Vibrations of Elastic Systems: With Applications to MEMS and NEMS

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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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 and, from Eq. (2.19), z(τ ) ∼ 1/k. This region is denoted the stiffness controlled region and is important in sensor design. The second region is when  = 1, where H() = 1/ (2ζ ) and z(τ ) ∼ 1/c. 12

ζ = 0.05 ζ = 0.1 ζ = 0.25

10

θ(Ω) (°)

H(Ω)

8 6 4 2 0

0

0.5

1 Ω (a)

1.5

2

180 160 140 120 100 80 60 40 20 0

ζ = 0.05 ζ = 0.1 ζ = 0.25

0

0.5

1 Ω (b)

1.5

2

Fig. 2.2 Response of a single degree-of-freedom system with viscous damping (a) Amplitude response and (b) Phase response

12

2 Spring-Mass Systems

This region is called the damping controlled region and is important in the design of energy harvesters. The third 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, we see from Eq. (2.36) that max average dissipated power simplifies to Pavg,max =

1

≈ 1. In this case, the maximum

PY PY Q = W. 4ζ 2

(2.39)

The difference between Eqs. (2.39) and (2.37) is less than 1% when ζ < 0.1 and as ζ decreases this difference deceases. We see from Eqs. (2.34) and (2.39) that for lightly damped systems, when the input powers are equal (PF = PY ), the maximum average dissipated powers are equal and vary inversely with the damping factor. In addition, these maximum values occur very close to the system’s natural frequency.

2.2

Some Preliminaries

15

2.2.4 Structural Damping Structural damping is a model that assumes that the dissipation in the system is due to losses in the material that provides the stiffness for the system. One structural damping model is to assume that the structural damping is independent of frequency. A model that satisfies this criterion is (Balachandran and Magrab, 2009, p. 249) Fs = kz + k

2η ∂z N ω ∂t

(2.40)

where η is an empirically determined constant. This model is restricted to systems undergoing harmonic oscillations. To obtain the governing equation of motion, we substitute Eq. (2.40) in Eq. (2.2), set Fr = 0, and employ the assumptions used to arrive at Eq. (2.5). These operations yield m

  d2 y d2z 2η dz + kz = f (t) − m 2 . + c + k 2 ω dt dt dt

(2.41)

We shall limit our discussion to the case where f (t) = 0 and, because of the restrictions on Eq. (2.40), it is assumed that y = Yo cos (ωt) . Then Eq. (2.41) becomes   d2 z 2η dz m 2 + c+k + kz = mω2 Yo cos (ωt) dt ω dt

(2.42)

or in terms of the non dimensional parameters,   d2 z 2η dz + z = Yo 2 cos (τ ) . + 2ζ +  dτ dτ 2

(2.43)

The solution to Eq. (2.43) is z(τ ) = Hv+s () cos (τ − θv+s ())

(2.44)

where the amplitude response and phase response, respectively, are 1 Hv+s () =  2 1 − 2 + (2ζ  + 2η)2 θv+s () = tan−1

2ζ  + 2η . 1 − 2

(2.45)

16

2 Spring-Mass Systems

10

180

ζ = 0.05 ζ = 0.1 ζ = 0.25

9 8

140

7

120

6

θs(Ω) (°)

Hs(Ω)

ζ = 0.05 ζ = 0.1 ζ = 0.25

160

5 4

100 80

3

60

2

40

1

20

0 0

0.5

1

1.5

0

2

0

0.5

1

Ω

Ω

(a)

(b)

1.5

2

Fig. 2.3 Response of a single degree-of-freedom system with structural damping (a) Amplitude response and (b) Phase response

If the viscous damping is neglected, then ζ = 0 and Eqs. (2.44) and (2.45), respectively, become z(τ ) = Hs () cos (τ − θs ())

(2.46)

and Hs () = 

1 2

1 − 2

θs () = tan−1

+ (2η)2

(2.47)

2η . 1 − 2

It is seen from Eq. (2.47) that the frequency at which the maximum value of the amplitude response function occurs is always at  = 1 irrespective of the value of η. Equation (2.47) is plotted in Fig. 2.3. The phase response differs from that due to viscous damping in that the phase angle does not go to zero for  << 1. However, as for the case of viscous damping, the phase angle is 90◦ when  = 1.

2.3 Squeeze Film Air Damping 2.3.1 Introduction The quality factor is frequently an important performance metric of micromechanical sensor systems. Since the quality factor is a function of how a system dissipates energy, knowledge of the damping mechanism affecting a particular micromechanical design is necessary. At the micromechanical scale, it has been found that the reaction forces generated by partially restricted viscous airflow can significantly affect a system’s response (Bao and Yang 2007; Pratap et al. 2007). We shall

2.3

Squeeze Film Air Damping

17

investigate this case, which is called squeeze film damping, at pressures that are at or slightly below atmospheric pressure. Squeeze film air damping is caused by the entrapment of air between two parallel surfaces that are moving relative to each other in the normal direction. The reactive pressure brought about by the relative movement of the surfaces consists of two components. One component is due to the viscous flow of air that is squeezed out of the volume between the two surfaces and the second component is due to compression of the air between the two surfaces. It will be shown that the former component results in a frequency-dependent damping and the latter component a frequency-dependent stiffness. We shall consider two geometries: a rigid rectangular plate and a rigid circular plate. The governing equation that is used to determine the pressure between the plates is the linearized Reynolds lubrication equation. The force in the air gap is determined by integrating the pressure over the area of the plate.

2.3.2 Rectangular Plates Consider two rigid rectangular plates of area A that are separated by a distance ho and that have a pressure in the gap of Pa . One plate is fixed and the other is oscillating harmonically at a frequency ω; that is, x = Xo cos ωt, where x is the displacement, Xo is the magnitude of the displacement and Xo << ho . The force on the thin film can be expressed as (Blech 1983) Fr,sf = kr,s Xo cos ωt − cr,d ωXo sin ωt = kr,s x + cr,d x˙ .

(2.48)

The elastic stiffness of the air film kr,s is kr,s =

Pa A Sr,k (β, σ ) N/m ho

(2.49)

Pa A Sr,d (β, σ ) Ns/m. ωho

(2.50)

and the damping coefficient cr,d is cr,d =

In Eqs. (2.49) and (2.50), the quantities Sr,k and Sr,d are given by ∞ 64σ 2  Sr,k (β, σ ) = π8

∞ 

1 Dmn (β, σ )

n=1,3,5 m=1,3,5 ∞ ∞   m2

+ (n/β)2 Dmn (β, σ ) n=1,3,5 m=1,3,5  2 2 2 2 2 2 4 + σ /π Dmn (β, σ ) = m n m + (n/β) Sr,d (β, σ ) =

64σ π6

(2.51)

18

2 Spring-Mass Systems

where σ =

12μωL2 Pa h2o

(2.52)

is the squeeze number. The quantity β = L/a is the aspect ratio of the rectangular plate such that for a narrow strip, β → ∞ and for a square plate β = 1. Since σ is a function of frequency, the stiffness and the damping coefficients are functions of frequency and, therefore, one can consider that the film acts as a viscoelastic material. The quantity μ is the dynamic viscosity of the gas between the two plates. If the gas is air at standard conditions, then μ = 1.83 × 10−5 Ns/m2 . Equations (2.49) to (2.52) are for plates that are vented on all four sides. The case when venting is restricted to fewer than four sides has also been investigated (Darling et al. 1998). A quantity that is important in evaluating squeeze film damping is the critical value of the squeeze number σ c . The critical squeeze number is defined as that value at a given value of β for which Sr,k (β, σc ) = Sr,d (β, σc ) .

(2.53)

It will be shown that for values of σ < σc , damping dominates and for values of σ > σc , stiffness dominates. For β → ∞ and σ << π 2 it is found that   6  ∞ ∞ 64σ 2 π 2 π σ2 64σ 2  1  1 = = 8 2 6 8 π n m π 8 960 120 n=1,3,5 m=1,3,5   4 ∞ ∞ π σ 64σ  1  1 64σ 2 π 2 . = = Sr,d (β, σ ) ≈ 6 2 4 6 8 96 12 π n m π Sr,k (β, σ ) ≈

n=1,3,5

(2.54)

m=1,3,5

If we use Eq. (2.54) in Eqs. (2.49) and (2.50), we obtain Pa A σ 2 << cr,d ho 120 Pa A σ → . ωho 12

kr,s → cr,d

(2.55)

Thus, the squeeze film stiffness can be ignored with respect to the squeeze film damping. In addition, it is seen that cr,d is independent of frequency; however, consideration of frequency was necessary to determine that σ << π 2 . For β → ∞ and large σ (Langlois 1962) 

2 Sr,k (β → ∞, σ ) → 1 − σ  2 . Sr,d (β → ∞, σ ) → σ

(2.56)

2.3

Squeeze Film Air Damping

19

For a given system, it is seen from Eq. (2.52) that for constant conditions σ is increased by increasing the oscillation frequency of the plate. Thus, from Eqs. (2.49), (2.50), and (2.56), it is found that as frequency becomes very large cr,d → 0 and kr,s → Pa A/ho . Therefore, when σ is very large, the air in the gap between the plates acts as a lossless compression spring and from Eq. (2.48) it is seen that the force and displacement are in phase. Values for cr,d and kr,s as a function of the squeeze number for two extreme values of 1/β are given in Fig. 2.4 along with the corresponding critical squeeze numbers. The preceding equations are based on the assumption that Pa is around one  atmosphere Pa ≈ 105 N/m2 . However, it has been shown that the viscosity of air damping is reduced appreciably when Pa is well below one atmosphere. One approach to take into account this effect so that the above results can still be used in rarified air is to introduce an effective viscosity μeff . This effective value is based on a Knudsen number Kn , which is given by Kn =

λ ho

(2.57)

In Eq. (2.57), λ is the mean free path of the air molecules and ho is the distance separating the two plates. The value of λ for air at standard conditions is λ ≈ 8×10−8 m. In addition, λ ∼ 1/Pa so that when Pa decreases, λ increases. For a separation distance of ho = 5 μm at standard conditions, Kn = 0.016. When Kn > 10, the 1 0.9

1/β = 0 1/β = 1

ksd

ksd/(PaA/ho), csd/(PaA/ωho)

0.8 0.7 0.6 0.5 σc = 10.1342

0.4 0.3

σc = 21.6349

csd

0.2 0.1 0 10−1

100

101

102

103

Squeeze number, σ

Fig. 2.4 Stiffness and damping coefficients for a rectangular plate as a function squeeze number for two values of β

20

2 Spring-Mass Systems

preceding results are not valid and a different theory to determine kr,s and cr,d has to be used (Hutcherson and Ye 2004). One of several proposed forms for μeff is (Veijola et al. 1995) μeff =

μ 1 + 9.638Kn1.159

0 ≤ Kn ≤ 10

(2.58)

and Eq. (2.52) becomes σ =

12μeff ωL2 . Pa h2o

(2.59)

For air at standard conditions, μeff = 0.9260μ = 1.71 × 10−5 Ns/m2 . Several other models for obtaining μeff have been proposed (Veijola 2004; Gallis and Torczynski 2004).

2.3.3 Circular Plates Consider two rigid circular plates of radius bo that are separated by an air gap of magnitude ho . One plate is fixed and the other is oscillating harmonically at a frequency ω; that is, x = Xo cos ωt, where x is the displacement, Xo is the magnitude of the displacement, and Xo << ho . Then, we can use Eq. (2.48) by replacing kr,s with kc,s and kr,d with kc,d where the circular plate spring constant is (Crandall 1918) kc,s =

Pa A Sc,k (σ ) N/m ho

(2.60)

Pa A Sc,d (σ ) Ns/m. ωho

(2.61)

and the damping coefficient cc,d is cc,d =

The quantities Sc,k and Sc,d are given by

where

 √  √ 1 2 Sc,k (σ ) = 1 + D (σ ) bei σ + C (σ ) ber σ E (σ ) σ  √  √ 2 1 D (σ ) ber σ − C (σ ) bei σ Sc,d (σ ) = E (σ ) σ

(2.62)

√ √ D (σ ) = ber1 σ + bei1 σ √ √ C (σ ) = ber1 σ − bei1 σ √ √ E (σ ) = ber2 σ + bei2 σ .

