer shell is given by

 (7.97) qn(1) = −co w(1) − w(2) and the pressure on the exterior surface of the inner shell is given by

 qn(2) = co w(1) − w(2) .

(7.98)

To obtain the governing equations, we write Eq. (7.35) in dimensional form and use Eqs. (7.97) and (7.98) to arrive at (s)

a2s

∂ 2 uθ ∂ 2 u(s) ∂ 2 u(s) ∂w(s) x x + K + a K + νa o s p s ∂x∂θ ∂x ∂x2 ∂θ 2   2 (s) 2ρ 1 − ν2 2 3 w(s)  a ∂ ux h ∂ 3 w(s) ∂ s 3 − as =0 + as Ko − 2 2 3 E 12as ∂x∂θ ∂x ∂t2 (s)

(s)

(s) ∂ 2 uθ ∂ 2 uθ ∂ 2 ux ∂w(s) + a2s Ko + + ∂x∂θ ∂θ ∂x2 ∂θ 2   2 as ρ 1 − ν 2 ∂ 2 u(s) h2 ∂ 3 w(s) θ − (1 + Ko ) − =0 2 ∂θ 2 12 E ∂x ∂t " (s) 3 (s) ∂ 3 u(s) ∂ 2 w(s) h2 ∂ 3 ux θ 3 ∂ ux 2 − a K + a + K −2 a (1 ) s o o s s 2 3 2 2 12as ∂x ∂x∂θ ∂x ∂θ ∂θ 2 #  (s) 4 (s) 4 (s) ∂ux ∂ 4 w(s) 4∂ w 2 ∂ w − νa − as + 2a + s s ∂x4 ∂x2 ∂θ 2 ∂θ 4 ∂x     (s) 2 2  a2s ρ 1 − ν 2 ∂ 2 w(s) ∂uθ (s) s−1 co as 1 − ν = (−1) − −w − w(1) − w(2) 2 ∂θ E ∂t Eh s = 1, 2. (7.99)

as Kp

The displacement of each shell is normalized with respect to a1 and we introduce the following quantities (s) u(s) ux w(s) (s) , u¯ θ = θ , w¯ (s) = , a1 a1 a1   ρa2 1 − ν 2 h2 t = , t12 = 1 , τ= 2 E t1 12a1

u¯ (s) x = h a1

c¯ o =

co a21 , E˜

η=

x , a1

αs =

as ≤1 a1

s = 1, 2.

(7.100)

434

7 Cylindrical Shells and Carbon Nanotube Approximations

Then, substituting Eq. (7.100) into Eq. (7.99), we obtain the following two sets of coupled partial differential equations (s)

αs2

(s) (s) ∂ 2 u¯ θ ∂ 2 u¯ x ∂ 2 u¯ x ∂w ¯ (s) + να + K + α K o s p s ∂η∂θ ∂η ∂η2 ∂θ 2  3 (s) 3 (s) 2 ¯ (s) ha ∂ w¯ ¯ x 3∂ w 2∂ u + 21 αs Ko − α − α =0 s s αs ∂η∂θ 2 ∂η3 ∂τ 2 (s)

αs Kp

(s)

(s) ∂ 2 u¯ θ ∂ 2 u¯ θ ∂ 2 u¯ x ∂ w¯ (s) + + + αs2 Ko 2 2 ∂η∂θ ∂η ∂θ ∂θ (s)

∂ 2 u¯ θ ∂ 3 w¯ (s) =0 − (1 + Ko ) ha1 2 − αs2 ∂η ∂θ ∂τ 2 " (s) 3 ¯ (s) ∂ 3 u¯ (s) ∂ 2 w¯ (s) ∂ 3 u¯ x ha1 x θ 3∂ u 2 − 2 α − α K + α + K (1 ) s o o s s αs2 ∂η3 ∂η∂θ 2 ∂η2 ∂θ ∂θ 2 #   (s) (s) 4 ¯ (s) ∂ u¯ θ ∂ 4 w¯ (s) ∂ u¯ x ∂ 4 w¯ (s) 2 ∂ w − να − αs4 + 2α + − s s ∂η4 ∂η2 ∂θ 2 ∂θ 4 ∂η ∂θ

 2 (s) ∂ w¯ = (−1)s−1 c¯ o αs2 w¯ (1) − w ¯ (2) − w¯ (s) − αs2 ∂τ 2 s = 1, 2.

(7.101)

To reduce the complexity of the solution somewhat and still retain the important characteristics of the interactions of a double-wall carbon nanotube, we shall consider the hinged boundary conditions given by Eq. (7.41). A solution that satisfies these boundary conditions at η = 0 and η = L/a1 is (Leissa 1973; Wang et al. 2005)

u¯ (s) x (η, θ , t) = (s)

u¯ θ (η, θ , t) = w¯ (s) (η, θ , t) =

∞ ∞   m = 1 n=1 ∞ ∞   m = 1 n=1 ∞  ∞ 

jτ A(s) mn cos (λm η) cos nθ e

jτ B(s) mn sin (λm η) sin nθ e

(7.102)

(s) Cmn sin (λm η) cos nθ e jτ

m = 1 n=1

where  = ωt1 , λm =

mπ a1 L

(7.103)

(s) (s) and A(s) mn , Bmn and Cmn are determined by substituting Eq. (7.102) into Eq. (7.101). This substitution results in the following system of equations

7.5

Natural Frequencies of Hinged Shells: Double-Wall Carbon Nanotube. . .



 [H] + 2 [K] {G}

435

(7.104)

where ⎡

(1) (1) a1mn b1mn

(1)

c1mn

0

0

0

⎤

⎢ ⎥ ⎢ (1) (1) ⎥ (1) 0 0 0 ⎢ a2mn b2mn c2mn ⎥ ⎢ ⎥ ⎢ (1) (1) (1) ⎥ 0 c¯ o ⎢ a3mn b3mn c3mn − c¯ o 0 ⎥ ⎢ ⎥ [H] = ⎢ ⎥ (2) (2) (2) ⎢ 0 ⎥ 0 0 a1mn b1mn c1mn ⎢ ⎥ ⎢ ⎥ (2) (2) (2) ⎢ 0 ⎥ 0 0 a b c 2mn 2mn 2mn ⎣ ⎦ (2) (2) (2) 2 2 0 0 α2 c¯ o a3mn b3mn c3mn − α2 c¯ o ⎫ ⎧ (1) ⎪ ⎡ ⎤ ⎪ Amn ⎪ ⎪ 100 0 0 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎢ ⎪ ⎥ (1) ⎪ ⎪ ⎪ B ⎪ ⎢0 1 0 0 0 0 ⎥ ⎪ mn ⎪ ⎪ ⎪ ⎢ ⎪ ⎥ ⎪ ⎪ ⎪ ⎢ ⎪ ⎥ (1) ⎨ 0 0 1 0 0 0 Cmn ⎬ ⎢ ⎥ ⎢ ⎥ [K] = ⎢ ⎥ {G} = ⎪ (2) ⎪ ⎢ 0 0 0 α22 0 0 ⎥ ⎪ A ⎪ ⎪ ⎪ ⎢ ⎪ mn ⎪ ⎥ ⎪ ⎪ ⎢ 0 0 0 0 α2 0 ⎥ ⎪ (2) ⎪ ⎪ ⎪ ⎪ ⎪ ⎣ ⎦ 2 B ⎪ ⎪ mn ⎪ ⎪ ⎪ ⎪ ⎪ (2) ⎪ ⎭ ⎩ 0 0 0 0 0 α22 Cmn

(7.105)

and 2 2 2 a(s) 1mn = − αs λm − n Ko (s)

b1mn = αs nλm Kp (s)

c1mn = ναs λm +

 h a1  2 3 3 −α n λ K + α λ s m o s m αs2

(s) a(s) 2mn = b1mn (s)

b2mn = − Ko αs2 λ2m − n2

(7.106)

(s)

c2mn = − n − (1 + Ko ) nha1 λ2m (s)

(s)

(s)

(s)

a3mn = c1mn b3mn = c2mn

    (s) c3mn = − 1 − ha1 −2n2 αs2 + αs2 λ4m + 2n2 λ2m + n4 αs2 .

It is seen that Eq. (7.104) is a standard eigenvalue formulation that can be solved by readily available procedures yielding, for each value of n, 6 natural frequency coefficients nm , m = 1, 2, . . . , 6.

436

7 Cylindrical Shells and Carbon Nanotube Approximations

A value for co has been estimated as2 (Ru 2001) co ≈ 0.3 (nm)−2 . Eh   Then, from Eq. (7.100), c¯ o ≈ 0.3a21 1 − ν 2 . Consequently, for 2 ≤ a1 ≤ 10 nm and ν = 0.3, 1 ≤ c¯ o ≤ 27. Using a value of c¯ o = 5, the numerical solutions to the characteristic equation obtained from Eq. (7.104) are shown in Fig. 7.9 for α2 = 0.85, ν = 0.3, n = 1, 2, 3, and a/h = 15 and 50. It is seen from these figures that the individual curves are similar to those obtained for a single-wall nanotube, even though the boundary conditions are different. It is noted that for each value of n, the natural frequency coefficients for the double-wall nanotube lie virtually mid way between the values of the natural frequency coefficients for the two uncoupled single-wall nanotubes, that is, for the cases when c¯ o = 0. The designation of being mid way on a logarithmic scale is equivalent to the geometric mean; that is, the coupled natural frequency coefficient is the square root of the product of the two uncoupled natural frequency values. Recall Eq. (2.22). 0.1

1 Outer shell; co = 0

Outer shell; co = 0

Inner shell; co = 0

Inner shell; co = 0 Double-wall shell; co = 5

0.01

n=2

0.1

n=3

Δn1

Δn1

Double-wall shell; co = 5 n=3

n=2

0.01 n=1 n=1 0.001

5

6 7 8 910

20 L /(ma1)

(a)

30

40 50 60 7080

0.001

5

6 7 8 910

20 L /(ma1)

30

40 50 60 7080

(b)

Fig. 7.9 Natural frequency coefficients n1 for a double-wall shell hinged at both ends according to the Flügge’s theory as a function of L/(ma1 ) = π/λm for v = 0.3, c¯ o = 5.0, α2 = 0.85, and n = 1, 2, 3 (a) a/h = 15 and (b) a/h = 50

References Gibson RF, Ayorinde EO, Wen Y-F (2007) Vibrations of carbon nanotubes and their composites: a review. Compos Sci Technol 67:1–28 Gupta SS, Batra RC (2008) Continuum structures equivalent in normal mode vibrations to singlewalled carbon nanotubes. Comput Mater Sci 43:715–723

2 A formula that relates the van der Waals interaction force between cylindrical surfaces and is an explicit function of the radii of the nanotubes can be found in (He et al. 2005).