(2.63)

2.3

Squeeze Film Air Damping

21

In these equations, the squeeze number σ is given by Eq. (2.59) with L replaced by bo . The Kelvin functions berν and beiν can be determined from

 berν (x) + jbeiν (x) = Jν xe3π j/4 where Jν (x) is the Bessel function of the first kind of order ν. It is convention to omit the subscript when ν = 0. These results have been extended to include a flow restriction coefficient that considers the pressure variation at the exit of the squeezed film from the circular plate (Perez and Shkel 2008). The critical squeeze number σ c is determined from Sc,k (σc ) = Sc,d (σc ) .

(2.64)

A graph of cc,d and kc,s as a function of the squeeze number is given in Fig. 2.5 along with the value of σ c .

2.3.4 Base Excitation with Squeeze Film Damping Consider a mass m that is supported by a spring with constant k as shown in Fig. 2.1. If the motion of the mass is resisted by squeeze film damping, then the reaction of the squeeze force on the mass is given by Eq. (2.48). In addition, it is assumed that 1 0.9

kc,sd /(PaA/ho) csd /(PaA/ωho)

Normalized magnitude

0.8 0.7 0.6 0.5 0.4

σc = 6.2332

0.3 0.2 0.1 0 10−2

10−1

100

101

102

103

Squeeze number, σ

Fig. 2.5 Stiffness and damping coefficients for a circular plate as a function squeeze number

22

2 Spring-Mass Systems

the base is subjected to a harmonic displacement of the form y(t) = Yo cos (ωt). To obtain the governing equation of motion, we substitute Eq. (2.48) in Eq. (2.2), set f (t) = 0, and employ the assumptions used to arrive at Eq. (2.5). Thus, m

dz d2 z + (k + ks ) z = mω2 Yo cos (ωt) + (c + cd ) 2 dt dt

(2.65)

where for a mass with a surface with a rectangular shape ks = kr,s and cd = cr,d , which are given by Eqs. (2.49) and (2.50), respectively, and for a mass with a surface of circular shape ks = kc,s and cd = cc,d , which are given by Eqs. (2.60) and (2.61), respectively. To bring some notational order to Eq. (2.65), we introduce the following non dimensional quantities rk =

Pa A , kho

σrn =

12μeff ωn L2 , Pa h2o

σcn =

12μeff ωn b2o Pa h2o

(2.66)

where rk is the ratio of the force on the plate due to atmospheric pressure to the force required to stretch the spring the length of the gap. As will be seen subsequently, when these results are numerically evaluated, rk is a very influential parameter. Equation (2.65) is written explicitly for masses with rectangular and circular surfaces as follows. Rectangular  dz   d 2 zr rk r S + 1 + r σ σ + 2ζ + (β, ) S (β, ) zr = 2 Yo cos (τ ) r,d rn k r,k rn  dτ dτ 2 (2.67) where Sr,k (β, σrn ) and Sr,d (β, σrn ) are given by Eq. (2.51). Circular  dz   rk d 2 zc c + 2ζ + ) S + 1 + rk Sc,k (σcn ) zc = 2 Yo cos (τ ) (σ c,d cn 2 dτ  dτ (2.68) where Sc,k (σcn ) and Sc,d (σcn ) are given by Eq. (2.62). We assume a solution to Eqs. (2.67) and (2.68) of the form z(τ ) = Zo cos (τ ). Then the solution to Eq. (2.67) is zr (τ ) = Yo 2 Hr () cos (τ − θr )

(2.69)

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

 2 −1/2 2  1 + rk Sr,k (β, σrn ) − 2 + 2ζ  + rk Sr,d (β, σrn )

θr () = tan−1

2ζ  + rk Sr,d (β, σrn ) . 1 + rk Sr,k (β, σrn ) − 2

(2.70)

2.3

Squeeze Film Air Damping

23

The solution to Eq. (2.68) is zc (τ ) = Yo 2 Hc () cos (τ − θc )

(2.71)

where amplitude response and phase response, respectively, are Hc () =

 2 −1/2 2  1 + rk Sc,k (σcn ) − 2 + 2ζ  + rk Sc,d (σcn )

θc () = tan−1

(2.72)

2ζ  + rk Sc,d (σcn ) . 1 + rk Sc,k (σcn ) − 2

When rk = 0, Eqs. (2.69) to (2.72) reduce to Eqs. (2.19) and (2.20) with Fo = 0. Representative values of Hr () and θ r () for a square plate with ζ = 0 are shown in Figs. 2.6 to 2.8. The results for Hr can be qualitatively explained as follows. From Eq. (2.70),it is seen that Hr will have a maximum value when approximately  = max = 1 + rk Sr,k (η, σrn max ). When σ >> σrn,c , where σ rn,c is the critical squeeze film number √ for a rectangular plate, we obtain from Eq. (2.56) that Sr,k ≈ 1 so that max ≈ 1 + rk . It is seen in these figures that this is approximately true when σrn = 100. From Fig. 2.4, it is seen that when σ = σrn,c , Sr,d (η, σrn.c max ) is a maximum. Hence, the maximum value of Hr () is a minimum when σ = σrn,c . An examination of Figs. 2.6 to 2.8 reveals these attributes. When the phase response is compared to that of a system with only viscous damping given in Fig. 2.2b, it is seen that with substantial squeeze film damping the phase responses appear to be dissimilar. However, on closer analysis, it is seen that the responses are more similar than they appear if it is realized that the resonance frequency has shifted to max . At this frequency, the phase angle for Q > 10 is very nearly 90◦ , the same as it is for the system with viscous damping. When Q < 10, it has been found numerically that the phase angle at max can be as much as 9◦ away from 90◦ . 100

180 σn = 3, Q = 96.8 σn = σc = 21.6, Q = 27.9 σn = 100, Q = 46.6

90 80

160 140 120

60

θr(Ω) (°)

Hr(Ω)

70 50 40

100 80 60

30 20

40

10

20

0 0.6

0.8

1

1.2

1.4

1.6

0 0.6

σn = 3 σn = σc = 21.63 σn = 100

0.8

1

1.2

Ω

Ω

(a)

(b)

1.4

1.6

Fig. 2.6 Values of Hr (), θ r (), and Q for a mass with a square surface (1/β = 1) subjected to squeeze film resistance for σ n = 3, 21.63, and 100, and rk = 0.1 (a) Amplitude response and (b) Phase response

24

2 Spring-Mass Systems

10

180

σn = 3, Q = 9.55

9

σn = σc = 21.6, Q = 3.33 σn = 100, Q = 8.16

8

160 140

7

120 θr(Ω) (°)

Hr(Ω)

6 5 4

100 80 60

3 2

40

σn = 3

1

20

σn = σc = 21.63 σn = 100

0

0 0

0.5

1

1.5

2

2.5

0

3

0.5

1

1.5

Ω

Ω

(a)

(b)

2

2.5

3

Fig. 2.7 Values of Hr (), θ r (), and Q for a mass with a square surface (1/β = 1) subjected to squeeze film resistance for σ n = 3, 21.63, and 100, and rk = 1.0 (a) Amplitude response and (b) Phase response 3.5

180

σn = 3, Q = 2.91 σn = σc = 21.6, Q = 1.82 σn = 100, Q = 5.92

3

160 140

2.5 θr(Ω) (°)

Hr(Ω)

120 2 1.5

100 80 60

1

40 0.5 0

σn = 3 σn = σc = 21.63 σn = 100

20 0

0.5

1

1.5

2

2.5

3

0

0

0.5

1

1.5

Ω

Ω

(a)

(b)

2

2.5

3

Fig. 2.8 Values of Hr (), θ r (), and Q for a mass with a square surface (1/β = 1) subjected to squeeze film resistance for σ n = 3, 21.63, and 100, and rk = 3.0 (a) Amplitude response and (b) Phase response

2.3.5 Time-Varying Force Excitation of the Mass Consider a mass whose planar shape parallel to the fixed surface is a thin strip that is subjected to a time varying force. In this case, the squeeze film damping force is estimated by (McCarthy et al. 2002) Fsq = −

μa3 L ∂h . h3 ∂t

(2.73)

If, at static equilibrium, the film’s gap is ho and the displacement of the mass is x, then h = ho − x. Noting that Fr (x, x˙ , x¨ ) = −Fsq , Eq. (2.6) becomes

2.3

Squeeze Film Air Damping

m

25

  d2x μeff a3 L dx + kx = f (t) + c + dt2 (ho − x)3 dt

(2.74)

and we have replaced the viscosity by its effective value μeff . To convert Eq. (2.74) to non dimensional form, we use Eqs. (2.9) and (2.11) and introduce the following non dimensional quantities f (t) fˆ (τ ) = kho ωn a3 Lμeff x γ = , y¯ = 3 ho kho

(2.75)

  d¯y d2 y¯ γ + 2ζ + + y¯ = fˆ (τ ). dτ 2 (1 − y¯ )3 dτ

(2.76)

τ = ωn t,

to obtain

The numerically obtained solution to Eq. (2.76) for fˆ (τ ) = fo u(τ ), where u (τ ) is the unit step function, is shown in Fig. 2.9 for ζ = 0, γ = 0.015 and 0.2, and γ = 0.015 fo = 0.65 ζequiv = 0.311 1

γ = 0.2 fo = 0.65 ζequiv = 0.879

0.8

y(τ)

y(τ)

0.6 0.5

0.4 0.2

0

0

5

10 τ

15

0

20

0

0.8

0.6

0.6

0.4 0.2 0

10 τ

15

20

γ = 0.2 fo = 0.3 ζequiv = 0.256

0.8

y(τ)

y(τ)

γ = 0.015 fo = 0.3 ζequiv = 0.0276

5

0.4 0.2

0

5

10 τ

15

20

0

0

5

10 τ

15

20

Fig. 2.9 Displacement response of a single degree-of-freedom system with nonlinear squeeze film damping that is subject to a unit step function of magnitude fo for γ = 0.015 and 0.2 and fo = 0.3 and 0.65: —— system with ζ = 0 and γ = 0; - - - system with γ = 0 and ζ = ζequiv

26

2 Spring-Mass Systems

fo = 0.3 and 0.65. We have also plotted in these figures the response when γ = 0 and ζ = ζequiv , where ζ equiv is that value that for which the maximum value y¯ max of the system with ζ = 0 and γ = 0 equals the maximum values of the system with γ = 0 and ζ = 0. The quantity ζ equiv is determined from a numerical solution to (Balachandran and Magrab 2009, p. 301) y¯ max

   2 −π ζequiv 1−ζequiv . = fo 1 + e

(2.77)

It is seen from these figures that the magnitude of fo causes the response to be nonlinear. When fo is sufficiently small, the systems with nonlinear squeeze film damping can be approximated by the system with constant viscous damping. Furthermore, when the effects of the nonlinear squeeze film damping are important, their damping is such that the system’s response time is faster; that is, it reaches its maximum value at an earlier time than that of a system with constant viscous damping.

2.4 Viscous Fluid Damping 2.4.1 Introduction There are occasions in atomic force microscopy (AFM) where the probe tip is immersed in a fluid. In order for AFM to produce correct results one needs to know the location of the resonance frequency and its magnitude. Consequently, a model that can incorporate the effects of a viscous fluid can be important. Consider the motion of a long rigid circular rod of length L oscillating harmonically at a frequency ω. The rod is in a viscous fluid that is itself infinite in extent. If the displacement transverse to the axis of the cylinder is denoted x and Xo is the magnitude of the displacement, then x = Xo e jωt . If the rod has a diameter b and is in a fluid of density ρ f (kg.m3 ) and dynamic viscosity μf (Ns/m2 ), then the hydrodynamic force Ff on the object is given by Eq. (2.281) of Appendix 2.1 (Chen et al. 1976; Sader 1998; Kirstein et al. 1998) Ff = ρf ALω2 cir (ω) Xo ejωt N

(2.78)

where A = π b2 /4 is the cross sectional area of the rod and the non dimensional quantity cir (ω) is called a hydrodynamic function and is a complex-valued function of ω. An expression for cir (ω) will be given subsequently. Another form of Eq. (2.78) is given by Eq. (2.280) of Appendix 2.1 as Ff = −ma x¨ − cv x˙

(2.79)

where the negative sign indicates the force is in a direction opposite to the displacement and

2.4

Viscous Fluid Damping

27

ma = ρf ALReal (cir (ω)) kg

(2.80)

cv = −ρf ALωImag (cir (ω)) kg/s.

The quantity ma is the added mass and the quantity cv is the damping coefficient of the viscous fluid. Thus, a rigid object oscillating in a viscous fluid will change the inertia of the object and add damping and both the added mass and the damping vary with frequency. From Eq. (2.278) of Appendix 2.1, the hydrodynamic function for a rod whose cross section is a circle of diameter b is  √ 4K1 j Re  √ (2.81) cir (ω) = 1 + √ j ReK0 j Re where K0 and K1 , respectively, are the modified Bessel functions of the second kind of order zero and order one, and Re =

ρf ωb2 4μf

(2.82)

is the Reynolds number. Although Eq. (2.81) is for a circular cross section, a correction function to make this function applicable to rectangular cross sections is given in Section 4.2.1 and Appendix 4.1 of Chapter 4. Equation (2.81) is plotted in Fig. 2.10. It seen that as Re increases, Real [cir (ω)] → 1 and Imag [cir (ω)] → 0; that is, damping effects diminish and

Real(Γcir) -Im(Γcir)

Real(Γcir), -Im(Γcir)

101

100

10−1

10−2 −1 10

100

101

102

Re Fig. 2.10 Real and imaginary parts of  cir as a function of Re

103

104

105

28

2 Spring-Mass Systems

the added mass term dominates: Ff → ma ω2 Xo ejω . We see from Eq. (2.82) that Re increases if either the frequency increases or the viscosity decreases; that is, the fluid becomes inviscid. Thus, for an inviscid (ideal) fluid the losses are zero, but there is an added mass that is proportional to the amount of volume displaced by the rod. It can be shown that for Re >> 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 << 1, then motion about the static equilibrium position can be determined from dwˆ dˆy d 2 wˆ + 2ζ + (1 − 2 h) w ˆ = 2ζ + yˆ . 2 dτ dτ dτ

(2.122)

A solution to Eq. (2.122) is obtained for the case where the base is excited harmonically. Thus, yˆ = Yo ejτ wˆ = Wo ejτ .

(2.123)

Substituting Eq. (2.123) into Eq. (2.122), yields Wo = Ha ()e−jθa () Yo

(2.124)

40

2 Spring-Mass Systems

where the amplitude response and phase response, respectively, are    1 + (2ζ )2 Ha () =   2 1 − 2 h − 2 + (2ζ )2   2ζ  2 h + 2 .  θa () = tan−1 1 − 2 h + 2 4ζ 2 − 1

(2.125)

Graphs of Ha () and θ a () for ζ = 0.05 and 2h = 0 and 0.035 are presented in Fig. 2.17. It is seen from this figure that the presence of this type of force shifts the resonance frequency to a lower value, increases the magnitude of the amplitude response at the resonance, and increases the phase lag. From Eq. (2.120), it is seen that the value of h can be adjusted by changing the value of do , which is the nominal distance of the mass from the attractive surface. If do is adjusted and measured so that the original amplitude at the driving frequency is attained, then one has a very good estimate of how close the vibrating mass was to the surface. The original amplitude is that amplitude for which the van der Waals force is negligible. Consequently, this type of measurement and control system can be used to measure properties of materials from its surface topography and forms the basis of atomic force microscopy. Referring to Fig. 2.17a, the procedure to estimate the distance for the surface is as follows. The mass is driven at a frequency that is slightly higher than the resonance frequency of the system when the effects of the van der Waals forces are negligible. For the system shown in Fig. 2.17a, it is chosen 2% higher. When the distance between the mass and the surface is such that the van der Waals force shifts the natural frequency to a lower value due to spring softening, it is seen that the amplitude response of the mass decreases. For the system shown in Fig. 2.17a, the non dimensional magnitude of this decrease is 1.24. One then increases the distance 12

200 2h = 0.035 2h = 0

180

10

160 ΔHa = 1.24

140 θa(Ω) (°)

Ha(Ω)

8 6 4

Δθa = 13.7°

120 100 80 60 40

2

1

Ω = 1.02

20

Ω = 1.02 0 0.5

2h = 0.035 2h = 0

1.5

0 0.5

1

Ω

Ω

(a)

(b)

1.5

Fig. 2.17 Response of a single degree-of-freedom system subject to a van der Waals force on the mass for ζ = 0.05 (a) Amplitude response and (b) Phase response

2.6

Energy Harvesters

41

between the mass and the surface until the magnitude of the mass’s displacement returns to its original value, in this case 9.16. The amount that the mass had to be moved is an estimate of the surface to the datum distance. A similar procedure can be developed when the change in phase angle is measured.

2.6 Energy Harvesters 2.6.1 Introduction In our context, the purpose of the energy harvester is to convert mechanical kinetic energy into electrical energy. Until recently, there have been very few generators that delivered power below 100 W. In this region, the primary source of energy comes from batteries. The obvious disadvantage of batteries is that they have to be replaced or recharged. There are many situations, however, where replacement or recharging of batteries is inconvenient or difficult. Examples of these situations include: devices implanted in the human body, such as pace makers; devices intended for operation for long periods of time such as monitoring/tracking devices; and systems that are physically remote such as ocean buoys. Another burgeoning area is the development of wireless devices that can transmit either continuously or intermittently information from distant locations. In order to take advantage of these devices’ inherent interconnection simplicity—no wires—they need a source of energy that is available for extended periods of time. For these and other applications, harvesting power from a continuously renewable energy source would be desirable. This continuously available energy can come for many different vibratory sources: industrial rotating machinery and manufacturing process equipment; moving vehicles such as railroad cars and aircraft; and ocean waves. There are many candidate solutions that have been explored and these are discussed in several review articles (Mitcheson et al. 2008; Arnold 2007; Anton and Sodano 2007; Beeby et al. 2006). We shall limit our discussion to two types of devices that have received considerable attention in the last decade: piezoelectric and electromagnetic. Piezoelectric materials have the ability to convert directly applied strain energy into usable electric energy and they can relatively easily be integrated into a system. On the other hand, magnetic power generation schemes using permanent magnets offer potentially high power density and efficiency. We shall discuss both of these systems in the context of a single degree-of-freedom system. The piezoelectric systems will also be discussed in Section 4.4 when they are coupled to thin elastic beams. However, the conversion circuitry needed to convert the output power to a form that can be used, for example, to charge a battery is not considered. The single degree-of-freedom system has been selected as a starting point because it exhibits the essential features of these types of systems and the factors that influence the power that can be generated. It has been found that vibration energy is

42

2 Spring-Mass Systems

best suited to those systems that are inertial; that is, ones that have their mechanical components attached to an inertial frame, which acts as the fixed reference. The inertial frame transmits the vibrations to the mass of the single degree-of-freedom system thereby producing a relative displacement between them. If the resonant frequency of the system appears in the frequency spectrum of the application environment, then the inertial frame’s vibration amplitude can be magnified and the power harvested can be maximized. In the case of the piezoelectric element, the strain is magnified and in the case of an electromagnetic element, the velocity is magnified. We shall introduce each of these harvesting devices independently; however, there have been several attempts at employing a unified approach to modeling various types of energy harvesters (Tilmans 1996; Roundy 2005; Halvorsen 2008).

2.6.2 Piezoelectric Generator Flexible piezoelectric materials are attractive for power harvesting applications because of their ability to withstand large amounts of strain, which provides more mechanical energy for conversion into electrical energy. There are two common ways in which piezoelectric materials have their voltages applied to them. These two means, which are called modes, are shown in Fig. 2.18. In the 33 mode, a force is applied in a direction perpendicular to the surfaces receiving the voltage; that is, parallel to the poling direction.2 In the 31 mode, a force is applied in the direction parallel to the surfaces receiving the voltage; that is, perpendicular to the poling direction. Associated with each mode is a coupling coefficient where, typically, the 33 mode coefficient is much greater that the 31 coefficient. The piezoelectric

3 2

F 1

P

Fig. 2.18 Two common modes of actuation of piezoelectric materials

F 33 mode

−

A V>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

and k33 is the electromechanical coupling coefficient, which relates the total energy placed into the piezoelectric element to the amount of energy that it converts. Comparing the second equation of Eq. (2.126) with the second equation of Eq. (2.128), it is seen that d33 = cE33 d33 sE33

 Do T 2 1 − k33 . = E = ε33 s33

e33 = S ε33

(2.130)

Typical values for the parameters appearing in the above relations are given in Table 2.1. Table 2.1 Representative values for several parameters of a piezoelectric material

Parameter

Value

Units

d33 S ε33 E s33 k33 ke (computed from k33 ) k31

400 × 10−12 1.3 × 10−8 16 × 10−12 0.75 1.134 0.35

m/V or C/N F/m m2 /N – – –

2.6

Energy Harvesters

45

When the 31 mode is being considered, the subscripts on the quantities are changed according to the following general form Tp = cEpq Sq − ekp Ek

(2.131)

SE . Di = eiq Sq + εik k

Thus, T1 = cE11 S1 − e31 E3

(2.132)

S E . D3 = e31 S1 + ε33 3

The governing equation of motion is given by Eq. (2.5) with f (t) = 0 and with the change of notation w (t) = z (t); thus, m

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

(2.133)

where FT (= Fr ) is the force that the piezoelectric element exerts on the mass m. We now put Eq. (2.126) into a more useful form. If A is the area of the piezoelectric element that is in contact with the mass and h its height, then by definition V = −E3 h is the voltage created by its deformation, w = S3 h is the displacement resulting from the application of the stress on the area A, FT = AT3 , and q = AD3 is the charge. In addition, we introduce the quantities S = Aε S h F Cpe 33

(2.134)

E = AcE /h N/m Kpe 33

S is the loaded capacitance of the piezoelectric and K E is the stiffness where Cpe pe of the piezoelectric element. Hence, we multiply Eq. (2.126) by A and make the appropriate substitutions to obtain

A E FT = e33 V + Kpe w h A S V + e33 w. q = −Cpe h

(2.135)

Using the first equation of Eq. (2.135) in Eq. (2.133) yields m

 dw d2 y d2 w A E w + e + c . + k + K V = −m 33 pe dt h dt2 dt2

(2.136)

We restrict our discussion to the case of harmonic oscillations of the form w = Wo ejωt ,

V = Vo ejωt ,

and

y = Yo ejωt

(2.137)

46

2 Spring-Mass Systems

where Yo is known and ω is the excitation frequency. Under these harmonic conditions, we consider the case where the output of the piezoelectric element is attached to three impedances configured as shown in the Fig. 2.19a. Impedance Zp = Rp +jXp represents the shunt resistance and/or capacitance, Zs = Rs +jXs represents the stray resistance and/or capacitance, and ZL = RL + jXL represents the load. It is recalled that for harmonic oscillations Z = R for a resistor R, Z = 1/ (jωC) for a capacitor C, and Z = jωL for an inductance L. The piezoelectric element sees a total impedance Z that is given by  Z=

1 1 + Zp Zs + ZL

−1

=

Zp (Zs + ZL ) . Zp + Zs + ZL

(2.138)

In addition, V dq = . dt Z

(2.139)

Returning to the second equation of Eq. (2.135), we differentiate it with respect to time and use Eq. (2.139) to obtain (Renno et al. 2009; duToit et al. 2005) S Cpe Z

AZ dw dV + V − e33 = 0. dt h dt

(2.140)

Hence, the coupled equations governing the piezoelectric harvester/generator and the single degree-of-freedom system are given by Eqs. (2.136) and (2.140). Substituting Eq. (2.137) into Eqs. (2.136) and (2.140), respectively, we arrive at

 e33 1 + kK − 2E + j2ζE E Wo + E Vo = 2E Yo c33 e33 (1 + jE Zn ) Vo − jE S Zn Wo = 0 ε33 where

 k kK = E , Kpe ω E = E , ωn

ωnE =

E Kpe

m c 2ζE = mωnE

(2.141)

rad/s (2.142)

and S S S Zn = ωnE Cpe Z = ωnE Cpe R + jωnE Cpe X = Rn + jXn

where R is the resistance and X is the reactance of the attached circuit.