References

437

He XQ, Kitipornchai S, Liew KM (2005) Buckling analysis of multi-walled carbon nanotubes: a continuum model accounting for van der Waals interaction. J Mech Phys Solids 53:303–326 Huang Y, Wu J, Hwang KC (2006) Thickness of graphene and single-wall carbon nanotubes. Phys Rev B 74:245413 Lee U, Oh H (2008) Evaluation of the structural properties of single-walled carbon nanotubes using a dynamic continuum modeling method. Mech Adv Mater Struct 15:79–87 Leissa AW (1973) Vibration of shells. NASA SP-288:Chapter 1 Li C, Chou T-W (2004) Vibrational behaviors of multiwalled-carbon-nanotube-based nanomechanical resonators. Appl Phys Lett 84(1):121–123 Natsuki T, Morinobu E (2006) Vibration analysis of embedded carbon nanotubes using wave propagation approach. J Appl Phys 99:034311 Popov VN, Van Doren VE, Balkanski M (2000) Elastic properties of single-walled carbon nanotubes. Phys Rev B 61(4):3078–3084 Qian D, Wagner JG, Liu WK, Yu MF, Ruoff RS (2002) Mechanics of carbon nanotubes. Appl Mech Rev 55(6):495–533 Ru CQ (2001) Degraded axial buckling strain of multiwalled carbon nanotubes due to interlayer slips. J Appl Phys 89(6):3426–3433 Sakhaee-Pour A, Ahmadian MT, Vafai A (2009) Vibrational analysis of single-walled carbon nanotubes using beam element. Thin-Walled Struct 47(6–7):646–652 Thostenson ET, Ren Z, Chou T-W (2001) Advances in the science and technology of carbon nanotubes and their composites: a review. Compos Sci Technol 61:1899–1912 Wang CY, Ru CQ, Mioduchowski A (2005) Free vibration of multiwall carbon nanotubes. J Appl Phys 97:114323

Appendix A Strain Energy in Linear Elastic Bodies

Stress-Strain Relations The relations between stress and strain in the Cartesian coordinate system are (Sokolnikoff 1956, Chapter 3)    E (1 − ν) εx + ν εy + εz , (1 + ν) (1 − 2ν)   E σy = (1 − ν) εy + ν (εx + εz ) , (1 + ν) (1 − 2ν)    E σz = (1 − ν) εz + ν εx + εy , (1 + ν) (1 − 2ν)

σx =

τxy = Gγxy τzy = Gγzy

(A.1)

τxz = Gγxz

where σx , σy , and σz are the normal stresses, εx , ε y , and ε z are the normal strains, τxy , τyz , and τzx are the shear stresses, γxy , γyz , and γzx are the shear strains, E is the Young’s modulus, ν is Poisson’s ratio, and G is the shear modulus which is related to the Young’s modulus as G=

E . 2 (1 + ν)

(A.2)

In Cartesian coordinates, the strains are related to the displacements by ∂u , ∂x ∂v εy = , ∂y ∂w , εz = ∂z

εx =

∂v ∂w + ∂z ∂y ∂w ∂u γzx = + ∂x ∂z ∂u ∂v γxy = + ∂y ∂x γyz =

(A.3)

where u = u (x, y, z), v = v (x, y, z), and w = w (x, y, z), respectively, are the displacements in the x, y, and z directions.

E.B. Magrab, Vibrations of Elastic Systems, Solid Mechanics and Its Applications 184, C Springer Science+Business Media B.V. 2012 DOI 10.1007/978-94-007-2672-7, 

439

440

Appendix A: Strain Energy in Linear Elastic Bodies

For the case where σz = 0, we obtain from Eq. (A.1) that    E (1 − ν) εx + ν εy + εz (1 + ν) (1 − 2ν)   E σy = (1 − ν) εy + ν (εx + εz ) (1 + ν) (1 − 2ν)    E 0= (1 − ν) εz + ν εx + εy . (1 + ν) (1 − 2ν) σx =

(A.4)

From the third equation of Eq. (A.4), it is seen that εz = −

 ν  εx + εy . 1−ν

(A.5)

Substituting Eq. (A.5) into the first two equations of Eq. (A.4), we obtain E 1 − ν2 E σy = 1 − ν2 σx =

  εx + νεy   εy + νεx .

(A.6)

Equation (A.6) is used in the determination of the strain energy of thin plates and shells. Equation (A.1) can be solved for the strains in terms of the stress, to yield 1 E 1 εy = E 1 εz = E

εx =

   σx − ν σy + σz ,   σy − ν (σx + σz ) ,    σz − ν σx + σy ,

τxy G τxz γxz = G τzy γzy = . G γxy =

(A.7)

Total Strain Energy The total strain energy in a deformed elastic body of volume V is 1 U= 2





 σx εx + σy εy + σz εz + τxy γxy + τyz γyz + τzx γzx dV.

(A.8)

V

We now examine several special cases of Eq. (A.8). Strain Energy: One-Dimensional Stretching or Contracting For this case, Eq. (A.8) becomes 1 U= 2



σx εx dV. V

(A.9)

Appendix A: Strain Energy in Linear Elastic Bodies

441

This expression is used for the derivation of the governing equation and boundary conditions of an Euler-Bernoulli beam. Strain Energy: One-Dimensional Stretching or Contracting with Shear For this case, Eq. (A.8) becomes U=

1 2



  σx εx + τxy γxy dV.

(A.10)

V

This expression is used for the derivation of the governing equations and boundary conditions of a Timoshenko beam. Strain Energy: Plane Stress For this case, σz = γyz , = γzx = 0 and Eq. (A.8) becomes U=

1 2



  σx εx + σy εy + τxy γxy dV.

(A.11)

V

This expression and Eq. (A.6) are used for the derivation of the governing equations and boundary conditions of thin plates and thin shells. Stress, Strain, and Displacements in Polar Coordinates In polar cylindrical coordinates, the strain-displacement relations are ∂ur , ∂r ∂uθ ur εθ = + , r∂θ r ∂uz , εz = ∂z εr =

∂uθ uθ ∂ur + − r∂θ ∂r r ∂uz ∂ur γzr = + ∂r ∂z ∂uz ∂uθ γzθ = + . ∂z r∂θ

γrθ =

(A.12)

These relations are used in the derivation of the governing equations describing the extensional motion of a circular plate.

Reference Sokolnikoff IS (1956) Mathematical theory of elasticity. McGraw-Hill, New York, NY

Appendix B Variational Calculus: Generation of Governing Equations, Boundary Conditions, and Orthogonal Functions

B.1 Variational Calculus B.1.1 System with One Dependent Variable Hamilton’s principle states that of all possible paths of motion to be taken by a system between two instants of time t1 and t2 , the actual path taken by the system gives a stationary (extremum) value to the integral

t2 I=

L (t) dt

(B.1)

t1

where, for the systems that we shall consider, L=T −U+W where T is the kinetic energy of the system, U is the potential energy of the system, and W is the external non conservative work performed on the system. With L in this form, the procedure that follows is referred to as the extended Hamilton’s principle. Consider the following three functions1   F = F x, y, t, u, u˙ , u,x , u,y , u,xx , u,xy , u,yy

 u,x , ~ u, ~ F(C1 ) = F (C1 ) x, y, t, ~ u˙ , ~ u˙ ,x

 F (C2 ) = F (C2 ) x, y, t, ~ u, ~ u,y , ~ u˙ , ~ u˙ ,y

(B.2)

where u˙ =

1

∂u , ∂t

u,α =

∂u ∂ 2u ∂ u˙ ∂ 2u , u,αβ = , u˙ ,α = = ∂α ∂α∂β ∂α ∂t∂α α = x, y β = x, y.

(B.3)

Portions of this appendix are based on (Weinstock, 1952). 443

444

Appendix B: Variational Calculus: Generation of Governing Equations, Boundary. . .

The definitions of ~ u and its derivatives are obtained from Eq. (B.3) by replacing u with ~ u in the appropriate expressions. The quantity F represents a function over the region R and u = u (x, y, t). The quantity ~ u and its derivatives are equal to u and its !   derivatives specified on C1 and C2 ; that is, ~ u = u (x, y, t)!Cj = u Cj The quantity F (C1 ) is a function that represents the difference between the kinetic energy and the potential energy of elements that are attached to the portion of the boundary denoted C1 and F (C2 ) is a function that represents the difference between the kinetic energy and the potential energy of elements that are attached to the remaining portion of the boundary denoted C2 . We assume that L can be expressed as2

L= Fdxdy + F (C1 ) + F (C2 ) . (B.4) R

Then, Eq. (B.1) becomes   I u, ~ u =

t2 :

; Fdxdy + F

t1

(C1 )

+F

(C2 )

dt.

(B.5)

R

  According to Hamilton’s principle, I u, ~ u will be an extremum with respect to those functions u (x, y, t) that describe the actual configuration at t1 and t2 by the particular function u (x, y, t) that describes the actual configuration for all t. Due to possible constraints along the boundary Cj , the u (x, y, t) eligible for an extremum   of I u, ~ u may be required to satisfy certain conditions on Cj .   To find the extremum of I u, ~ u given by Eq. (B.5), we construct the following functions u¯ = u + εη u + εη u¯ = ~ ~ ~

(B.6)

where ε x1 . Thus, along the edges xj , ds = y = y2 − y1 . Similarly, C2 is composed of two parallel lines, one located at y1 and the other located at y2 : y2 > y1 . Thus, along the edges yj , ds = x = x2 − x1 . Using these two values of θ , Eqs. (B.16) and (B.27) can be combined and written as BC (η (x, y, t), u (x, y, t)) =

2         η xj , y, t H3 u xj , t ,1 + (−1)j H1 (x, y, j) y j=1

+

2 

       η,x xj , y, t H3 u,x xj , t , 1 + (−1)j H2 (x, j) y (B.28)

j=1

+

2         η x, yj , t H3 u yj , t , 2 + (−1)j−1 H1 (y, x, j) x j=1

+

2  j=1

       η,y x, yj , t H3 u,y yj , t , 2 + (−1)j H2 (y, j) x

Appendix B: Variational Calculus: Generation of Governing Equations, Boundary. . .

451

where  ∂Fu,γβ ∂Fu,γ γ − H1 (γ , β, j) = Fu,γ − ∂γ ∂β ! H2 (γ , j) = Fu ! ,γ γ

γ =γj

(B.29)

γ =γj

and H3 is given by Eq. (B.17). In the particular case where η (x, y, t) = u (x, y, t), Eqs. (B.20) becomes

t2 :

δ [I (¯u)] = t1

  u L1 (u (x, y, t)) − L2 (˙u (x, y, t)) dxdy

R

;

(B.30)

+ BC (u (x, y, t), u (x, y, t)) dt = 0. Thus, for Eq. (B.30) to be an extremum, L1 (u (x, y, t)) − L2 (˙u (x, y, t)) = 0 or, using Eq. (B.21), Fu −

∂ 2 Fu,yy ∂ 2 Fu,xy ∂Fu,y ∂Fu,x ∂ 2 Fu,xx ∂Fu˙ + + − + − =0 ∂x ∂y ∂x2 ∂y2 ∂x∂y ∂t

(B.31)

is the non trivial requirement on the interior region R and BC (u (x, y, t), u (x, y, t)) = 0

(B.32)

on the boundary C. Equation (B.31) is known as the Euler-Lagrange equation. In order for BC (u (x, y, t), u (x, y, t)) = 0, it is found from Eq. (B.28) that for a rectangular region the following boundary conditions must be satisfied. At x = xj , j = 1, 2       either u xj , y, t = 0 or H3 u xj , t , 1 + (−1)j H1 (x, y, j) y = 0

(B.33a)

      either u,x xj , y, t = 0 or H3 u,x xj , t , 1 + (−1)j H2 (x, j) y = 0

(B.33b)

and

At y = yj , j = 1, 2       either u x, yj , t = 0 or H3 u yj , t , 2 + (−1)j−1 H1 (y, x, j) x = 0

(B.34a)

Appendix B: Variational Calculus: Generation of Governing Equations, Boundary. . .

452

and       either u,y x, yj , t = 0 or H3 u,y yj , t , 2 + (−1)j H2 (y, j) x = 0

(B.34b)

where it is re-stated that y = y2 − y1 and x = x2 − x1 . Hence, Eqs. (B.33) and (B.34) are the boundary conditions for the system described by Eq. (B.31). There are several special cases of Eqs. (B.31), (B.33), and (B.34). These are summarized in Table B.1. In arriving at Cases 3 and 4 in this table, Eq. (B.27) was used to obtain the form of BC . Returning to Eqs. (B.15) and (B.20), it has been shown that for a rectangular region whose edges are aligned with the coordinate axes, 

t2 :  ∂Fu˙ η Fu − + η,x Fu,x + ηy Fu,y + η,xx Fu,xx + η,xy Fu,xy + η,yy Fu,yy dxdy ∂t t1 R ; + B12 (C1 , C2 ) dt

t2 :

=





;

η L1 (u (x, y, t)) − L2 (˙u (x, y, t)) dxdy + BC (η (x, y, t), u (x, y, t)) dt = 0. t1

R

(B.35)

B.1.2 A Special Case for Systems with One Dependent Variable In order to make the preceding results less general and more applicable to our needs, it is assumed that F (Cj ) is a symmetric quadratic. Although this form is restrictive, it will turn out that it is sufficiently general to provide the basis for deriving the equations for very general boundary conditions for thin beams and thin plates. In addition, this specific form will be instrumental in our determining when we can generate orthogonal functions. Thus,  1 a11 u˙ 2 (x1 , t) + a12 u˙ 2x (x1 , t) + a21 u˙ 2 (x2 , t) + a22 u˙ 2x (x2 , t) 2  1 − A11 u2 (x1 , t) + A12 u2x (x1 , t) + A21 u2 (x2 , t) + A22 u2x (x2 , t) 2 (B.36)  1 = b11 u˙ 2 (y1 , t) + b12 u˙ 2x (y1 , t) + b21 u˙ 2 (y2 , t) + b22 u˙ 2x (y2 , t) 2  1 − B11 u2 (y1 , t) + B12 u2y (y1 , t) + B21 u2 (y2 , t) + B22 u2y (y2 , t) 2

F (C1 ) =

F (C2 )

5

4

3

2

1

Case

F (t, u, u˙ )

F (C2 ) = 0

  F x, t, u, u˙ , u,x

 F (C1 ) x, t, ~ u, ~ u˙

  F x, y, t, u, u˙ , u,x , u,y

 F (C1 ) x, y, t, ~ u, ~ u˙

 F (C2 ) x, y, t, ~ u, ~ u˙

F (C2 ) = 0

F, F (C1 ) , and F(C2 )   F x, y, t, u, u˙ , u,x , u,y , u,xx , u,xy , u,yy

 F (C1 ) x, y, t, ~ u, ~ u˙ , ~ u,x , ~ u˙ ,x

 F (C2 ) x, y, t, ~ u˙ , ~ u,y , ~ u, ~ u˙ ,y   F x, t, u, u˙ , u,x , u,xx

 F (C1 ) x, t, ~ u, ~ u,x , ~ u˙ ,x u˙ , ~

Fu −

Fu −

Fu −

Fu −

∂Fu˙ =0 ∂t

∂Fu,x ∂Fu˙ − =0 ∂x ∂t

∂Fu,y ∂Fu,x ∂Fu˙ − − =0 ∂x ∂y ∂t

∂ 2 Fu,xx ∂Fu,x ∂Fu˙ + =0 − ∂x ∂x2 ∂t

Eq. (B.31)

Euler-Lagrange equation

N/A

j=1

⎤ ⎡ 1) 2 ∂Fu(C  !   ˙ (xj ,t) (C1 ) j ! ⎣ + (−1) Fu,x x=xj ⎦ u xj , t Fu(x ,t) − j ∂t

j=1

⎡ ⎤ 1) 2 ∂Fu(C  !   ˙ (xj ,t) (C1 ) j ! ⎣ + (−1) Fu,x x=x y⎦ u xj , y, t Fu x ,t − (j ) j ∂t j=1 ⎡ ⎤ 2) 2 ∂Fu(C  !   y ,t ˙ ( ) j (C ) + + (−1)j−1 Fu,y !y=y x⎦ u x, yj , t ⎣Fu y2 ,t − (j ) j ∂t

j=1

⎤ ⎡ 1)  2 ∂Fu(C    ∂Fu,xx ˙ (xj ,t) (C ) j 1 ⎦ + (−1) Fu,x − u xj , t ⎣Fu(x ,t) − j ∂t ∂x x=xj j=1 ⎤ ⎡ 1) 2 ∂Fu(C  !   x ,t ˙ ( ) ,x j (C ) + + (−1)j Fu,xx !x=xj ⎦ u,x xj , t ⎣Fu 1(x ,t) − ,x j ∂t

Eq. (B.28) with η = u

BC (u,u)

Table B.1 Equations (B.2), (B.31), and (B.28) and their special cases. When N > 1, the appropriate case is selected for each dependent variable ui and u in the table is replaced with ui , i = 1, 2, . . . , N

Appendix B: Variational Calculus: Generation of Governing Equations, Boundary. . . 453

454

Appendix B: Variational Calculus: Generation of Governing Equations, Boundary. . .

where aij , bij , Aij , and Bij are known constants. The notation in Eq. (B.36) indicates, for example, that u (x1 , t) is constant along the edge x = x1 ; that is, it is independent of a particular value of y, y1 < y < y2 . Since Eq. (B.36) will be used in Eq. (B.17), it is found from Eq. (B.36) that (C )

(C )

Fu˙ (x11 ,t) = a11 u˙ (x1 , t),

Fu˙ (x12 ,t) = a21 u˙ (x2 , t)

(C )

(C )

Fu˙ ,x1(x1 ,t) = a12 u˙ ,x (x1 , t),

Fu˙ ,x1(x2 ,t) = a22 u˙ ,x (x2 , t)

2) Fu(C ˙ (y1 , t), ˙ (y1 ,t) = b11 u

Fu˙ (y22 ,t) = b21 u˙ (y2 , t)

(C )

(C )

(B.37)

(C )

Fu˙ ,y2(y1 ,t) = b12 u˙ ,y (y1 , t),

Fu˙ ,y 2(y2 ,t) = b22 u˙ ,y (y2 , t)

and (C1 ) Fu(x = −A11 u (x1 , t), 1 ,t)

Fu,x 1(x1 ,t) = −A12 u,x (x1 , t)

(C1 ) Fu(x = −A21 u (x2 , t), 2 ,t)

Fu,x 1(x2 ,t) = −A22 u,x (x2 , t)

(C2 ) Fu(y = −B11 u (y1 , t), 1 ,t)

Fu,y2(y1 ,t) = −B12 u,y (y1 , t)

(C )

Fu(y22 ,t) = −B21 u (y2 , t),

(C ) (C ) (C )

(B.38)

(C )

Fu,y2(y2 ,t) = −B22 u,y (y2 , t) .

From Eq. (B.37), it is determined that ∂ (C1 ) F ∂t u˙ (x1 ,t) ∂ (C1 ) F ∂t u˙ ,x (x1 ,t) ∂ (C2 ) F ∂t u˙ (y1 ,t) ∂ (C2 ) F ∂t u˙ ,y (y1 ,t)

= a11 u¨ (x1 , t), = a12 u¨ ,x (x1 , t), = b11 u¨ (y1 , t), = b12 u¨ ,y (y1 , t),

∂ (C1 ) F ∂t u˙ (x2 ,t) ∂ (C1 ) F ∂t u˙ ,x (x2 ,t) ∂ (C2 ) F ∂t u˙ (y2 ,t) ∂ (C2 ) F ∂t u˙ ,y (y2 ,t)

= a21 u¨ (x2 , t) = a22 u¨ ,x (x2 , t) (B.39) = b21 u¨ (y2 , t) = b22 u¨ ,y (y2 , t) .

Consequently, from Eqs. (B.17), (B.38), and (B.39), it is found that     H 3 u xj , t , 1 =     H3 u,x xj , t , 1 =     H3 u yj , t , 2 =     H3 u,y yj , t , 2 =

    −Aj1 u xj , t − aj1 u¨ xj , t     −Aj2 u,x xj , t − aj2 u¨ ,x xj , t     −Bj1 u yj , t − bj1 u¨ yj , t     −Bj2 u,y yj , t − bj2 u¨ ,y yj , t j = 1, 2

and, therefore, Eq. (B.16) can be written as

(B.40)

Appendix B: Variational Calculus: Generation of Governing Equations, Boundary. . .

B12 (C1 , C2 ) = −

455

2        η xj , y, t Aj1 u xj , t + aj1 u¨ xj , t j=1

−

2 

      η,x xj , y, t Aj2 u,x xj , t + aj2 u¨ ,x xj , t

j=1 2        − η x, yj , t Bj1 u yj , t + bj1 u¨ yj , t

(B.41)

j=1

−

2 

      η,y x, yj , t Bj2 u,y yj , t + bj2 u¨ ,y yj , t .

j=1

Using Eq. (B.40), the boundary conditions given by Eqs. (B.33a,b) and (B.34a,b) become as follows. At x = xj , j = 1, 2 either or

  u xj , y, t = 0

     − Aj1 u xj , t + aj1 u¨ xj , t + (−1)j H1 (x, y, j) y = 0

(B.42a)

and either or

  u,x xj , y, t = 0

     − Aj2 u,x xj , t + aj2 u¨ ,x xj , t + (−1)j H2 (x, j) y = 0

(B.42b)

At y = yj , j = 1, 2 either or

  u x, yj , t = 0

     − Bj1 u yj , t + bj1 u¨ yj , t + (−1)j−1 H1 (y, x, j) x = 0

(B.43a)

and either or

  u,y x, yj , t = 0

     − Bj2 u,y yj , t + bj2 u¨ ,y yj , t + (−1)j H2 (y, j) x = 0

(B.43b)

There are several special cases of Eqs. (B.42a,b) and (B.43a,b). These are summarized in Table B.2 using the corresponding cases appearing in the last column of Table B.1.

B.1.3 Systems with N Dependent Variables For the case of N dependent variables u1 (x, y, t), u2 (x, y, t), . . . , uN (x, y, t), we consider the following functions

5

4

3

2

1

Case



F (t, u, u˙ )

F x, y, t, u, u˙ , u,x , u,y   F (C1 ) (x, y, t, ~ u, ~ u˙ ) Aj2 = aj2 = 0   F (C2 ) (x, y, t, ~ u, ~ u˙ ) Bj2 = bj2 = 0   F x, t, u, u˙ , u,x

   F (C1 ) x, t, ~ Aj2 = aj2 = 0 u, ~ u˙ F (C2 ) = 0



F (C2 ) = 0

F, F (C1 ) , and F (C2 )   F x, y, t, u, u˙ , u,x , u,y , u,xx , u,xy , u,yy

 F (C1 ) x, y, t, ~ u, ~ u˙ , ~ u,x , ~ u˙ ,x

 (C ) 2 x, y, t, ~ F u, ~ u˙ , ~ u,y , ~ u˙ ,y   F x, t, u, u˙ , u,x , u,xx

 F (C1 ) x, t, ~ u, ~ u˙ , ~ u,x , ~ u˙ ,x



x=xj

j



j

N/A

Either u xj , t = 0 !     or − Aj1 u xj , t − aj1 u¨ xj , t + (−1)j Fu,x !x=x = 0



Either u xj , y, t = 0 !     or − Aj1 u xj , t − aj1 u¨ xj , t + (−1)j Fu,x !x=x y = 0



,y

y=yj

N/A

N/A

  Either u x, yj , t = 0     or − Bj1 u yj , t − bj1 u¨ yj , t + ! x = 0 (−1)j−1 Fu !