(2.143)

2.6

Energy Harvesters

47

Solving Eq. (2.141) for Wo and Vo , it is found that Yo 2E (1 + jE Zn ) m A + jB j3 Zn Vref V Vo = E A + jB

Wo =

(2.144)

where   A = 1 + kK − 2E (1 − E Xn ) − 2ζE 2E Rn − E Xn ke2    B = E 2ζE (1 − E Xn ) + Rn 1 + kK − 2E + ke2 Vref =

Yo e33 S ε33

(2.145)

V

and ke is the alternative electromechanical coupling coefficient given by ke2 =

e233 S cE33 ε33

(2.146)

which is a non dimensional quantity. Using Eqs. (2.129) and (2.130) in Eq. (2.146), the alternative electromechanical coupling coefficient is related to the electromechanical coupling coefficient k33 by ke2 =

2 k33 2 1 − k33

(2.147)

Eq. (2.129). The coefficient k33 is less than one; therefore, where k33 is given by √ ke > 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 << 1, 1 → 0 and 2 → 1. Thus, we see that the effect of attaching one single degree of freedom system to another single degreeof-freedom system is to create a system that has its natural frequencies below and above the respective natural frequencies of the uncoupled systems. The numerical evaluation of Eq. (2.208) is shown in Fig. 2.26. k32 = 0

k32 = 0

5

0.5

Ω2

Ω1

1

0 2

4

1 ωr

0 2

0 0

ωr

mr

k32 = 3

0 0

2 mr

k32 = 3

5 Ω2

4 Ω1

4

1

2

2 0 2

4

1 ωr

2 0 0

mr

0 2

4

1 ωr

0 0

2 mr

Fig. 2.26 Natural frequency coefficients for a two degree-of-freedom system as a function of mr and ωr for two values of k32

62

2 Spring-Mass Systems

Frequency Response Functions The frequency response functions Hln (j) are defined as Hln (j) = k1 Gln (j)

l, n = 1, 2.

(2.211)

Thus, E(j) D(j) C(j) H21 (j) = k1 G21 (j) = D(j) B(j) H12 (j) = k1 G12 (j) = = H21 (j) mr D(j) A(j) . H22 (j) = k1 G22 (j) = mr D(j) H11 (j) = k1 G11 (j) =

(2.212)

The magnitude |Hln (j) | is called the amplitude response function. The amplitude response functions |Hl1 (j) | for k3 = c3 = 0 are shown Figs. 2.27 and 2.28 along with the corresponding values of 1 and 2 obtained from Eq. (2.208). It is seen that each of the curves for |H11 (j) | has a minimum between 1 and 2 that occurs close to the value predicted by Eq. (2.205); that is, at  = ωr . Vibration Absorber When it is necessary to limit the magnitude of the displacement response of a single degree-of-freedom system with a fixed base whose mass is being excited at its resonance frequency, one technique is to attach another single degree-of-freedom 12

Ω 1 Ω2 0.6 0.584 1.03

ωr

10

1

25

ωr = 0.6 ωr = 1 ωr = 1.5

0.854 1.17

1.5 0.928 1.62

ωr

Ω1 Ω2 0.6 0.584 1.03 1

20

ωr = 0.6 ωr = 1 ωr = 1.5

0.854 1.17

1.5 0.928 1.62

|H21(jΩ)|

|H11(jΩ)|

8 6

15

10

4

5

2 0

0

0.5

1

1.5

2

0

0

0.5

1

Ω

Ω

(a)

(b)

1.5

2

Fig. 2.27 Amplitude response function for k3 = c3 = 0, ζ1 = ζ2 = 0.05, ωr = 0.5, and 1.5 and mr = 0.1 (a) |H11 ( j)| and (b) |H21 ( j)|. The quantities 1 and 2 are the natural frequencies of the system

2.7

Two Degree-of-Freedom Systems

12

ωr

10

1

Ω1 Ω2 0.6 0.536 1.12

63 25

ωr = 0.6 ωr = 1 ωr = 1.5

0.707 1.41

1.5 0.772 1.94

ωr

Ω1 Ω2 0.6 0.536 1.12 1

20

ωr = 0.6 ωr = 1 ωr = 1.5

0.707 1.41

1.5 0.772 1.94

|H21(jΩ)|

|H11(jΩ)|

8 6

15

10

4 5

2 0

0

0.5

1

1.5

2

0

0

0.5

1

Ω

Ω

(a)

(b)

1.5

2

Fig. 2.28 Amplitude response function for k3 = c3 = 0, ζ1 = ζ2 = 0.05, 1.0, ωr = 0.5, and 1.5 and mr = 0.5 (a) |H11 ( j)| and (b) |H21 ( j)|. The quantities 1 and 2 are the natural frequencies of the system

system to the mass of the original system. This forms a two degree-of-freedom system that can be described by the previous results by setting X3 = 0 and k3 = c3 = 0. We notice from Eqs. (2.201) to (2.203) that X1 (j) =

g1 −2 + 2jζ2 ωr  + ωr2 g1 E(j) = . k1 D(j) k1 D(j)

(2.213)

If we assume that c2 = 0 and select  = ωr , then from Eq. (2.213) it is seen that X1 = 0. Since the original system was oscillating at  = 1, ωr = 1 or ωn1 = ωn2 ; that is, the uncoupled natural frequencies of each single degree-of-freedom system should be equal. Under these assumptions and assuming that c1 = 0, the displacement of m2 becomes X2 (j) =

g1 C(j) g1 ωr2 g1 = →− . k1 D(j) k1 D(j) k2

(2.214)

The final result was obtained by setting ζ1 = ζ2 = c3 = k3 = 0 and  = ωr = 1 in Eqs. (2.203) and (2.204). To show the effects of c2 on |H11 (j) |, we examine the case for ζ2 = 0.05, 0.15, and 0.25 when X3 = k3 = c3 = c1 = 0. The results are shown in Fig. 2.29 for mr = 0.05 and ωr = 1. It is seen from the figure that irrespective of the value of ζ 2 there are two frequencies at which the amplitude response curves always have the same amplitude values. These values are determined by first rewriting H11 (j) as follows H11 (, ζ2 ) =

E (, ζ2 ) D (, ζ2 )

(2.215)

64

2 Spring-Mass Systems 16 ζ2 = 0.05 ζ2 = 0.15

14

ζ2 = 0.25

12

|H11(jΩ)|

10

Ω1 = 0.919

8 Ω2 = 1.08

6 4 2 0 0.4

0.6

0.8

1

1.2

1.4

1.6

Ω

Fig. 2.29 Amplitude response function |H11 ( j)| for X3 = k3 = c3 = c1 = 0, mr = 0.05, ωr = 1, and ζ2 = 0.05, 0.15, and 0.25. The values of  at which |H11 ()| intersect irrespective of the value of ζ 2 is obtained from Eq. (2.217)

to explicitly indicate the dependence of H11 (j) on ζ 2 . Then, to find the values of  at which the amplitude response is independent of ζ 2 , we find the value of , ˆ for which denoted , !

! !

! ! ˆ ζ2 !! = !!H11 , ˆ ζ2 !! (2.216) !H11 , ˆ it is found that where ζ2 = ζ2 . Solving Eq. (2.216) for ,     2  1 ˆ 1,2 = 1 + ωr2 (1 + mr ) ∓ ωr2 − 1 + mr (2 + mr ) .  2 + mr

(2.217)

Harmonic Base Excitation When the base of a two degree-of-freedom system is excited harmonically, we set g1 = g2 = 0. In addition, for convenience, we set k3 = c3 = 0. Then, from Eqs. (2.202) and (2.203), the displacements become E(j) (2jζ1  + 1) X1 (j) = X3 D(j) C(j) (2jζ1  + 1) X2 (j) = . X3 D(j) Representative results for Eq. (2.218) are shown in Fig. 2.30.

(2.218)

2.7

Two Degree-of-Freedom Systems

65

mr = 0.2 ωr = 0.4

mr = 0.8 ωr = 0.4 15

Magnitude

Magnitude

15

10

5

0

0

0.5

1 Ω

1.5

10

5

0

2

0

0.5

mr = 0.2 ωr = 1

2

1.5

2

20

20

15 Magnitude

Magnitude

1.5

mr = 0.8 ωr = 1

25

15 10

10 5

5 0

1 Ω

0

0.5

1 Ω

1.5

2

0

0

0.5

1 Ω

Fig. 2.30 Amplitude ratios of a two degree-of-freedom system subjected to a harmonic base excitation of magnitude X3 for k3 = c3 = 0, ζ1 = ζ2 = 0.05, mr = 0.2 and 0.8, and ωr = 0.4 and 1.0: —— |X1 ( j)/X3 |; – – – |X2 ( j)/X3 |

2.7.3 Enhanced Energy Harvester To increase the performance of a piezoelectric energy harvester of the type discussed in Section 2.6.2 and shown in Fig. 2.19a, one can attach the base of this system to the mass of another single degree-of-freedom system (Cornwell et al. 2005; Ma et al. 2010; Aldraihem and Baz 2011). This forms a base-excited two degree-offreedom system shown in Fig. 2.24 with k3 = c3 = f1 (t) = f2 (t) = 0. For this system to represent an energy harvester, the spring k2 in Fig. 2.24 is replaced with a piezoelectric element and its electrical load as shown in Fig. 2.19a. As was done in Section 2.6.2, we shall only consider the case when Z = RL , where RL is the load resistor. The governing equations can be obtained by replacing the force term k2 (x2 − x1 ) in Fig. 2.25 with the force term FT given by Eq. (2.135) with w = x2 − x1 ; that is, A E Kpe (x2 − x1 ) + e33 V h

66

2 Spring-Mass Systems

E is given by Eq. (2.134) and V is the voltage generated by the deformation where Kpe of the piezoelectric element. The other terms are defined in Section 2.6.2. Then Eq. (2.190) becomes

 d2 x1 dx2 dx1 E E + + c + K x2 + k − Kpe (c ) 1 2 1 pe x1 − c2 2 dt dt dt dx3 A − k1 x3 − e33 V = 0 − c1 dt h d2 x2 dx2 dx1 A E E + Kpe − Kpe m2 2 + c2 x2 − c2 x1 + e33 V = 0. dt dt h dt m1

(2.219)

It is noted that the second equation of Eq. (2.219) is identical to Eq. (2.136) when in Eq. (2.219) w = x2 − y, x1 = y, m2 = m, c2 = c and in Eq. (2.136) k = 0. The voltage and the displacements are related by Eq. (2.140) with w = x2 − x1 ; that is, S RL Cpe

dV ARL + V − e33 dt h



dx2 dx1 − dt dt

 =0

(2.220)

S is given by Eq. (2.134). where Cpe Substituting the quantities in Eq. (2.191) into Eqs. (2.119) and (2.220), respectively, we obtain

 d2 x1 dx1 dx2 2 x1 − 2ζ2 mr ωr + + 2ζ m ω ω + 1 + m (2ζ ) 1 2 r r r r 2 dτ dτ dτ dx 3 −mr ωr2 x2 − e˜ 33 V = 2ζ1 + x3 dτ d2 x2 dx2 dx1 e˜ 33 + 2ζ2 ωr V=0 + ωr2 x2 − 2ζ2 ωr − ωr2 x1 + dτ 2 dτ dτ mr

(2.221)

and dV rL dτ

+V −

e33 rL S ε33



dx2 dx1 − dτ dτ

 =0

(2.222)

where A C/N hk1 S R ω . rL = Cpe L n1

e˜ 33 = e33

(2.223)

In Eq. (2.191), we note from the change in notation that [see Eq. (2.142)]  ωn2 =

E Kpe

m2

= ωnE

rad/s,

E Kpe

k1

= ωr2 mr .