N/A

Eq. (B.43a,b)

Eq. (B.42a,b)

  Either u xj , t = 0      ∂Fu,xx =0 or − Aj1 u xj , t − aj1 u¨ xj , t + (−1)j Fu,x − ∂x x=xj   and either u,x xj , t = 0 !     =0 or − Aj2 u,x xj , t − aj2 u¨ ,x xj , t + (−1)j Fu !

At y = yj , j = 1, 2

At x = xj , j = 1, 2

,xx

Table B.2 Equations (B.42a,b) and (B.43a,b) and their special cases for F (Cj ) given by Eq. (B.36). Case numbers correspond to those of Table B.1. When N > 1, the appropriate case is selected for each dependent variable ui and u in the table is replaced with ui , i = 1, 2, . . . , N

456 Appendix B: Variational Calculus: Generation of Governing Equations, Boundary. . .

Appendix B: Variational Calculus: Generation of Governing Equations, Boundary. . .

 G = G x, y, t, u1 , u˙ 1 , u1,x , u1,y , u1,xx , u1,xy , u1,yy , ...,  uN , u˙ N , uN,x , uN,y , uN,xx , uN,xy , uN,yy

 G(C1 ) = G(C1 ) x, y, t, ~ u1 , ~ u1,x , ~ uN , ~ uN,x , ~ u˙ 1 , ~ u˙ 1,x , ..., ~ u˙ N , ~ u˙ N,x

 u1 , ~ u1,y , ~ uN , ~ uN,y , ~ G(C2 ) = G(C2 ) x, y, t, ~ u˙ 1 , ~ u˙ 1,y , ..., ~ u˙ N , ~ u˙ N,y

457

(B.44)

where G represents a function over the region R, ~ uj and its derivatives are equal to

u and its derivatives evaluated on C1 and C2 , G(C1 ) is a function that represents the difference between the kinetic energy and the potential energy of elements that are attached on the portion of the boundary denoted C1 , G(C2 ) is a function that represents the difference between the kinetic energy and the potential energy of elements that are attached on the remaining portion of the boundary denoted C2 , and u˙ j =

∂uj , ∂t

uj,α =

∂uj , ∂α

uj,αβ =

∂ 2 uj ∂ 2 uj ∂ u˙ j , u˙ j,α = = ∂α∂β ∂α ∂t∂α α = x, y β = x, y.

To obtain ~ uj,α , one replaces uj with ~ uj in the relations given above. It is assumed that L can be expressed as

L= Gdxdy + G(C1 ) + G(C2 ) .

(B.45)

R

Then, Eq. (B.1) becomes   I u, ~ u =

t2 :

; Gdxdy + G

t1

(C1 )

+G

(C2 )

dt.

(B.46)

R

Using the procedure that was used to arrive at Eq. (B.15), it is found that

t2 :

δ [I] = t1

 η1 Gu1 + η˙ 1 Gu˙ 1 + η1,x Gu1,x + η1,y Gu1,y + η1,xx Gu1,xx

R

+ η1,yy Gu1,yy + η1,xy Gu1,xy + ... + ηN GuN + η˙ N Gu˙ N + ηN,x GuN,x  + ηN,y GuN,y + ηN,xx GuN,xx + ηN,yy GuN,yy + ηN,xy GuN,xy dxdy ; (j)

(N)

+ B12 (C1 , C2 ) + ... + B12 (C1 , C2 ) dt

(B.47)



t2 : N

  ∂Gu˙ j + ηj,x Guj,x + ηj,y Guj,y + ηj,xx Guj,xx = ηj Guj − ∂t t1

j=1 R

+ ηj,xy Guj,xy + ηj,yy Guj,yy dxdy +

N  j=1

; (j) B12 (C1 , C2 )

dt = 0

Appendix B: Variational Calculus: Generation of Governing Equations, Boundary. . .

458

where (j)

B12 (C1 , C2 ) =

2 

 uj (xl , t), 1

(j) 

ηj (xl , y, t) H3

l=1

+

2 

 uj,x (xl , t), 1

(j) 

ηj,x (xl , y, t) H3

l=1

+

2 

(B.48)

 (j)  ηj (x, yl , t) H3 uj (yl , t), 2

l=1

+

2 

 uj,y (yl , t), 2

(j) 

ηj,y (x, yl , t) H3

l=1

and (j)

H3 (α, l) = Gα(Cl ) −

l) ∂Gα(C ˙ . ∂t

(B.49)

In arriving at Eq. (B.48), the following expanded notation has been used: ηj (C1 ) = ηj (xl , y, t), ηj,x (C1 ) = ηj,x (xl , y, t), ηj (C2 ) = ηj (x, yl , t), and ηj,y (C2 ) = ηj,y (x, yl , t). Assuming a rectangular geometry with the edges of the region aligning with the coordinate axes, we use the procedures that were employed to arrive at Eq. (B.20). Then, Eq. (B.47) becomes

δ [I] =

t2 : N

t1

 ηj L1j (u1 (x, y, t), ..., uN (x, y, t))

j=1 R

 − L2j (˙u1 (x, y, t), ..., u˙ N (x, y, t)) dxdy ;

(B.50)

+ BN C (ηl (x, y, t), ul (x, y, t)) dt = 0 where L1j (u1 (x, y, t), ..., uN (x, y, t)) = Guj − + L2j (˙u1 (x, y, t), ..., u˙ N (x, y, t)) =

∂Guj,x ∂x

∂ 2 Guj,yy

∂Gu˙ j ∂t

∂y2

−

+

∂Guj,y

∂y ∂ 2 Guj,xy ∂x∂y

j = 1, 2, ..., N

+

∂ 2 Guj,xx ∂x2 (B.51)

Appendix B: Variational Calculus: Generation of Governing Equations, Boundary. . .

459

and   BN C ηj (x, y, t), uj (x, y, t) = " 2 N      (j)  (j) ηj (xl , y, t) H3 uj (xl , t), 1 + (−1)l H1 (x, y, l) y j=1

+

l=1

2 

   (j)  (j) ηj,x (xl , y, t) H3 uj,x (xl , t), 1 + (−1)l H2 (x, l) y

l=1

+ +

2 

   (j)  (j) ηj (x, yl , t) H3 uj (yl , t), 2 + (−1)l−1 H1 (y, x, l) x

l=1

#    (j)  l (j) ηj,y (x, yl , t) H3 uj,y (yl , t), 2 + (−1) H2 (y, l) x

2 

(B.52)

l=1

where  ∂Guj,γβ ∂Guj,γ γ − = Guj,γ − ∂γ ∂β ! (j) H2 (γ , l) = Guj,γ γ !γ =γ .

(j) H1 (γ , β, l)

γ =γl

(B.53)

l

In the particular case where ηj (x, y, t) = uj (x, y, t), Eq. (B.50) becomes

t2 : N

 δ [I] = uj L1j (u1 (x, y, t), ..., uN (x, y, t)) t1

j=1 R

 − L2j (˙u1 (x, y, t), ..., u˙ N (x, y, t)) dxdy ;

(B.54)

+ BN C (ul (x, y, t), ul (x, y, t)) dt = 0. For Eq. (B.54) to be an extremum on the interior, L1j (u1 (x, y, t), ..., uN (x, y, t)) − L2j (˙u1 (x, y, t), ..., u˙ N (x, y, t)) = 0 j = 1, 2, ..., N or, using Eq. (B.51), G uj −

∂Guj,x ∂x

−

∂Guj,y ∂y

+

∂ 2 Guj,xx ∂x2

+

∂ 2 Guj,yy ∂y2

+

∂ 2 Guj,xy ∂x∂y

−

∂Gu˙ j

=0 ∂t j = 1, 2, ..., N

(B.55)

and on the boundary   BN C uj (x, y, t), uj (x, y, t) = 0.

(B.56)

Appendix B: Variational Calculus: Generation of Governing Equations, Boundary. . .

460

Equation (B.55) is the for N dependent variables.  Euler-Lagrange equation  u y, t), u y, t) = 0, it is seen from Eq. (B.52) that the In order for BN (x, (x, j j C following boundary conditions must be satisfied. At x = xl , l = 1, 2 either or

uj (xl , y, t)  (j)  l (j) H3 uj (xl , t), 1 + (−1) H1 (x, y, l) y

=0

uj,x (xl , y, t)  (j)  l (j) H3 uj,x (xl , t), 1 + (−1) H2 (x, l) y

=0

=0

j = 1, 2, ..., N

(B.57a)

and either or

=0

j = 1, 2, ..., N

(B.57b)

At y = yl , l = 1, 2 either or

uj (x, yl , t) = 0   (j) (j) H3 uj (yl , t), 2 + (−1)l−1 H1 (y, x, l) x = 0

j = 1, 2, ..., N

(B.58a)

and either or

uj,y (x, yl , t)  (j)  (j) H3 uj,y (yl , t), 2 + (−1)l H2 (y, l) x

=0 =0

j = 1, 2, ..., N

(B.58b)

There are several special cases of Eqs. (B.55), (B.57a,b), and (B.58a,b). These can be obtained from Table B.1 in the following manner. First, u in Table B.1 is replaced by uj . Then, the governing equation and general form of the corresponding boundary conditions are determined from whether the uj are functions of x and y or just a function of x and from an examination of each uj to determine which case in Table B.1 applies. As will be seen in Section B.3 and Chapter 7, it will be found that, for example, uk is represented by Case 1 and that ul is represented by Case 3. For a rectangular region oriented so that its edges are aligned with the coordinate axes, it is found from Eqs. (B.47) and (B.50) that  N

   ∂Gu˙ j + ηj,x Guj,x + ηj,y Guj,y + ηj,xx Guj,xx ηj Guj − ∂t j=1 R

+ηj,xy Guj,xy + ηj,yy Guj,yy dxdy +

N  j=1

=

N





ηj L1j (u1 (x, y, t), ..., uN (x, y, t))

j=1 R

 − L2j (˙u1 (x, y, t), ..., u˙ N (x, y, t)) dxdy + BN C (ηl (x, y, t), ul (x, y, t)) = 0.

(j)

B12 (C1 , C2 ) (B.59)

Appendix B: Variational Calculus: Generation of Governing Equations, Boundary. . .

461

B.1.4 A Special Case for Systems with N Dependent Variables In a manner similar to that employed in Section Sec B.1.2, it is assumed that G(Cl ) is a symmetric quadratic. It will turn out that this form is general enough to provide the basis for deriving very general boundary conditions for Timoshenko beams and cylindrical shells. Thus, G(C1 ) =

G(C2 )

1  al1j u˙ 2j (xl , t) + al2j u˙ 2j,x (xl , t) 2 j=1 l=1  − Al1j u2j (xl , t) − Al2j u2j,x (xl , t) N

2

(B.60)

1  bl1j u˙ 2j (yl , t) + bl2j u˙ 2j,x (yl , t) = 2 j=1 l=1  − Bl1j u2j (yl , t) − Bl2j u2j,x (yl , t) 2

N

where alkj , blkj , Alkj , and Blkj , are known constants. Since Eq. (B.60) will be used in Eq. (B.49), it is found from Eq. (B.60) that 1) G(C ˙ j (x1 , t), u˙ j (x1 ,t) = a11j u

(C )

(C )

Gu˙ j (x1 2 ,t) = a21j u˙ j (x2 , t) (C )

Gu˙ j,x1(x1 ,t) = a12j u˙ j,x (x1 , t),

Gu˙ j,x1(x2 ,t) = a22j u˙ j,x (x2 , t)

2) G(C ˙ j (y1 , t), u˙ j (y1 ,t) = b11j u

Gu˙ j (y2 2 ,t) = b21j u˙ j (y2 , t)

(C )

Gu˙ j,y2(y1 ,t) = b12j u˙ j,y (y1 , t),

(B.61)

(C )

(C )

Gu˙ j,y2(y2 ,t) = b22j u˙ j,y (y2 , t)

j = 1, 2, ..., N

and (C )

(C )

Guj (x1 1 ,t) = −A11j uj (x1 , t),

Guj (x1 2 ,t) = −A21j uj (x2 , t)

1) G(C uj,x (x1 ,t) = −A12j uj,x (x1 , t),

Guj,x1(x2 ,t) = −A22j uj,x (x2 , t)

(C )

(C )

(C )

Guj (y2 1 ,t) = −B11j uj (y1 , t),

Guj (y2 2 ,t) = −B21j uj (y2 , t)

Gu(Cj,y2(y) 1 ,t) = −B12j uj,y (y1 , t),

Guj,y2(y2 ,t) = −B22j uj,y (y2 , t)

(C )

j = 1, 2, ..., N. (B.62)

From Eq. (B.61), it is found that ∂ (C1 ) G ∂t u˙ j (x1 ,t) ∂ (C1 ) G ∂t u˙ j,x (x1 ,t) ∂ (C2 ) G ∂t u˙ j (y1 ,t) ∂ (C2 ) G ∂t u˙ j,y (y1 ,t)

= a11j u¨ j (x1 , t), = a12j u¨ j,x (x1 , t), = b11j u¨ j (y1 , t), = b12j u¨ j,y (y1 , t),

∂ (C1 ) G ∂t u˙ j (x2 ,t) ∂ (C1 ) G ∂t u˙ j,x (x2 ,t) ∂ (C2 ) G ∂t u˙ j (y2 ,t) ∂ (C2 ) G ∂t u˙ j,y (y2 ,t)

= a21j u¨ j (x2 , t) = a22j u¨ j,x (x2 , t) = b21j u¨ j (y2 , t) = b22j u¨ j,y (y2 , t) j = 1, 2, ..., N. (B.63)

Appendix B: Variational Calculus: Generation of Governing Equations, Boundary. . .

462

Consequently, from Eqs. (B.49), (B.62), and (B.63), it is found that (n)

H3 (un (xl , t), 1) = − Al1n un (xl , t) − al1n u¨ n (xl , t)   H3(n) un,x (xl , t), 1 = − Al2n un,x (xl , t) − al2n u¨ n,x (xl , t) (n)

H3 (un (yl , t), 2) = − Bl1n un (yl , t) − bl1n u¨ n (yl , t)   H3(n) un,y (yl , t), 2 = − Bl2n un,y (yl , t) − bl2n u¨ n,y (yl , t)

(B.64)

and, therefore, Eq. (B.48) can be written as (n)

B12 (C1 , C2 ) = −

2 

ηn (xl , y, t) [Al1n un (xl , t) + al1n u¨ n (xl , t)]

l=1

−

2 

  ηn,x (xl , y, t) Al2n un,x (xl , t) + al2n u¨ n,x (xl , t)

l=1

−

2 

  ηn (x, yl , t) Bl1n un (yl , t) + bl1n u¨ n (yl , t)

(B.65)

l=1

−

2 

  ηn,y (x, yl , t) Bl2n un,y (yl , t) − bl2n u¨ n,y (yl , t) .

l=1

Using Eq. (B.64) in Eqs. (B.57a,b) and (B.58a,b), the boundary conditions can be written as follows. At x = xl , l = 1, 2 either − [Al1n un (xl , t) + al1n u¨ n (xl , t)] + (−1)

l

or

un (xl , y, t) (n) H1 (x, y, l) y

=0 =0

(B.66a)

n = 1, 2, ..., N and either or





un,x (xl , y, t) = 0

− Al2n un,x (xl , t) + al2n u¨ n,x (xl , t) + (−1)l H2(n) (x, l) y = 0

(B.66b)

n = 1, 2, ..., N At y = yl , l = 1, 2 either or





un (x, yl , t) = 0

− Bl1n un (yl , t) + bl1n u¨ n (yl , t) + (−1)l−1 H1(n) (y, x, l) x = 0 n = 1, 2, ..., N

(B.67a)

Appendix B: Variational Calculus: Generation of Governing Equations, Boundary. . .

463

and either or





un,y (x, yl , t) = 0

− Bl2n un,y (yl , t) − bl2n u¨ n,y (yl , t) + (−1) H2(n) (y, l) x = 0 l

(B.67b)

n = 1, 2, ..., N There are several special cases of Eqs. (B.66) and (B.67). These can be obtained from Table B.2 in the following manner. First, u in Table B.2 is replaced by uj . Then, the general form of the corresponding boundary conditions is determined from whether the uj are functions of x and y or just a function of x and from an examination of each uj to determine which case in Table B.2 applies. As will be seen in Section B.3 and Chapter 7, it will be found that, for example, uk is represented by Case 1 and that ul is represented by Case 3.

B.2 Orthogonal Functions One method used to solve a class of partial differential equations is the separation of variables and the generation of orthogonal functions. It will be shown in this section that when the form of F and G on the interior and F (Cj ) and G(Cj ) on the boundary are symmetric quadratics and the motion of the system is harmonic, we shall be able to generate orthogonal functions.

B.2.1 Systems with One Dependent Variable A set of functions {ψn (x)} , n = 1, 2, . . . , on the interval x1 ≤ x ≤ x2 that has the property

x2 p (x) ψn (x) ψm (x) dx = δnm Nn

(B.68)

x1

is said to be orthogonal with respect to the weight function p (x). The quantity δ nm is the Kronecker delta and

x2 Nn =

p (x) ψn2 dx

(B.69)

x1

is the norm of this set of functions. In order to be able to use the definition of orthogonality, we must restrict the following discussion to systems that are independent of y. It will turn out that this restriction is a minor one, since our primary solution method for linear systems will be separation of variables. This, in effect, uncouples the system with two independent spatial variables into two ‘separate’ solutions, one in each of the spatial variables.

464

Appendix B: Variational Calculus: Generation of Governing Equations, Boundary. . .

Hence, we assume that F and F (C1 ) are symmetric quadratics given by 3 3 1  1 Dln ϕl ϕn F = p (x) u˙ 2 − 2 2 l=1 n=1  1 F (C1 ) = a11 u˙ 2 (x1 , t) + a12 u˙ 2,x (x1 , t) + a21 u˙ 2 (x2 , t) + a22 u˙ 2,x (x2 , t) 2  1 − A11 u2 (x1 , t) + A12 u2,x (x1 , t) + A21 u2 (x2 , t) + A22 u2,x (x2 , t) 2

(B.70)

where ϕ1 = u, ϕ2 = u,x , and ϕ3 = u,xx , Dln = Dln (x), by definition Dln = Dnl , and aij and Aij , are known constants. It is seen that F (C1 ) in Eq. (B.70) is the same as F (C1 ) in Eq. (B.36). We simplify Eq. (B.35) to be a function of x only and arrive at − =0

(B.71)

where 

x2  ∂Fu˙ = + η,x Fu,x + η,xx Fu,xx dx + B12 (C1 ) η Fu − ∂t x1

x2

(B.72) η [L1 (u (x, t)) − L2 (˙u (x, t))] dx + BC (η (x, t), u (x, t))

= x1

and from the simplification of Eq. (B.21) L1 (u (x, t)) = Fu − ∂Fu˙ L2 (˙u (x, t)) = . ∂t

∂Fu,x ∂ 2 Fu,xx + ∂x ∂x2

(B.73)

The quantities BC and B12 , respectively, are obtained from Case 2 of Table B.1 and from Eq. (B.41) as ⎡ ⎤ (C )  2 ∂Fu˙ (x1 ,t)    (C1 ) ∂F u,xx j ⎦ η xj , t ⎣Fu x ,t − BC (η (x, t), u (x, t)) = + (−1)j Fu,x − (j ) ∂t ∂x x=xj j=1 ⎤ ⎡ 1) 2 ∂Fu(C  !   (C1 ) ˙ ,x (xj ,t) η,x xj , t ⎣Fu x ,t − + + (−1)j Fu,xx !x=xj ⎦ ,x ( j ) ∂t j=1

B12 (C1 ) = −

2        η xj , t Aj1 u xj , t + aj1 u¨ xj , t j=1

−

2 

      η,x xj , t Aj2 u,x xj , t − aj2 u¨ ,x xj , t .

j=1

(B.74)

Appendix B: Variational Calculus: Generation of Governing Equations, Boundary. . .

465

In anticipation of our subsequent needs, we rewrite B12 (C1 ) given in Eq. (B.74) as B12 (C1 ) = −B (η, u) − B (η, u¨ )

(B.75)

where B (η, u) = A11 u (x1 , t) η (x1 , t) + A21 u (x2 , t) η (x2 , t) + A12 u,x (x1 , t) η,x (x1 , t) + A22 u,x (x2 , t) η,x (x2 , t) = B (u, η)

(B.76)

and B (η, u¨ ) = a11 u¨ (x1 , t) η (x1 , t) + a21 u¨ (x2 , t) η (x2 , t) + a12 u¨ ,x (x1 , t) η,x (x1 , t) + a22 u¨ ,x (x2 , t) η,x (x2 , t) .

(B.77)

To determine the conditions under which orthogonal functions can be generated, we start by setting  = 0. The quantity  will equal zero when on the interior L1 (u (x, t)) − L2 (˙u (x, t)) = 0

(B.78)

where L1 and L2 are given by Eq. (B.73) and when on the boundary BC (u, u) = 0

(B.79)

where BC is given by Eq. (B.74). We are interested in a solution to Eq. (B.78) of the form ¯ ¯ (x) ejωt . u (x, t) = U

(B.80)

Then Eq. (B.78) becomes ∂Fu˙ ∂t = L1 (u (x, t)) − p (x) u¨   ¯ =0 ¯ + ω¯ 2 p (x) U = L1 U

L1 (u (x, t)) − L2 (˙u (x, t)) = L1 (u (x, t)) −

(B.81)

and the boundary conditions given by Eq. (B.79) become   ¯ U ¯ = 0. BC U,

(B.82)

The homogeneous differential equation given by Eq. (B.81) and the homogeneous boundary conditions given by Eq. (B.82) form a standard eigenvalue problem, where the eigenvalue to be determined is ω. ¯ A solution to this homogeneous differential equation and the homogeneous boundary conditions is the eigenvector ¯ (x) = U (n) (x), n = 1, 2, . . . , which corresponds to the eigenvalue ω¯ = ωn . U

Appendix B: Variational Calculus: Generation of Governing Equations, Boundary. . .

466

Thus, when U(n) is a solution to Eq. (B.81) and the boundary conditions given by Eqs. (B.82),  = 0. For the applications in Chapters 3 to 7, the eigenvector is the mode shape and the eigenvalue is the natural frequency. Returning to Eqs. (B.71) and (B.72) with  = 0, we set η = uˆ , where uˆ has properties similar to that of u, and obtain 

x2      ∂Fu˙ uˆ Fu − + uˆ ,x Fu,x + uˆ ,xx Fu,xx dx − B uˆ , u − B uˆ , u¨ = 0 = ∂t x1

(B.83) where we have used Eq. (B.75). From Eq. (B.70) and the definitions of ϕj , j = 1, 2, 3, it is seen that ∂Fu˙ = p (x) u¨ , ∂t Fu,x = −

3 

D2n ϕn ,

Fu = −

3 

D1n ϕn

n=1

Fu,xx = −

n=1

3 

(B.84) D3n ϕn

n=1

and, therefore, Eq. (B.83) can be written as

x2 , −p (x) uˆ u¨ − ϕˆ1 x1

3  n=1

D1n ϕn − ϕˆ 2

    − B uˆ , u − B uˆ , u¨ = 0

3 

D2n ϕn − ϕˆ3

n=1

3 

D3n ϕn dx

n=1

or

x2 , 3  3      p (x) u¨ uˆ + Dln ϕn ϕˆl dx + B uˆ , u + B uˆ , u¨ = 0.