(2.224)

2.7

Two Degree-of-Freedom Systems

67

We now consider the case of the system undergoing harmonic oscillations of the form V = Vo ejτ ,

xl = Xl ejτ

l = 1, 2, 3

(2.225)

where X3 is known,  = ω/ωn1 , and ω is the excitation frequency. Substituting Eq. (2.225) into Eq. (2.222) and solving for Vo gives Vo =

e33 S ε33

R(j) (X2 − X1 )

(2.226)

jrL . 1 + jrL

(2.227)

where R (j) =

It is noted that R(j) has the form of the frequency response function of a high pass RC filter. When the piezoelectric is short circuited, rL = 0 and therefore, R(j) = 0. When the circuit is an open circuit, then rL → ∞, and therefore, R(j) → 1. Substituting Eqs. (2.225) and (2.226) into Eq. (2.221) gives 

A(j) + α33 R(j) −B(j) − α33 R(j) −C(j) − α33 R(j)/mr E(j) + α33 R(j)/mr



X1 X2



=

 X3 (1 + 2jζ1 ) 0 (2.228)

where A(j), B(j), C(j), and E(j) are given in Eq. (2.201) with c32 = k32 = 0 and α33 =

e33 e˜ 33 S ε33

= ke2 ωr2 mr .

(2.229)

In arriving at Eq. (2.229), Eqs. (2.146) and (2.224) were used. Solving Eq. (2.228) for X1 and X2 , it is found that X1 =

X3 (1 + 2jζ1 ) (E(j) + α33 R(j)/mr ) Do (j)

X3 (1 + 2jζ1 ) X2 = (C(j) + α33 R(j)/mr ) Do (j)

(2.230)

where Do (j) = (A(j) + α33 R(j)) (E(j) + α33 R(j)/mr ) − (B(j) + α33 R(j)) (C(j) + α33 R(j)/mr ) .

(2.231)

When the piezoelectric element is replaced with an ordinary spring, a33 = 0 and Eqs. (2.230) and (2.231), respectively, reduce to those given by Eqs. (2.218) and (2.204).

68

2 Spring-Mass Systems

From Eqs. (2.226) and (2.230), it is found that Vo =

X3 e33 R(j) (1 + 2jζ1 ) S ε33 Do (j)

(C(j) − E(j)) .

(2.232)

Using Eq. (2.152), the average power P in the load resistor RL can be expressed as rL ke2 P () = P ref 2ωr

! !2 ! j (1 + 2jζ ) ! 1 ! !   − E(j)) (C(j) ! ! ! Do (j) 1 + jrL !

(2.233)

where E W. P ref = ωn2 X32 Kpe

(2.234)

The form of P ref was chosen so that Eq. (2.233) could be compared directly with the power attained by a single degree-of-freedom system given by Eq. (2.152) since, as indicated in Eq. (2.224), ωn2 = ωnE . To determine the value of rL that maximizes P , we use Eqs. (2.201) and (2.227) to rewrite Eq. (2.233) as rL co () P () = !  ! P ref !Do (j) 1 + jr !2 L

rL co () = ! !  !f1 r + f2 + j g1 r + g2 !2 L L

(2.235)

where f1 =  {(Bi − Ai ) α33 /mr + Bi Cr + Br Ci − Ai Er − Ar Ei + (Ci − Ei ) α33 } f2 = Bi Ci − Br Cr − Ai Ei + Ar Er g1 =  {(Ar − Br ) α33 /mr + Bi Ci − Br Cr − Ai Ei + Ar Er + (Er − Cr ) α33 } g2 = Ai Er − Br Ci − Bi Cr + Ar Ei (2.236) and !2 ke2 !! ! !j (1 + 2jζ1 ) (Cr − Er + j (Ci − Ei ))! 2ωr Ar + jAi = −2 + 1 + mr ωr2 + 2j (ζ1 + ζ2 mr ωr )  co () =

Br + jBi = mr ωr2 + 2jζ2 mr ωr  Cr + jCi = ωr2 + 2jζ2 ωr  Er + jEi = −2 + ωr2 + 2jζ2 ωr .

(2.237)

2.7

Two Degree-of-Freedom Systems

69

The value of rL that maximizes P avg is determined from d drL



P P ref

 = 0.

(2.238)

Thus, from Eqs. (2.238) and (2.235), it is found that  () = rL,opt

f22 + g22 f12 + g21

.

(2.239)

It is seen from Eq. (2.239) that rL,opt is a function of the non dimensional parameters that have been introduced to describe the system: ζ 1 , ζ 2 , ωr , mr , and ke . For a given set of these parameters, the value of rL,opt () will produce the maximum value of P () Pref at each value of . From a plot of P () P ref as a function of  using the corresponding value rL,opt (), one can determine the maximum value of power, which occurs at  = max and is denoted P avg (max ) P ref . To illustrate this, Eqs. (2.239) and Eq. (2.235) are evaluated for ζ1 = ζ2 = 0.03, k3 = 0.75, mr = 0.5, and ωr = 0.5 and 0.8 and the results are shown in Fig. 2.31. Each of these curves represents the envelop of the maximum power; that is, for the given

ωr=0.5

25

P′(Ωmax=1.08)/P′ref=23.4

ωr=0.8

P′/P′ref

20

15

10 P′(Ωmax=0.744)/P′ref=7.01 5

0

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Ω

Fig. 2.31 Maximum power frequency response envelop of a two degree-of-freedom system piezo (), ζ1 = ζ2 = 0.03, k33 = 0.75, mr = 0.5, and electric harvester as a function of  for rL = rL,opt ωr = 0.5 and 0.8

70

2 Spring-Mass Systems

values of mr , ωr , and a specific value of rL , no part of the power versus  curve will lie outside the respective curves shown in Fig. 2.31. To compare the two degree-of-freedom harvester with the single degree-offreedom harvester, Eqs. (2.235) and (2.170) are plotted in Fig. 2.32 for mr = 0.5, ωr = 0.5, ζ1 = ζ2 = ζE = 0.03 and k33 = 0.75. In order to place these results on the same graph, it is noted that  in Eq. (2.235) and E in Eq. (2.170) are related by  = ωr E . From Fig. 2.32, it is seen that the maximum average power for a two degree-of-freedom harvester is 23.4, which occurs when  = max = 1.08 and rL = 0.327. The maximum average power for a single degree-of-freedom system is 10.87, which occurs when E = E,max = 1.51 and rL = 9.398. Then the ratio of the load resistance RL,1dof of the single degree-of-freedom harvester  tothat of the two degree-of-freedom harvester RL,2dof is RL,1dof /RL,2dof = = 9.398/(0.5 × 0.327) = 57.5. Furthermore, the maximum power of rL / ωr rL the two degree-of-freedom harvester is approximately 23.4/10.87 = 2.1 times larger than that of a single degree-of-freedom harvester. Consequently, attaching a single degree-of-freedom energy harvester to another single degree-of-freedom system can substantially increase the power output of the energy harvester. The effects of mr and ωr on the maximum power of a two degree-of-freedom harvester compared to that of a single degree-of-freedom harvester are shown in Fig. 2.33 for ζ1 = ζ2 = ζE = 0.03 and k33 = 0.75. The horizontal dashed line in this figure indicates the maximum power that is available from a single

25 2 dof 1 dof

P′(Ωmax=1.08)/P′ref = 23.4 at r′L,opt(Ωmax= 1.08)= 0.327

20

15

= ωrΩE,max= 0.756)/P′ref= 10.87

P′(Ω

P′/P′ref

max

at r

(Ω

opt

max

=ω Ω

r E,max

= 0.756)= 9.398

10

5

0 0.4

0.6

0.8

1 Ω

1.2

1.4

1.6

Fig. 2.32 Average power as a function of frequency from a single and a two degree-of-freedom harvester for mr = 0.5, ωr = 0.5, ζ1 = ζ2 = ζE = 0.03 and k33 = 0.75

2.7

Two Degree-of-Freedom Systems

71

50 45 40

P′(Ωmax)/P′ref

35 30 25 20

0.2

mr=

0.5

1

15 10 5 0 0.2

0.4

0.6

0.8

1

1.2

ωr

Fig. 2.33 Maximum power ratio for a two degree-of-freedom piezoelectric harvester as a function of mr and ωr for ζ1 = ζ2 = 0.03 and k33 = 0.75. Dashed line indicates maximum power ratio of 10.89 for a single degree of freedom system for ζE = ζ2 = 0.03 and k33 = 0.75 obtained from Fig. 2.23 or Fig. 2.32

degree-of-freedom harvester. The values of power above this line indicate that the two degree-of-freedom harvester produces more power than the single degree-offreedom harvester. From these curves, it is seen that for this lightly damped two degree-of-freedom system, larger amounts of power than that produced by a single degree-of-freedom system can be obtained by carefully selecting ωr and mr .

2.7.4 MEMS Filters There is a class of devices called mechanical filters that have been used in signal processing for the last 60 years (Johnson 1983). These are resonant electromechanical systems that exhibit such characteristics as narrow bandwidth, low loss, and good stability. These devices are getting considerable attention again with the advent of small-scale systems being developed in the field of MEMS (Lin et al. 1998; Wang and Nguyen 1997; Bannon et al. 2000; Elka and Bucher 2008; Chivukila and Rhoads 2010). One form of these devices is a two-stage MEMS filter whose mechanical model can be represented by two masses and three springs as shown in Fig. 2.24 with X3 = 0. In one type of device, the inertial element m1 is driven by an electrostatic comb transducer. Another electrostatic comb transducer senses the motion of the other inertial element m2 . Between the two masses is a weak

72

2 Spring-Mass Systems

coupling spring k2 . The masses and the geometric parameters are chosen so that m1 = m2 (mr = 1) , k1 = k3 , c1 = c3 , and c2 = 0. The goal is to select the appropriate values for mj , kj , and cj to create a filter whose amplitude response of m2 is relatively uniform within a specified bandwidth. The degree of uniformity is called pass band ripple, and the corresponding magnitudes are usually expressed in decibels (dB). The filter is described by the transfer function involving the displacement response of m2 and the force applied to mass m1 ; that is, we are interested in the frequency response function H21 (j). Thus, the input to the filter is g1 (g2 = 0), and we are interested in the displacement X2 . If we let k21 = k2 /k1 and note that ωr2 = k21 , ωr2 k32 = 1, 2ζ2 ωr c32 = 2ζ1 , then from Eqs. (2.212) and (2.201) H21 (j) =

X2 (j) k21 = D1 (j) g1 k 1

(2.240)

where



D1 (j) = 4 − 4jζ1 3 − 2 1 + k21 + 2ζ12 2 + 4jζ1 (1 + k21 )  + 1 + 2k21 . (2.241)

The damping coefficients for micromechanical filters vary, but typically they are very low and on the order of 0.002, or less. To give an idea of the amplitude response of m2 , we select k21 = 0.006, and ζ1 = 0.0015. The numerically obtained results are shown in Fig. 2.34. 0

Ripple = 1.94 dB

20log10[|H21(jΩ)|/H21,max]

−5 −10 −15 −20 −25 −30 −35 −40 −45 0.97

0.98

0.99

1 Ω

1.01

1.02

1.03

Fig. 2.34 Response of a two-stage micromechanical filter for k21 = 0.006, and ζ1 = 0.0015

2.7

Two Degree-of-Freedom Systems

73

2.7.5 Time-Domain Response In this section, we shall consider the time-domain response of a two degree-offreedom system in which the displacement of the base is a half-sine wave of frequency ωo and duration to = π/ωo . In terms of the non dimensional quantities, this wave form can be expressed as x3 (τ ) = Xo sin (o τ ) [u (τ ) − u (τ − τo )]

(2.242)

where o = ωo /ωn1 and τo = ωn1 to . The Laplace transform of Eq. (2.242) is obtained from Eq. (C.1) of Appendix C. Thus,   Xo o 1 + e−π s/o X3 (s) = s2 + 2o

(2.243)

where X3 (s) is the Laplace transform of x3 (t). Then, with f1 = f2 = 0 and all the initial conditions assumed to be zero, we use Eq. (2.243) in Eq. (2.195) and arrive at   Xo o (2ζ1 s + 1) 1 + e−π s/o K1 (s) = (2ζ1 s + 1) X3 (s) = s2 + 2o

(2.244)

K2 (s) = 0. Substituting Eq. (2.244) into Eq. (2.196), we obtain   Xo o (2ζ1 s + 1) E(s) 1 + e−π s/o   X1 (s) = s2 + 2o D(s)   Xo o (2ζ1 s + 1) C(s) 1 + e−π s/o   . X2 (s) = s2 + 2o D(s)

(2.245)

The numerically obtained inverse Laplace transform4 of Eq. (2.245) is shown in Fig. 2.35.