(B.85)

l=1 n=1

x1

If, in Eq. (B.85), u is replaced with uˆ and uˆ is replaced with u, then Eq. (B.85) becomes

x2 , 3 3 

    ¨ Dln ϕˆn ϕl dx + B u, uˆ + B u, u¨ˆ = 0. p (x) uˆ u +

(B.86)

l=1 n=1

x1

Upon subtracting Eq. (B.86) from Eq. (B.85), we obtain

x2 x1



   p (x) u¨ uˆ − u¨ˆ u dx + B uˆ , u¨ − B u, u¨ˆ = 0

(B.87)

Appendix B: Variational Calculus: Generation of Governing Equations, Boundary. . .

467

    where we have used Eq. (B.76); that is, B u, uˆ = B uˆ , u , and the fact that since Dln = Dnl 3  3 

Dln ϕˆl ϕn −

l=1 n=1

3  3  l=1 n=1

Dln ϕl ϕˆn =

3  3 

  Dln ϕˆl ϕn − ϕn ϕˆl = 0.

l=1 n=1

If it is assumed that u (x, t) = U (n) (x) ejωn t and uˆ (x, t) = U (m) (x) ejωm t , where and ωm , m = 1, 2, . . ., are solutions to Eqs. (B.81) and (B.82), then substituting these expressions into Eq. (B.87) yields U(m) (x)

⎛ ⎞

 x2

 2 ⎝ ωn2 − ωm p (x) U (n) (x) U (m) (x) dx + B U (n) , U (m) ⎠ = 0

(B.88)

x1

where

 B U (n) , U (m) = a11 U (n) (x1 ) U (m) (x1 ) + a21 U (n) (x2 ) U (m) (x2 ) + a12 U,x(n) (x1 ) U,x(m) (x1 ) + a22 U,x(n) (x2 ) U,x(m) (x2 )

 = B U (m) , U (n)

(B.89)

and U,x(k) = dU (k)/dx. Since, ωn = ωm , it is seen that Eq. (B.88) can be written as

x2

 p (x) U (n) (x) U (m) (x) dx + B U (n) , U (m) = 0

ωn = ωm

x1

= Nn

(B.90)

ωn = ωm

where

x2 Nn =

2

  p (x) U (n) (x) dx + B U (n) , U (n)

(B.91)

x1

and

    2 2 2 B U (n) , U (n) = a11 U (n) (x1 ) + a21 U (n) (x2 ) + a12 U,x(n) (x1 ) (B.92) 2  + a22 U,x(n) (x2 ) . Equation (B.90) is the orthogonality condition for a system with one dependent variable. Thus, if F and F (C1 ) can be expressed as symmetric quadratics of the forms given by Eq. (B.70), then the solution to the governing equation given by Eq. (B.81) and the boundary conditions given by Eq. (B.82) will be an orthogonal function. Notice

Appendix B: Variational Calculus: Generation of Governing Equations, Boundary. . .

468

that we have arrived at this result without having to specify the specific form of the differential equation and the boundary conditions, only that they can be obtained from a symmetric quadratic that is a function of one or more of the quantities given by ϕ n .   In the case when F (C1 ) = 0, B U (n) , U (n) = 0 and Eqs. (B.90) and (B.91), respectively, become

x2

p (x) U (n) (x) U (m) (x) dx = 0

ωn  = ω m

x1

(B.93)

= Nn ωn = ωm where

x2 Nn =

 2 p (x) U (n) (x) dx.

(B.94)

x1

B.2.2 Systems with N Dependent Variables It is assumed that G and G(C1 ) are the following symmetric quadratics 1  1  pln (x) u˙ j u˙ n − Eln θl θn 2 2 N

G=

N

j=1 n=1

G(C1 ) =

3N 3N

l=1 n=1

N 2  1  al1j u˙ 2j (xl , t) + al2j u˙ 2j,x (xl , t) − Al1j u2j (xl , t) − Al2j u2j,x (xl , t) 2 j=1 l=1

(B.95) where alkj and Alkj are known constants, θ1 = u1 , θ2 = u1,x , θ3 = u1,xx , . . . , θ3N−2 = uN , θ3N−1 = uN,x , θ3N = uN,xx , and by definition pln (x) = pnl (x) and Eln = Enl . It is seen that G(C1 ) in Eq. (B.95) is the same as G(C1 ) in Eq. (B.60). We simplify Eq. (B.59) to be a function of x only and arrive at − =0

(B.96)

where ⎫ ⎧  N ⎨ x2   ⎬  ∂Gu˙ j (j) = ηj Guj − + ηj,x Guj,x + ηj,xx Guj,,xx dx + B12 (C1 ) ⎭ ⎩ ∂t j=1 x1 ⎧ N ⎨ x2   = ηj L1j (u1 (x, t), ..., uN (x, t)) (B.97) ⎩ j=1 x1 ⎫  ⎬ − L2j (˙u1 (x, t), ..., u˙ N (x, t)) dx + BN C (ηl , ul ) ⎭

Appendix B: Variational Calculus: Generation of Governing Equations, Boundary. . .

469

and from the simplification of Eq. (B.51) L1j (u1 (x, t), ..., uN (x, t)) = Guj − L2j (˙u1 (x, t), ..., u˙ N (x, t)) =

∂Gu˙ j ∂t

∂Guj,x ∂x

+

∂ 2 Guj,xx ∂x2

(B.98)

j = 1, 2, ..., N.

(j)

The quantities BN C (ηl , ul ) and B12 (C1 ), respectively, are obtained from Case 2 of Table B.1 with u = uj and Eq. (B.65) as ⎧ ⎡  1) N ⎨ 2 ∂Fu(C    ∂Fuj,xx ˙ j (xl ,t) (C1 ) l ⎣ BN , u η , t) − − F η = + (−1) F (x j j j l u j,x C (x ,t) u j l ⎩ ∂t ∂x j=1 l=1 ⎤⎫ ⎡ (C ) 2 ⎬ ∂Fu˙ j,x1(xl ,t)  ! (C ) ηj,x (xl , t) ⎣Fuj,x1(xl ,t) − + + (−1)l Fuj,xx !x=x ⎦ l ⎭ ∂t

⎤ ⎦ x=xl

l=1

(j)

B12 (C1 ) = −

2 

  ηj (xl , t) Al1j uj (xl , t) + al1j u¨ j (xl , t)

l=1

−

2 

  ηj,x (xl , t) Al2j uj,x (xl , t) + al2j u¨ j,x (xl , t) .

l=1

(B.99) In anticipation of our subsequent needs, we rewrite

(j) B12

given in Eq. (B.99) as

    (j) B12 (C1 ) = −B j ηj , uj − B j ηj , u¨ j

(B.100)

where   B j ηj , uj = A11j uj (x1 , t) ηj (x1 , t) + A21j uj (x2 , t) ηj (x2 , t) A12j uj,x (x1 , t) ηj,x (x1 , t) + A22j uj,x (x2 , t) ηj,x (x2 , t)   = B uj , ηj

(B.101)

and   B j ηj , u¨ j = a11j u¨ j (x1 , t) ηj (x1 , t) + a21j u¨ j (x2 , t) ηj (x2 , t) + a12j u¨ j,x (x1 , t) ηj,x (x1 , t) + a22j u¨ j,x (x2 , t) ηj,x (x2 , t) .

(B.102)

To determine the conditions under which orthogonal functions can be generated, we start by setting  = 0. The quantity  will equal zero when on the interior L1j (u1 (x, t), ..., uN (x, t)) − L2j (˙u1 (x, t), ..., u˙ N (x, t)) = 0 j = 1, 2, ..., N. (B.103)

Appendix B: Variational Calculus: Generation of Governing Equations, Boundary. . .

470

where L1j and L2j are given by Eq. (B.98) and when on the boundary   BN C uj , uj = 0

(B.104)

  where BN C uj , uj is given by Eq. (B.99). We are interested in a solution to Eq. (B.103) of the form ¯ ¯ j (x) ejωt . uj (x, t) = U

(B.105)

Then Eq. (B.103) becomes L1j (u1 (x, t), ..., uN (x, t)) −

∂Gu˙ j ∂t

= L1j (u1 (x, t), ..., uN (x, t)) −

N 

plj (x, y) u¨ l

l=1

= L1j (U1 , ..., UN ) + ω¯ 2

N 

plj (x) Ul = 0

l=1

j = 1, 2, ..., N (B.106) and the boundary conditions become   ¯ ¯ BN C Uj , Uj = 0.

(B.107)

The homogeneous differential equations given by Eq. (B.106) and the homogeneous boundary conditions given by Eq. (B.107) form a standard eigenvalue problem, where the eigenvalue to be determined is ω. ¯ A solution to this set of homogeneous differential equations and homogeneous boundary conditions is the ¯ j (x) = Uj(n) (x), n = 1, 2, ..., which corresponds to the eigenvalue eigenvector U

ω¯ = ωn . Thus, when Uj(n) are solutions to Eq. (B.106) and the boundary conditions given by Eqs. (B.107),  = 0. Returning to Eqs. (B.96) and (B.97) with  = 0, we set ηj = uˆ j , where uˆ j has properties similar to that of uj , and obtain ⎧ ⎫

x2 ⎨ N N    ∂Gu˙ j ⎬     = B j uˆ j , uj + B j uˆ j , u¨ j = 0 uˆ j − dx − ⎩ ∂t ⎭ x1

j=1

(B.108)

j=1

where =

N    uˆ j Guj + uˆ j,x Guj,x + uˆ j,xx Guj,,xx .

(B.109)

j=1

Using Eq. (B.95) and the definitions of ϕj , j = 1, 2, . . . , 3N and introducing the subscript notation n (j, k) = 3 (j − 1) + k, Eq. (B.109) can be rewritten as

Appendix B: Variational Calculus: Generation of Governing Equations, Boundary. . .

=

471

N    uˆ j Guj + uˆ j,x Guj,x + uˆ j,xx Guj,,xx j=1

=

N    θˆn(j,1) Gϕn(j,1) + θˆn(j,2) Gϕn(j,2) + θˆn(j,3) Gϕn(j,3) j=1

=−

N 

, θˆn(j,1)

j=1

=−

3N 

En(j,1)m θm + θˆn(j,2)

m=1

N  3N 

=−

En(j,2)m θm + θˆn(j,3)

m=1

En(j,1)m θˆn(j,1) θm −

j=1 m=1 3N 3N  

3N 

N  3N 

-

3N 

En(j,3)m θm

m=1

En(j,2)m θˆn(j,2) θm −

j=1 m=1

N  3N 

En(j,3)m θˆn(j,3) θm

j=1 m=1

Ejm θˆj θm .

j=1 m=1

(B.110) Also from Eq. (B.95), we have that ∂Gu˙ j ∂t

=

N 

pjn (x) u¨ n .

(B.111)

n=1

Using Eqs. (B.110) and (B.111), Eq. (B.108) can be written as ⎧

x2 ⎨ N N  x1

⎩

pjn (x) uˆ j u¨ n +

j=1 n=1

3N 3N   j=1 n=1

⎫ ⎬

Ejn θˆj θn dx + ⎭

N       B j uˆ j , uj + B j uˆ j , u¨ j = 0. j=1

(B.112) If, in Eq. (B.112), uj is replaced with uˆ j and uˆ j is replaced with uj , then Eq. (B.112) becomes ⎡ ⎤

x2  N 3N N  3N  N 

     ⎣ B j uj , uˆ j + B j uj , u¨ˆ j = 0. plj (x) u¨ˆ l uj + Ejn θˆn θj ⎦ dx + j=1 l=1

x1

j=1 n=1

j=1

(B.113) Upon subtracting Eq. (B.112) from Eq. (B.113), we obtain

x2 , N N  x1

l=1 n=1

N  



   B n un , u¨ˆ − B n uˆ n , u¨ n = 0 pln (x) u¨ˆ l un − u¨ l uˆ n dx + ~n n=1

(B.114)

Appendix B: Variational Calculus: Generation of Governing Equations, Boundary. . .

472

    where we have used Eq. (B.101); that is, B j uj , uˆ j = B j uˆ j , uj , and the fact that since Eln = Enl 3N  3N 

Eln θˆl θj −

j=1 l=1

3N  3N 

Elj θl θˆj =

j=1 l=1

3N  3N 

 Elj θˆl θj − θj θˆl = 0.

l=1 j=1 (n)

(m)

If it is assumed that uj (x, t) = Uj (x) ejωn t and uˆ (x, t) = Uj

(x) ejωm t , where

Uj(m)

and ωm , m = 1, 2, . . ., are solutions to Eqs. (B.106) and (B.107), then substituting these expressions into Eq. (B.114) yields ⎧x ⎡ ⎫ ⎤ N  N N  ⎨ 2 

⎬

 (n) (m) (n) (m) 2 ⎣ =0 plj (x) Ul Uj ⎦ dx + B j Uj , Uj ωn2 − ωm ⎩ ⎭ x1

l=1 j=1

j=1

(B.115) where we have used the symmetric quadratic assumption that pij (x) = pji (x) and

 (n) (m) (n) (m) (n) (m) = a11j Uj (x1 ) Uj (x1 ) + a21j Uj (x2 ) Uj (x2 ) B j Uj , Uj (n)

(m)

(n)

(m)

+ a12j Uj,x (x1 ) Uj,x (x1 ) + a22j Uj,x (x2 ) Uj,x (x2 )

 = B j Uj(m) , Uj(n) .

(B.116)

Since ωn = ωm , it is seen that Eq. (B.115) can be written as ⎡ ⎤

x2  N  N N

  (n) (m) (n) (m) ⎣ =0 plj (x) U U ⎦ dx + B j U , U l

x1

l=1 j=1

j

j

j

ωn = ωm

j=1

= Mn

ωn = ωm (B.117)

where ⎡ ⎤

x2  N N  N

  (n) (n) (n) (n) Mn = ⎣ plj (x) Ul Uj ⎦ dx + B j Uj , Uj x1

l=1 j=1

(B.118)

j=1

and 2 2 2

    (n) (n) (n) (n) (n) = a11j Uj (x1 ) + a21j Uj (x2 ) + a12j Uj,x (x1 ) B j Uj , Uj  2 (n) + a22j Uj,x (x2 ) . (B.119) Equation (B.117) is the orthogonality condition for a system with N dependent variables.

Appendix B: Variational Calculus: Generation of Governing Equations, Boundary. . .

473

B.3 Application of Results to Specific Elastic Systems In Chapters 3 to 7, the minimization functions for different elastic systems are obtained. In order to use these functions to derive the governing equations and the boundary conditions in the manner presented in this appendix, we shall summarize the preceding results so that they can be used directly in the appropriate chapter. Based on the form of the minimization function, there are two important factors that determine which of the preceding results is used: the number of dependent variables N and the number of spatial variables, either x or x and y (or x and θ ). It is mentioned that the specialization to the specific elastic systems given below is still in a general form. Consequently, in certain cases some of the terms may be zero. These aspects become apparent after each specific minimization function is derived in the chapters indicated. N = 1: Thin Rectangular Plates For the case of thin plates, it is found in Chapter 6 that N = 1, the spatial variables are x and y, and F, F (C1 ) , and F(C2 ) are given by Case 1 of Table B.1. Then the governing equation is obtained from Eq. (B.31); that is, Fu −

∂ 2 Fu,yy ∂ 2 Fu,xy ∂Fu,y ∂Fu,x ∂ 2 Fu,xx ∂Fu˙ + + − + − = 0. 2 2 ∂x ∂y ∂x ∂y ∂x∂y ∂t

(B.120)

The boundary conditions can be obtained from Eqs. (B.29), (B.42a,b), and (B.43a,b), which lead to the following expressions. At x = xj , j = 1, 2 either

   ∂Fu,xx j − Aj1 u xj , t + aj1 u¨ xj , t + (−1) y Fu,x − ∂x 

or





  u xj , y, t = 0 ∂Fu,xy =0 − ∂y x=xj (B.121a)

and either or

  u,x xj , y, t = 0 !      − Aj2 u,x xj , t + aj2 u¨ ,x xj , t + (−1)j y Fu,xx !x=x = 0

(B.121b)

j

At y = yj , j = 1, 2 either or

   ∂Fu,yy j−1 − Bj1 u yj , t + bj1 u¨ yj , t + (−1) x Fu,y − ∂y 





  u x, yj , t = 0 ∂Fu,yx − =0 ∂x y=yj (B.122a)

Appendix B: Variational Calculus: Generation of Governing Equations, Boundary. . .

474

and either or

  u,y x, yj , t = 0 !      − Bj2 u,y yj , t + bj2 u¨ ,y yj , t + (−1)j x Fu,yy !y=y = 0 j

(B.122b)

N = 1: Euler-Bernoulli Beams For the case of thin beams, it is found in Chapter 3 that N = 1, the spatial variable is x; that is, u = u (x, t), and F and F (C1 ) are given by Case 2 of Table B.1. Then the governing equation is obtained from Case 2 of Table B.1; that is, Fu −

∂Fu,x ∂ 2 Fu,xx ∂Fu˙ − + = 0. 2 ∂x ∂x ∂t

(B.123)

The boundary conditions can be obtained from Case 2 of Table B.2 and are repeated below for convenience. At x = xj , j = 1, 2 either or

  u xj , t = 0

      ∂Fu,xx − Aj1 u xj , t + aj1 u¨ xj , t + (−1)j Fu,x − ∂x

=0

(B.124a)

x=xj

and either or

  u,x xj , t = 0 !      − Aj2 u,x xj , t + aj2 u¨ ,x xj , t + (−1)j Fu,xx !x=x = 0 j

(B.124b)

N = 3: Cylindrical Shells For the case of thin cylindrical shells, it is found in Chapter 7 that N = 3 and the spatial variables are x = x and y = θ . The three dependent variables are denoted as u1 = ux , u2 = uθ , and u3 = w. It is also found in Chapter 7 that u1 = ux and u2 = uθ are described by Case 3 in Tables B.1 and B.2 and u3 = w is described by Case 1 in these tables. Then, from Table B.1, it is found that the governing equations can be obtained from the following three equations ∂Gux,θ ∂Gux,x ∂Gu˙ x − − =0 ∂x ∂θ ∂t ∂Guθ,x ∂Guθ,θ ∂Gu˙ θ Guθ − − − =0 ∂x ∂θ ∂t ∂ 2 Gw,θθ ∂ 2 Gw,xθ ∂ 2 Gw,xx ∂Gw˙ + + + − = 0. ∂x∂θ ∂t ∂x2 ∂θ 2 Gu x −

Gw −

∂Gw,θ ∂Gw,x − ∂x ∂θ

(B.125)

Appendix B: Variational Calculus: Generation of Governing Equations, Boundary. . .

475

In Chapter 7, only complete cylindrical shells are considered; thus, there will be no boundary conditions to be specified along a θ -edge. Consequently, the boundary conditions are only those given in Table B.2 for x1 and x2 . Since u1 = ux and u2 = uθ are governed by Case 3 and u3 = w is governed by Case 1, the boundary conditions for these quantities at x = xl , l = 1, 2 and are as follows. either or

ux (xl , θ , t) = 0 ! − [Al11 ux (xl , t) + al11 u¨ x (xl , t)] + (−1)l y Gux,x !x=x = 0

(B.126a)

uθ (xl , θ , t) = 0 ! − [Al12 uθ (xl , t) + al12 u¨ θ (xl , t)] + (−1) y Guθ,x !x=x = 0

(B.126b)

l

and either or

l

l

and either



¨ (xl , t)] + (−1)l y Gw,x − [Al13 w (xl , t) + al13 w

or

w (xl , θ , t) = 0 ∂Gw,xθ ∂Gw,xx − =0 − ∂x ∂θ x=xj (B.126c)

and either or

w,x (xl , θ , t) = 0 !  l ¨ ,x (xl , t) + (−1) y Gw,xx !x=x = 0 − Al23 w,x (xl , t) + al23 w 

(B.126d)

l

N = 2: Timoshenko Beams For the case of Timoshenko beams, it is found in Chapter 5 that N = 2 and the spatial variable is x; that is, uj = uj (x, t). The two dependent variables are denoted as u1 = w and u2 = ψ. It is also shown in Chapter 5 that u1 = w and u2 = ψ are described by Case 4 in Tables B.1 and B.2. From Table B.1, it is found that the governing equations can be obtained from the following two equations ∂Gw,x ∂Gw˙ − =0 ∂x ∂t ∂Gψ˙ ∂Gψ,x Gψ − − = 0. ∂x ∂t Gw −

(B.127)

The boundary conditions can be obtained from Case 4 of Table B.2 and are repeated below for convenience.

Appendix B: Variational Calculus: Generation of Governing Equations, Boundary. . .

476

At x = xl , l = 1, 2 either or

w (xl , t) = 0 ! − [Al11 w (xl , t) + al11 w ¨ (xl , t)] + (−1) Gw,x !x=x = 0

(B.128a)

ψ (xl , t) = 0 ! − Al12 ψ (xl , t) + al12 ψ¨ (xl , t) + (−1) Gψ,x !x=x = 0

(B.128b)

l

l

and either or





l

l

Reference Weinstock R (1952) Calculus of variations: with applications to physics and engineering. McGraw-Hill, New York, NY

Appendix C Laplace Transforms and the Solutions to Ordinary Differential Equations

C.1 Definition of the Laplace Transform The Laplace transform of a function g(t) is defined as

∞ G (s) =

e−st g (t) dt

(C.1)

0

where the variable s is a complex variable represented as s = σ + jω, where j = √ −1. In writing this integral transform definition, it is assumed that the function g (t) is defined for all values of t > 0 and that this function is such that this integral exists; that is,

∞

|g (t)| e−at dt < ∞

(C.2)

0

where a is a positive real number. This restriction means that a function g (t) that satisfies Eq. (C.2) does not increase with time more rapidly than the exponential function e−at . In addition, the function g (t) is required to be piecewise continuous. For the functions g (t) considered in this book, these conditions are satisfied. We shall confine our interest to the Laplace transform of a second-order equation with constant coefficients and a fourth-order equation with constant coefficients. In practice, the application of Laplace transforms is implemented with the use of tables of Laplace transform pairs, several of which are given in Table C.1. A large compendium of Laplace transforms and their inverse transforms is available (Roberts and Kaufman 1966). We shall now illustrate the method by examining separately these two equations of different order.