2.7.6 Design of an Atomic Force Microscope Motion Scanner A new mechanical design of the (x,y,z)-motion scanner of an atomic force microscope has been built and analyzed (Schitter et al. 2007). The vertical stage (z-direction) of the scanner is composed of a piezoelectric element placed between two masses such that mass m1 shown Fig. 2.24 is elastically supported by the spring k1 , and k2 represents the stiffness of the piezoelectric element to which a mass m2 4

The MATLAB function ilaplace from the Symbolic Math toolbox was used.

74

2 Spring-Mass Systems ω = 0.1

ω = 0.4 r

Amplitude

Ω = 0.05 o

0 0

50

100

5

150

Ωo= 0.15

Amplitude

−5

0 −5 0

50

100

2

150

Ωo= 0.4

Amplitude

Amplitude

Amplitude

Amplitude

r

5

0 −2 0

50

τ

100

150

2

Ω = 0.05 o

0 −2

0

50

100

2

150

Ωo= 0.15

0 −2 0

50

100

2

150

Ωo= 0.4

0 −2 0

50

τ

100

150

Fig. 2.35 Displacement response of m1 and m2 when a half-sine wave displacement is applied to the base of m1 for ζ1 = ζ2 = 0.1 and ωr = 0.05 and 0.4: – – – x1 (τ )/Xo ; —— x2 (τ )/Xo ; - - - Base input

is attached. It is assumed that c2 = 0. When the piezoelectric element is actuated by a voltage, an equal force fo (τ ) is applied to both masses such that f1 (τ ) = −fo (τ ) and f2 (τ ) = fo (τ ). We are interested in the response of m2 . Therefore, we use Eqs. (2.194) to (2.196) with X3 = 0 and the initial conditions equal to zero to obtain (−mr C(s) + A(s)) Fo (s) k1 mr D(s) 2  s + 2ζ1 s + 1 Fo (s) = k1 mr D(s)

X2 (s) =

(2.246)

where D(s) is given by Eq. (2.197) and Fo (s) is the Laplace transform of fo (τ ). If it is assumed that the force is a step function of the form fo (τ ) = Go u(τ ), then from Laplace transform pair 6 of Table C.1 of Appendix C, it is found that Fo (s) = Go /s. A numerically obtained inverse Laplace transform5 of Eq. (2.246) for mr = 0.1, ζ1 = ζ2 = 0.1, and ωr = 0.2 and 1 is shown in Fig. 2.36. It is seen that when ωr = 1 the duration of the transient response and it its peak amplitude are greatly decreased.

5

The MATLAB function ilaplace from the Symbolic Math toolbox was used.

450

18

400

16

350

14

75

12

300

x2(τ)/(Go/k1)

x2(τ)/(Go/k1)

Appendix 2.1 Forces on a Submerged Vibrating Cylinder

250 200 150

10 8 6 4

100

2

50

0

0

−2

0

50

100

150

0

50

100

τ

τ

(a)

(b)

150

Fig. 2.36 Response of m2 of a two degree-of-freedom system to a unit step force of magnitude Go applied to m1 and m2 for mr = 0.1, ζ1 = ζ2 = 0.1, and for (a) ωr = 0.2 and (b) ωr = 1

Appendix 2.1 Forces on a Submerged Vibrating Cylinder Consider a very long rigid solid circular cylinder of diameter b and length L that is immersed in an incompressible viscous fluid of density ρ f and dynamic viscosity μf . It is assumed that the fluid can be modeled by the linearized momentum equations, which in terms of the scalar stream function ψ = ψ (r, θ , t) is given by (Stokes 1901, pp. 38–47)   ρf ∂ψ =0 ∇ 2 ∇ 2ψ − μf ∂t

(2.247)

where ∇2 =

∂2 1 ∂ 1 ∂2 + . + r ∂r r2 ∂θ 2 ∂r2

(2.248)

A solution to Eq. (2.247) is ψ = ψ1 + ψ2 , where ψ 1 and ψ 2 , respectively, are solutions to ∇ 2 ψ1 = 0 ρf ∂ψ2 = 0. ∇ 2 ψ2 − μf ∂t

(2.249)

The velocities of the fluid are given by ur =

1 ∂ψ , r ∂θ

uθ = −

∂ψ . ∂r

(2.250)

We assume that the origin of the polar coordinate system is at the center of the cylinder, and that the cylinder is undergoing forced harmonic oscillations at frequency ω of the form

76

2 Spring-Mass Systems

ur (b/2, θ , t) = uo cos θ ejωt uθ (b/2, θ , t) = −uo sin θ ejωt

(2.251)

where uo is the magnitude of the velocity. It is further assumed that the surrounding fluid is infinite in extent, that is, as r → ∞, ur → 0 and uθ → 0. In view of the boundary conditions given by Eq. (2.251), we assume solutions to Eq. (2.249) of the form ψk = k (r) sin θ e jωt

k = 1, 2.

(2.252)

Substituting Eq. (2.252) into Eq. (2.249) yields ∇r2 1 = 0

(2.253)

and ∇r2 2 −

jωρf 2 = 0 μf

(2.254)

where ∇r2 =

∂2 1 ∂ 1 + − 2. 2 ∂r r ∂r r

(2.255)

C1 + C2 r. r

(2.256)

The solution to Eq. (2.253) is 1 =

To satisfy the requirements that as r → ∞, ur → 0 and uθ → 0, we must set C2 = 0. Then, Eq. (2.256) becomes 1 =

C1 . r

(2.257)

To determine the solution to Eq. (2.254), we rewrite it as r

2∂

2 2 ∂r2

∂2 +r − ∂r



2λr b

2

 + 1 2 = 0

(2.258)

where λ= Re =

√ j Re ρf ωb2 . 4μf

(2.259)

Appendix 2.1 Forces on a Submerged Vibrating Cylinder

77

and Re is the Reynolds number. The solution to Eq. (2.258) is 2 = C3 I1 (2λr/b) + C4 K1 (2λr/b)

(2.260)

where I1 (x) is the modified Bessel function of the first kind of order 1 and K1 (x) is the modified Bessel function of the second kind of order 1. To satisfy the requirements that as r → ∞, ur → 0 and uθ → 0, we must set C3 = 0. Then, Eq. (2.260) becomes 2 = C4 K1 (2λr/b) .

(2.261)

Thus, from Eqs. (2.257) and (2.261), ψ =  (r) sin θ e jωt = (1 + 2 ) sin θ e jωt   C1 = + C4 K1 (2λr/b) sin θ e jωt . r

(2.262)

Then, from Eq. (2.250) and (2.262), ur = uθ =

1 r 



 C1 + C4 K1 (2λr/b) cos θ ejωt r   C1 2λC4 b + K + K sin θ ejωt . (2λr/b) (2λr/b) 1 0 b 2λr r2

(2.263)

The constants C1 and C4 are determined by substituting Eq. (2.263) into the boundary conditions given by Eq. (2.251). This substitution results in 4C1 2 + C4 K1 (λ) 2 b b  4C1 2λC4 1 K1 (λ) + K0 (λ) . −uo = 2 + b λ b uo =

(2.264)

Solving Eq. (2.264) for C1 and C4 , we obtain   uo b2 2 K1 (λ) +1 C1 = 4 λ K0 (λ) uo b C4 = − . λK0 (λ)

(2.265)

Then Eqs. (2.261) and (2.257), respectively, become 2 = −uo b

K1 (2λr/b) λK0 (λ)

(2.266)

78

2 Spring-Mass Systems

and u o b2 4r

1 =



 2 K1 (λ) +1 . λ K0 (λ)

(2.267)

The force F acting on the cylinder of length L at r = b/2 is determined from b F=L 2

2π [Pr cos θ − Pθ sin θ ] dθ

(2.268)

0

where 

 ∂ur ∂r r=b/2 " #   

u  1 ∂ur ∂uθ θ Pθ = μ f + − r ∂θ r=b/2 ∂r r=b/2 r r=b/2 Pr = − (p)r=b/2 + 2μf

(2.269)

and p is the pressure in the fluid. We now use our previous results to evaluate each of the terms appearing in Eq. (2.269). From the velocity relations given by Eq. (2.250) and the boundary conditions given by Eq. (2.251), it is seen that 



∂ur ∂r

1 ∂ur r ∂θ



      1 1 ∂uθ ∂ 2ψ 1 ∂ψ = + − −ur − r r ∂θ ∂r∂θ r=b/2 r ∂θ   ∂ 2 = −uo cos θ e jωt + uo (sin θ) e jωt = 0 b ∂θ    1 ∂ 2ψ 2 ∂  = = uo cos θ e jωt 2 2 r ∂θ r=b/2 b ∂θ  uθ  2 . = −uo sin θ e jωt = b r r=b/2 =

r=b/2

 r=b/2

r=b/2

(2.270) Furthermore, from Eqs. (2.248) to (2.250) and Eq. (2.270), 

∂uθ ∂r

 r=b/2

  2     ρf ∂ψ2 1 ∂ψ 1 ∂ 2ψ ∂ ψ = − 2 = + 2 2 − ∂r r=b/2 r ∂r r=b/2 r ∂θ r=b/2 μf ∂t

u 

u  ρf ∂ψ2 θ θ + − = − r r=b/2 r r=b/2 μf ∂t =−

ρf ∂ψ2 . μf ∂t (2.271)

Appendix 2.1 Forces on a Submerged Vibrating Cylinder

79

Using Eqs. (2.270) and (2.271) in Eq. (2.269), we obtain Pr = − (p)r=b/2 ∂ψ2 . Pθ = −ρf ∂t

(2.272)

Then, Eq. (2.268) becomes b F = −L 2

2π

b (p)r=b/2 cos θ dθ + Lρf 2

0

2π 0

∂ψ2 sin θ dθ. ∂t

(2.273)

We now use integration by parts on the first integral of Eq. (2.273) to obtain ⎤ ⎡ !2π 2π  dp  b⎣ ! sin θ dθ ⎦ sin θ ! − F1 = −L 0 2 dθ r=b/2 0 (2.274)

2π   b dp =L sin θ dθ . 2 dθ r=b/2 0

It can be shown that (Stokes 1901, p. 39)   ∂ψ1 ∂p ∂ = ρf r . ∂θ ∂t ∂r Therefore, Eq. (2.274) becomes b F1 = Lρf 2

2π  0

  ∂ ∂ψ1 sin θ dθ r ∂t ∂r r=b/2

(2.275)

and Eq. (2.273) can be written as b∂ F = Lρf 2 ∂t

2π  0

∂ψ1 r + ψ2 ∂r

 sin θ dθ.

(2.276)

r=b/2

Substituting Eqs. (2.252), (2.266), and (2.267) into Eq. (2.276), we arrive at 

2π  ∂1 r sin2 θ dθ F = jωLρf e + 2 2 ∂r r=b/2 0   b ∂1 jωt = jωLρf π e + 2 r 2 ∂r r=b/2     2 b 2 K1 (λ) K1 (2λr/b) jωt b = −jωuo Lρf π e +1 + 2 4r λ K0 (λ) λK0 (λ) r=b/2 jωt b

= −jωuo e jωt Mf cir (ω)

(2.277)

80

2 Spring-Mass Systems

where b2 Mf = ρ f π L   4 4 K1 (λ) cir (ω) = 1 + λ K0 (λ)

(2.278)

The quantity Mf is the mass of the fluid displaced by the cylinder and  cir is a complex quantity called a hydrodynamic function. The force acting on the cylinder is obtained by taking the real part of F, which is   Ff = Real (F) = ωuo Mf Real (cir ) sin (ωt) + Imag (cir ) cos (ωt) .