477

478

Appendix C: Laplace Transforms and the Solutions to Ordinary Differential Equations

C.2 Solution to a Second-Order Equation Consider the following second-order equation dx f (t) d2 x + 2ζ ωn + ωn2 x = dt m dt2

(C.3)

where ζ < 1, x = x (t) and t is time. If X (s) denotes the Laplace transform of x (t) and F (s) denotes the Laplace transform of f (t), then from pair 2 of Table C.1 it is seen that the Laplace transform of Eq. (C.3) is s2 X (s) − x˙ (0) − sx (0) + 2ζ ωn [sX (s) − x (0)] + ωn2 X (s) =

1 F (s) m

which, upon rearrangement, becomes X (s) =

sx (0) 2ζ ωn x (0) + x˙ (0) F (s) + + D (s) D (s) mD (s)

(C.4)

In Eq. (C.4), x (0) is the value of x at t = 0, x˙ (0) is the value of first derivative of x at t = 0, and D (s) = s2 + 2ζ ωn s + ωn2 .

(C.5)

Using transform pairs 4, 8, and 10 of Table C.1, the inverse transform of Eq. (C.4) is x˙ (0) + ζ ωn x (0) −ζ ωn t x (t) = x (0) e−ζ ωn t cos (ωd t) + e sin (ωd t) ωd

t 1 e−ζ ωn η sin (ωd η) f (t − η) dη + mωd 0 (C.6) x˙ (0) + ζ ωn x (0) −ζ ωn t −ζ ωn t cos (ωd t) + e sin (ωd t) = x (0) e ωd

t 1 + e−ζ ωn (t−η) sin (ωd (t − η)) f (η) dη mωd 0

 where ωd = ωn 1 − ζ 2 and we have used the relation sin (ωd t − ϕ) = sin (ωd t) cos (ϕ) − cos (ωd t) sin (ϕ)  = ζ sin (ωd t) − 1 − ζ 2 cos (ωd t)

Appendix C: Laplace Transforms and the Solutions to Ordinary Differential Equations

479

since, from Table C.1, ϕ = cos−1 ζ = sin−1



1 − ζ 2 ζ < 1.

Equation (C.6) can be written in another form by using the identity 

a2 + b2 sin (ωt ± ψ) b ψ = tan−1 . a

a sin (ωt) ± b cos (ωt) =

(C.7)

Thus, Eq. (C.6) becomes

−ζ ωn t

x (t) = Ao e

1 sin (ωd t + ϕd ) + mωd

t

e−ζ ωn η sin (ωd η) f (t − η) dη

(C.8)

0

where Ao and ϕ d , respectively, are given by  Ao =



x˙ (0) + ζ ωn x (0) ωd ωd x (0) . x˙ (0) + ζ ωn x (0)

2

x2 (0) +

φd = tan

−1

(C.9)

C.3 Solution to a Fourth-Order Equation Consider the following fourth-order equation 

d4 y d2 y 4 y = f (x) − 2β + Kδ − x y + k −  (x ) 1 dx4 dx2

(C.10)

where y = y (x), x is a spatial coordinate, and δ (x) is the delta function. If Y (s) denotes the Laplace transform of y (x) and F (s) denotes the Laplace transform of f (x), then from pairs 2 and 5 of Table C.1 it is seen that the Laplace transform of Eq. (C.10) is s4 Y (s) − s3 y (0) − s2 y (0) − sy (0) − y (0) + Ky (x1 ) e−x1 s  

− 2β s2 Y (s) − sy (0) − y (0) + k − 4 Y (s) = F (s) which, upon rearrangement, yields

(C.11)

480

Appendix C: Laplace Transforms and the Solutions to Ordinary Differential Equations

Y (s) =



 1  3 s − 2βs y (0) + s2 − 2β y (0) + sy (0) D (s)  + y (0) − Ky (x1 ) e−x1 s + F (s)

(C.12)

where the prime denotes the derivative with respect to x,  

D (s) = s4 − 2βs2 + k − 4 = s2 − δ 2 s2 + ε 2  ε 2 = − β + β 2 + 4 − k  δ 2 = β + β 2 + 4 − k

(C.13)

and it is assumed that β 2 + 4 − k > 0. It is noted that ε2 δ 2 = 4 − k and ε 2 − δ 2 = −2β and, therefore, when β = 0, ε 2 = δ 2 . To obtain the inverse Laplace transform, we use partial fractions on the following quantities to find that s3 − 2βs 1 ˆ¯ (s) =   = 2 Q 2 2 2 2 ε + δ2 s −δ s +ε 1 s2 − 2β  = 2 Rˆ¯ (s) =  2 ε + δ2 s − δ 2 s2 + ε 2 s 1  = 2 Sˆ¯ (s) =  2 2 2 2 ε + δ2 s −δ s +ε 1 1  = 2 Tˆ¯ (s) =  2 ε + δ2 s − δ 2 s2 + ε 2

, , , ,

sε 2 sδ 2  +  s2 − δ 2 s2 + ε 2 δ2 ε2  +  s2 − δ 2 s2 + ε 2 s s  −  2 2 2 s −δ s + ε2

-

(C.14)

1 1  −  . s2 − δ 2 s2 + ε 2

Using pairs 11 to 14 in Table C.1, it is found that the inverse Laplace transform of Eq. (C.14) is 1 δ 2 + ε2 1 Rˆ (x) = 2 δ + ε2 1 Sˆ (x) = 2 δ + ε2 1 Tˆ (x) = 2 δ + ε2

ˆ (x) = Q

  δ 2 cos (εx) + ε 2 cosh (δx)  2 δ ε2 sin (εx) + sinh (δx) ε δ [− cos (εx) + cosh (δx)]  1 1 − sin (εx) + sinh (δx) . ε δ

(C.15)

Appendix C: Laplace Transforms and the Solutions to Ordinary Differential Equations

481

The derivatives of the functions defined in Eq. (C.15) are ˆ (x) = δ 2 ε2 Tˆ (x) Q ˆ (x) Rˆ (x) = Q

 Sˆ (x) = Rˆ (x) + δ 2 − ε2 Tˆ (x)

Tˆ (x) = Sˆ (x) ˆ (x) = δ 2 ε2 Sˆ (x) Q Rˆ (x) = δ 2 ε2 Tˆ (x) 

ˆ (x) + δ 2 − ε 2 Sˆ (x) Sˆ (x) = Q 

Tˆ (x) = Rˆ (x) + δ 2 − ε2 Tˆ (x)

(C.16)

  ˆ (x) = δ 2 ε2 Rˆ (x) + δ 2 − ε 2 Tˆ (x) Q Rˆ (x) = δ 2 ε2 Sˆ (x)   2 

Tˆ (x) Sˆ (x) = δ 2 − ε 2 Rˆ (x) + δ 2 ε2 + δ 2 − ε 2 

ˆ (x) + δ 2 − ε 2 Sˆ (x) Tˆ (x) = Q where the prime denotes the derivative with respect to x. Using pairs 3 and 4 of Table C.1 and Eqs. (C.14) and (C.15), the inverse Laplace transform of Eq. (C.12) is ˆ (x) + y (0) Rˆ (x) + y (0) Sˆ (x) + y (0) Tˆ (x) y (x) = y (0) Q

x − Ky (x1 ) Tˆ (x − x1 ) u (x − x1 ) + f (η) Tˆ (x − η) dη.

(C.17)

0

where u (x) is the unit step function. When β = k = 0, Eq. (C.17) can be written as y (x) = y (0) Q (x) + y (0) R (x)/ + y (0) S (x)/2 + y (0) T (x)/3 − Ky (x1 ) T ( [x − x1 ]) u (x − x1 ) /3

x 1 + 3 f (η) T ( [x − η]) dη  0

(C.18)

482

Appendix C: Laplace Transforms and the Solutions to Ordinary Differential Equations

where 1 [cos (x) + cosh (x)] 2 1 R (x) = [sin (x) + sinh (x)] 2 1 S (x) = [− cos (x) + cosh (x)] 2 1 T (x) = [− sin (x) + sinh (x)] . 2

Q (x) =

(C.19)

The derivatives of Eq. (C.19) are Q (x) = T (x)

Q (x) = 2 S (x)

Q (x) = 3 R (x)

R (x) = Q (x)

R (x) = 2 T (x)

R (x) = 3 S (x)

S (x) = R (x)

S (x) = 2 Q (x)

S (x) = 3 T (x)

T (x) = S (x)

T (x) = 2 R (x)

T (x) = 3 Q (x)

(C.20)

where the prime denotes the derivative with respect to x. Equations (C.17) to (C.20) are used extensively in Chapter 3. The following set of transformed quantities appears in Chapter 5 [see Eq. (5.66)]: , s3 α2 s 1 β 2s  = 2  +  Qαβ (s, α, β) =  2 α + β 2 s2 − α 2 s − α 2 s2 + β 2 s2 + β 2 , 1 β2 s2 α2  = 2  +  Rαβ (s, α, β) =  2 α + β 2 s2 − α 2 s − α 2 s2 + β 2 s2 + β 2 , s 1 s s  = 2  −  Sαβ (s, α, β) =  2 α + β 2 s2 − α 2 s − α 2 s2 + β 2 s2 + β 2 , 1 1 1 1  = 2  −  . Tαβ (s, α, β) =  2 α + β 2 s2 − α 2 s − α 2 s2 + β 2 s2 + β 2 (C.21) Using pairs 11 to 14 in Table C.1, it is found that the inverse Laplace transform of Eq. (C.21) is 1 + β2 1 Rαβ (x, α, β) = 2 α + β2 1 Sαβ (x, α, β) = 2 α + β2 1 Tαβ (x, α, β) = 2 α + β2

Qαβ (x, α, β) =

α2



β 2 cos (βx) + α 2 cosh (αx)



[β sin (βx) + α sinh (αx)] (C.22) [− cos (βx) + cosh (αx)]  1 1 − sin (βx) + sinh (αx) . β α

Appendix C: Laplace Transforms and the Solutions to Ordinary Differential Equations

483

The first derivative of the functions appearing in Eq. (C.22) are 

Q αβ (x, α, β) = Vαβ (x, α, β) = α 2 − β 2 Rαβ (x, α, β) + α 2 β 2 Tαβ (x, α, β) R αβ (x, α, β) = Qαβ (x, α, β) Sαβ (x, α, β) = Rαβ (x, α, β)

Tαβ (x, α, β) = Sαβ (x, α, β) .

(C.23)

C.4 Table of Laplace Transform Pairs Table C.1 Laplace transform pairs G (s)

g (t)

Description

1

G (s/a)

ag (at)

Scaling of variable

2

sn G (s) n  sn−k gk−1 (0) −

dn g gn (t) = n dt

nth-order derivative, n = 1, 2, ...

3

e−to s G (s)

g (t − to ) u (t − to )

Shifting

4

G (s) H (s)

t

Convolution

k=1

g (η) h (t − η) dη 0

or

t g (t − η) h (η) dη 0

5

g (to ) e−sto e−sto

g (t) δ (t − to )

Delta function

s 1 s−a

u (t − to )

Unit step function

eat

Exponential

8

1 s2 + 2ζ ωn s + ωn2

1 −ζ ωn t e sin (ωd t) ωd

9

ωn2   s s2 + 2ζ ωn s + ωn2

1−

6 7

10 11

s s2 + 2ζ ωn s + ωn2 s s2 + ω2

−

ωn −ζ ωn t e sin (ωd t + φ) ωd

ωn −ζ ωn t e sin (ωd t − φ) ωd

cos (ωt)

When t is time, g (t) is impulse response of single degree-of-freedom system:  ωd = ωn 1 − ζ 2 When t is time, g (t) is step response of single degree-of-freedom system: φ = cos−1 ζ ζ < 1 φ = cos−1 ζ

ζ