(2.279)

If the harmonic displacement of the cylinder in the x-direction is given by x = Xo sin (ωt), then Eq. (2.279) can be written as Ff = −Mf Re (cir ) x¨ + Mf Im (cir ) x˙

(2.280)

where the over dot indicate the derivate with respect to time and we have recognized that for harmonic oscillations uo = ωXo . On the other hand, if the harmonic displacement of the cylinder in the x-direction is given by x = Xo ejωt , then Eq. (2.277) can be written as F = Xo ω2 e jωt Mf cir (ω) .

(2.281)

References Abramovitch DY, Andersson SB, Pao LY, Schitter G (2007) A tutorial on the mechanisms, dynamics, and control of atomic force microscopes. Proceedings of the 2007 American Control Conference, pp 3488–3502 Aldraihem O, Baz A (2011) Energy harvester with a dynamic magnifier. J Intell Mater Syst Struct 22(6):521–530 ANSI/IEEE Std 176-1987 (1988) IEEE standard on piezoelectricity. The Institute of Electrical and Electronics Engineers, New York, NY Anton SR, Sodano HA (2007) A review of power harvesting using piezoelectric materials (2003–2006). Smart Mater Struct 16:R1–R21 Arnold DP (2007) Review of microscale magnetic power generation. IEEE Trans Magn 43(11):3940–3951 Balachandran B, Magrab EB (2009) Vibrations, 2nd edn. Cengage, Toronto, ON Bannon FD, Clark JR, Nguyen CT-C (2000) High-Q HF microelectromechanical filters. IEEE J Solid-State Circuits 35(4):512–536 Bao M, Yang H (2007) Review: squeeze film air damping in MEMS. Sens Actuators A 136(1):3–27 Batra RC, Porfiri M, Spinello D (2007) Review of modeling electrostatically actuated microelectromechanical systems. Smart Mater Struct 16(6):R23–R31 Beeby SP, Tudor MJ, White NM (2006) Energy harvesting vibration sources for microsystems applications. Meas Sci Technol 17:R175–R195

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Ma PS, Kim JE, Kim YY (2010) Power-amplifying strategy in vibration-powered energy harvesters. Proc SPIE 7643:76430O-1 Martin MJ, Houston BH (2007) Gas damping of carbon nanotube oscillators. Appl Phys Lett 91(10):103116 Martin Y, Williams CC, Wickramasinghe HK (1987) Atomic force microscope-force mapping and profiling on a sub 100-Å scale. J Appl Phy 61:4723–4729 McCarthy B, Adams G, McGruer NE (2002) A dynamic model, including contact bounce, of an electrostatically actuated microswitch. J Microelectromech Syst 11(3): 276–283 Mitcheson PD, Yeatman EM, Rao GK, Holmes AS, Green TC (2008) Energy harvesting from human and machine motion for wireless electronic devices. Proceeding of the IEEE 96(9):1457–1486 Nielson GN, Barbastathis G (2006) Dynamic pull-in of parallel-plate and torsional electrostatic mems actuators. J Microelectromech Syst 15(4):811–821 Perez MA, Shkel AM (2008) The effect of squeeze film constriction on bandwidth improvement in interferometric accelerometers. J Micromech Microeng 18:055031 Pratap R, Mohite S, Pandey AK (2007) Squeeze films effects in MEMS devices. J Indian Inst Sci 87:75–94 Raman A, Melcher J, Tung R (2008) Cantilever dynamics in atomic force microscopy. Nanotoday 3(1–2):20–27 Renno JM, Daqaq MF, Inman DJ (2009) On the optimal energy harvesting from a vibration source. J Sound Vib 320:386–405 Roundy S (2005) On the effectiveness of vibration-based energy harvesting. J Intell Mater Syst Struct 16:809–823 Rützel S, Lee SI, Raman A (2003) Nonlinear dynamics of atomic-force-microscope probes driven in Lennard-Jones potentials. Proc R Soc London A 459:1925–1948 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 Schitter G, Åström KJ, DeMartini BE, Thurner PJ, Turner KL, Hansma PK (2007) Design and modeling of a high-speed AFM-scanner. IEEE Trans Control Syst Technol 15(5):906–915 Soliman MSM, Abdel-Rahman EM, El-Saadany EF, Mansour RR (2008) A wideband vibrationbased energy harvester. J Micromech Microeng 18:115021 Spreemann D, Hoffmann D, Folkmer B, Manoli Y (2008) Numerical optimization approach for resonant electromagnetic vibration transducer designed for random vibration. J Micromech Microeng 18:104001 Stark RW, Schitter G, Stemmer A (2003) Tuning the interaction forces in tapping mode atomic force microscopy. Phys Rev B 68:085401-1 Stephen NG (2006) On energy harvesting from ambient vibration. J Sound Vib 293:409–425 Stokes SS (1901) Mathematical and physical papers, Vol. III. Cambridge University Press, London Tilmans HAC (1996) Equivalent circuit representation of electromechanical transducers: I. Lumped parameter systems. J Micromech Microeng 6(1):157–176 Tuma JJ (1979) Engineering mathematics handbook, 2nd edn. McGraw Hill, New York Veijola T (2004) Compact models for squeezed-film dampers with inertial and rarefied gas effects. J Micromech Microeng 14:1109–1118 Veijola T, Kuisma H, Lahdenpera J, Ryhanen T (1995) Equivalent-circuit model of the squeezed gas film in a silicon accelerometer. Sens Actuators A 48:239–248 Wang K, Nguyen CT-C (1997) High-order microelectromechanical electronic filters. Proceedings, 10th Annual International Workshop on Micro Electro Mechanical Systems, IEEE Robotics and Automation Society, pp 25–30 Younis MI, Miles R, Jordy D (2006) Investigation of the response of microstructures under the combined effect of mechanical shock and electrostatic forces. J Micromech Microeng 16: 2463–2474 Zhang WM, Meng G, Zhou J-B, Chen J-Y (2009) Nonlinear dynamics and chaos of microcantilever-based TM-AFMs with squeeze film effects. Sensors 9:3854–3874 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 3

Thin Beams: Part I

The natural frequencies and mode shapes of thin 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. Several of the variable cross section geometries considered have application to MEMS probes and to atomic force microscopes. The in-span and boundary attachments include springs, concentrated masses, and single and two degree-of-freedom systems. The effects of a proof mass on the natural frequencies of a beam are illustrated by considering a cantilever beam in which the center of gravity of the attached mass is a finite distance from the end of the beam and by considering a rigid mass of finite length supported by a beam at each of its ends. The natural frequencies of elastically connected beams, which have been used to model double-wall carbon nanotubes, are determined. The responses of these systems to externally applied displacements at the boundary and to forces applied to its interior are also obtained.

3.1 Introduction Beams are ubiquitous structural elements that appear in many different forms and comprise many different artifacts such as: supporting members in buildings and bridges, propellers for airplanes and boats, turbine blades, and automobile chasses. The beam model is used as a first approximation to analyze complicated structures such as an aircraft wing and a ship’s interaction with waves. It is an important component of rotating machinery: the shaft. In recent years, a piezoelectric element has been attached to the surface of a cantilever and used as an energy harvester. Cantilever beams are also being used at the MEMS scale as sensors and beams clamped at both ends have been used to model carbon nanotube wire resonators (Ekinci and Roukes 2005). In this chapter, we shall consider beams mostly in its more classical role and in the next chapter we shall examine beams in their new role within MEMS structures. In this new role, complicating factors that, for the most part, are only important at the MEMS scale are considered.

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_3, 

83

84

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Thin Beams: Part I

3.2 Derivation of Governing Equation and Boundary Conditions 3.2.1 Contributions to the Total Energy Consider   a beam of length L (m) that has a rectangular cross section with area A = bh m2 where b is the thickness and h the depth, a density ρ (kg/m3 ), and a Young’s modulus E (N/m2 ). Let the beam be subjected to a moment M (Nm) at each end. Referring to Fig. 3.1, it is seen that the face BB of the beam located toward the center of curvature will be contracted while that on the opposite face AA will be extended. Under the application of the moments, there is a fiber located a distance a from the z coordinate axis that experiences zero axial strain. This fiber is called the neutral axis. The deformation of the beam is assumed to be described by Euler-Bernoulli beam theory (Popov 1990), which is applicable to thin elastic beams whose length to depth ratio is greater than 10. In keeping with this theory, it is assumed that under the application of a moment the neutral axis remains unaltered, that the plane sections of the beam normal to the neutral axis remain plane and normal to the deformed neutral axis, and that the transverse normal, such as BA, experience zero strain along the normal direction. We shall now derive the governing equation of motion using the method described in Appendix B by determining the energy contributions of various factors that will be considered in the subsequent analyses. The beam under consideration is shown in Fig. 3.2. For this beam, the factors are the strain energy due to the deformation of the beam, the kinetic energy of the beam and that of a concentrated mass attached at an in-span location, the energy due to an externally applied transverse force and an externally applied axial load, the energy due to an externally applied electrostatic force, the stored energy of an elastic foundation and a translation spring attached at an in-span location, and the kinetic energy and potential energy of a single degree-of-freedom system attached at an in-span location. In addition, it is assumed that at each end of the beam there are attached translation and torsion springs and masses in which the effects of their rotational inertia are included. Each of these contributions is now determined.

A a1 h M

a b1

A

Δs

o

x

Δso w(x,t)

R M

B z

B

Neutral axis Δθ

Fig. 3.1 Deformation of a beam subjected to end moments

3.2

Derivation of Governing Equation and Boundary Conditions

Fig. 3.2 Nomenclature for a beam with externally applied forces, an electrostatic potential, and in-span and boundary attachments

ML, JL

f (x, t)

Mi

ktL

mo

z(t)

MR, JR ktR

ko E, A, I, ρ

p kL

85

ki

Vo

...

...

p

x

kR

kf

w(x, t)

Lm Ls Lo L

Strain Energy For a fiber located at a distance z from the neutral axis, as shown in Fig. 3.1, the strain experienced along the length of the beam is given by εx =

s − so z−a =− so R

(3.1)

where R is the radius of curvature, so is the length of a fiber along the neutral axis, s is the length of a fiber that is located at a distance z from the neutral axis, a is the distance from the coordinate axis to the neutral axis, and we have used geometry to write s = (R − z + a) θ and so = Rθ. If we assume that the σy = σz = 0, then from Eq. (A.7) of Appendix A the corresponding axial stress σx acting on the fiber is σx = Eεx = −

E (z − a) . R

(3.2)

According to the convention shown in Fig. 3.1, a positive displacement w is in the positive z-direction. Therefore, the fibers above the neutral axis experience a positive σx , which denotes tension, and the fibers below the neutral axis experience a negative σx , which denotes compression. At an internal section of the beam, a moment balance about the neutral axis leads to

y2 b1 M=− −y1 −a1

y2 b1 σx (z − a) dzdy = −y1 −a1

E (z − a)2 EI dzdy = R R

(3.3)

where y1 and y2 are the spatial limits corresponding to integration along the y-direction, we have used Eq. (3.2), and

86

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Thin Beams: Part I

y2 b1 I=

(z − a)2 dzdy.

(3.4)

−y1 −a1

Note that a1 + b1 = h and that y1 + y2 = b. The quantity I represents the area moment of inertia of the beam’s cross-section about the neutral axis. In the case of beams with rectangular, circular, and elliptical cross sections, a = 0 and I is the area moment of inertia through its centroid. In general, the limits of the double integral in Eq. (3.4) do not have to be constants; that is, a1 = a1 (x), b1 = b1 (x), y1 = y1 (x), and y2 = y2 (x). For this case, the area moment of inertia varies along the length, and therefore, in general, I = I (x). In this chapter, we shall only be considering beams for which a = 0. In Section 4.4, the determination of a is shown for a laminated beam [see Eqs. (4.139) and (4.140)]. If the axial displacement of the beam is u, then it has been shown (Hodges 1984) that the curvature of the beam κ = 1/R, where R is the radius of curvature, can be given by    1 + ∂u ∂x ∂ 2 w ∂x2 − ∂w ∂x ∂ 2 u ∂x2 . κ=  2  2 3/2 1 + ∂u ∂x + ∂w ∂x 

(3.5)

For small strains, however,  2  2 1 + ∂u ∂x + ∂w ∂x ≈ 1 and Eq. (3.5) reduces to       κ = 1 + ∂u ∂x ∂ 2 w ∂x2 − ∂w ∂x ∂ 2 u ∂x2 .

(3.6)

When the axial displacements and its derivatives are negligible, Eq. (3.6) gives κ=

1 ∼ ∂ 2w = 2. R ∂x

(3.7)

Using Eq. (3.7) in Eqs. (3.1) and (3.2), respectively, we obtain, εx = − (z − a)

∂ 2w ∂x2

(3.8)

∂ 2w . ∂x2

(3.9)

and σx = −E (z − a)

3.2

Derivation of Governing Equation and Boundary Conditions

87

From Eq. (3.3), it is found that M = EI (x)

∂ 2w . ∂x2

(3.10)

Thus, the magnitude of the axial strain, axial stress, and bending moment are proportional to the second spatial derivative of the beam displacement. The statement that the bending moment is linearly proportional to the second spatial derivative of the beam displacement is the Euler-Bernoulli law, which is the underlying basis for the theory of linear elastic thin beams. Equation (3.10) was obtained by considering only the effects of moments on the ends of the beam. If, in addition, there is a transverse load f (x,t), then there are vertical shear forces within the beam that resist this force. Referring to Fig. 3.3, if the sum of the moments about point o is taken along the y-direction, and if the rotary inertia of the beam element is neglected, the result is M + (V + V) x = M + M which leads to M = V + V. x In the limit x → 0, the shear force increment V → 0, and we have ∂M M = =V x→0 x ∂x lim

which, after making use of Eq. (3.10), results in   ∂M ∂ ∂ 2w V= = EI (x) 2 . ∂x ∂x ∂x

(3.11)

f (x, t) V + ΔV

o Δso

M

x Δw

V

z

Δx

M + ΔM

Fig. 3.3 Deformation of an element of a beam subjected to a transverse load

88

3

Thin Beams: Part I

Upon substituting Eqs. (3.8) and (3.9) into Eq. (A.9) of Appendix A, the strain energy in the beam is given by

1 U= 2 =

1 2

L b1 y2 0 −a1 −y1

1 σx εx dydzdx = 2



L EI (x) 0

2 ∂ 2w ∂x2

L b1 y2 

∂ 2w ∂x2

0 −a1 −y1

2 E (z − a)2 dydzdx (3.12)

dx.

Kinetic Energy: Beam Element and In-Span Concentrated Mass The kinetic energy of an elemental volume of the beam is  2 ∂w 1 ρdxdydz 2 ∂t and, therefore, the total kinetic energy of the beam is

1 T= 2



L b1 y2 ρ 0 −a1 −y1

∂w ∂t

2

1 dydzdx = 2



L ρA (x) 0

∂w ∂t

2 dx

(3.13)

where

b1 y2 A (x) =

dydz. −a1 −y1

The kinetic energy of a concentrated mass Mi (kg) located at x = Lm , 0 ≤ Lm ≤ L, is given by TMi =

Mi 2



∂w (Lm , t) ∂t

2

which can be written as

TMi

1 = 2



L Mi 0

where δ (x) is the delta function.

∂w (x, t) ∂t

2 δ (x − Lm ) dx

(3.14)

3.2

Derivation of Governing Equation and Boundary Conditions

89

External Forces: Transverse and Axial The work done by an applied transverse load per unit length f (x,t) (N/m) is given by

L W=

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

(3.15)

0

If the beam is also under the action of an axial tensile force p (x,t) (N), then the length of the neutral axis no longer remains constant, but extends to a new length. If we assume that the deformation is small in magnitude and does not affect the loading p (x,t), then the change in length  can be approximated by   = ds − dx = dx2 + dw2 − dx   2 dw = dx 1 + − dx dx   1 dw 2 ∼ dx = 2 dx

(3.16)

where we have assumed that dw/dx << 1. Therefore, the external work of the tensile axial force is given by

L Wp = −

1 p (x, t)  = − 2

0



L p (x, t) 0

∂w ∂x

2 dx.

(3.17)

Since the tensile force acts to oppose the beam’s transverse displacement w, the work done has a minus sign. If the axial force is compressive, then p (x,t) is replaced by –p (x,t). When p (x,t) is compressive, one has to make sure that its magnitude does not exceed the load at which the beam will buckle.1 1 If the ends of the beam are constrained from moving and the beam is subject to a uniform change in temperature T, then the magnitude of the axial tensile force generated by the thermally-induced strain is

EAo αT 1 − 2ν where ν is Poisson’s ratio and α is the coefficient of thermal expansion. In keeping with the sign convention for p, if we consider the change in axial force due to temperature, then in Eq. (3.17) we replace p with p→−

EAo αT 1 − 2ν

where a decrease in temperature means that T < 0. In other words, a decrease in temperature causes an axial tensile force in the beam.

90

3

Thin Beams: Part I

An axial force also can be generated in the beam when the beam deforms due to the externally applied force f (x,t) and the boundaries of the beam are constrained such that no axial motion is allowed. In this case, the beam undergoes an axial extension T , which creates an axial tensile force Po as if the beam were a rod in tension; that is,

Ao E Ao ET = Po (w (t)) = L L

L

Ao E (ds − dx) = L

0

=

Ao E 2L

L  0

∂w ∂x

L

⎛ ⎝ 1+



dw dx

2

⎞ − 1⎠dx

0

2 dx (3.18)

where it has been assumed that ∂w/∂x << 1 and we have restricted this case to that of a beam of constant cross section Ao . The notation Po (w (t)) indicates that while it is a function of w, it is only a function of its temporal portion since the integral has removed the spatial dependency. An application that includes Po (w (t)) is given in Section 4.3.3. Consequently, we modify Eq. (3.17) as

1 Wp = − 2

L 0



   ∂w 2 dx. p (x, t) + Po (w (t)) ∂x

(3.19)

Elastic Foundation, In-Span Translation Spring, and Spring of an In-span Single Degree-of-Freedom System When a beam is placed on a linear elastic foundation, the transverse displacement of the beam creates a force in the foundation with the magnitude kf w (x, t), where kf is the spring constant per unit length of the foundation (N/m2 ). The stored energy of the elastic foundation is 1 Ukf = 2

L kf w2 (x, t) dx.

(3.20)

0

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

1 2 ki w (Ls , t) 2

(3.21)

3.2

Derivation of Governing Equation and Boundary Conditions

1 Uk i = 2

91

L ki w2 (x, t) δ (x − Ls ) dx.

(3.22)

0

When a single degree-of-freedom system is attached at x = Lo , 0 ≤ Lo ≤ L and the displacement of the mass is denoted z (t), the energy stored by the spring ko is Uk o =

1 ko (w (Lo , t) − z (t))2 2

(3.23)

where z (t) is the displacement of the mass of the single degree-of-freedom system. Equation (3.23) can written as

Uko

1 = 2

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

(3.24)

0

Electrostatic Force Consider a capacitor that is formed by the beam and a fixed flat surface that is a nominal distance go beneath the beam and parallel to it. If a voltage of magnitude Vo = Vo (t) is used to create a capacitor C formed by these two surfaces, then the electric energy due to the applied voltage Vo is WE =

1 2 CV . 2 o

(3.25)

The value of the capacitor is determined by the distance the beam’s surface is from the fixed surface and is inversely proportional to go . However, when considering beams, the capacitance is not inversely proportional to go for two reasons: the distance of the beam’s surface from the fixed beam varies along the length of the beam and there are fringe effects that have to be taken into account. An empirical relationship that has been shown (Batra et al. 2006) to agree within 2% to a numerical solution of a model that includes the fringe effects is

L Fr (go ) dx F

C = εo

(3.26)

0

where C is the capacitance between the beam and the fixed plate, εo = 8.854 × 10−12 F/m is the permittivity of free space, and2 2 This model is proposed as a more accurate replacement for the Meijs-Fokkema model, which is given by    0.25  0.5  0.5 b b h h Fr (go ) = + 0.77 + 1.06 + 1.06 . go go b go

92

3

 Fr (go ) =

b go



 − 0.36 + 0.85

b go

0.24

 + 2.5

h go

Thin Beams: Part I

0.24 .

(3.27)

This model is applicable for the following geometric ranges: 0.2 ≤ h/b ≤ 2 and 0.4 ≤ h/go ≤ 5. It is noted that the third term of Eq. (3.27) accounts for the finite width of the beam and the fourth term accounts for the finite height of the beam. This relation has been developed for the case where the fixed beam has the same geometric cross section as the flexible beam. Since go = do − w is the spatially dependent spacing forming the plates of the capacitor, Eqs. (3.26) and (3.27), respectively, become

L Fr (w) dx

C = εo

(3.28)

0

and   0.24 0.24 h b b Fr (w) = + 2.5 − 0.36 + 0.85 do − w do − w do − w  0.24 b b = − 0.36 + c1 do − w do − w

(3.29)

where  0.24 h . b

c1 = 0.85 + 2.5

(3.30)

Substituting Eq. (3.28) into Eq. (3.25), we obtain εo Vo2 WE = 2

L Fr (w) dx

(3.31)

0

where εo Vo2 has the units of force (N). When one neglects the correction for the fringe effects, Eq. (3.29) reduces to Fr (w) =

b . do − w

(3.32)

Single Degree-of-Freedom System The kinetic energy of the mass mo (kg) of a single degree-of-freedom system is given by Tmo =

mo 2



∂z ∂t

2 (3.33)

3.2

Derivation of Governing Equation and Boundary Conditions

93

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. (3.23); that is, Uko =

1 ko (w (Lo , t) − z (t))2 . 2

(3.34)

Thus, the difference between the kinetic energy and the potential energy in the single degree-of-freedom system is F¯ = Tmo − Uko =

mo 2



∂z ∂t

2

1 − ko (w (Lo , t) − z (t))2 . 2

(3.35)

Attachments on the Boundaries As shown in Fig. 3.2, it is assumed that at the end of the beam at x = 0 the following elements are attached: a mass ML that has a rotational inertia JL , a translational spring with constant kL (0 ≤ kL ≤ ∞), and a torsion spring with constant ktL (0 ≤ ktL ≤ ∞). At the other end of the beam at x = L it is assumed that the following elements are attached: a mass MR that has a rotational inertia JR , a translational spring with constant kR (0 ≤ kR ≤ ∞), and a torsion spring with constant ktR (0 ≤ ktR ≤ ∞). Then, the difference between the kinetic energy and the potential energy of these elements is F (C1 ) =

 1 ML w˙ 2 (0, t) + JL w˙ 2x (0, t) + MR w˙ 2 (L, t) + JR w˙ 2x (L, t) 2  1 − kL w2 (0, t) + ktL w2x (0, t) + kR w2 (L, t) + ktR w2x (L, t) . 2

(3.36)

where the subscript x indicates the derivative with respect to x and the over dot indicates the derivative with respect to time. Minimization Function From Eqs. (3.12) to (3.15) and Eqs. (3.19), (3.20), (3.22), (3.24), and (3.31), it is found that the kinetic energy is given by

1 T= 2

L 0

=

1 2



∂w ρA (x) ∂t

2

1 dx + 2

Mi

∂w ∂t



2

0

L (ρA (x) + Mi δ (x − Lm )) 0



L

∂w ∂t

2 δ (x − Lm ) dx (3.37)

dx

94

3

Thin Beams: Part I

the potential energy by 1 U= 2

 2 2

L , ∂ w EI (x) + kf w2 (x, t) + ki w2 (x, t) δ (x − Ls ) ∂x2 0

(3.38)

-

+ ko (w (x, t) − z (t))2 δ (x − Lo ) dx and the external non conservative work by

L W=

f (x, t) w (x, t) dx − 0

+

1 2

L 0

εo Vo2

   ∂w 2  dx p (x, t) + Po (w (t)) ∂x (3.39)

L Fr (w) dx.

2 0

Thus,

T −U+W +F

(C1 )

L =

Fdx + F (C1 )

(3.40)

0

where  2 2  2 1 1 ∂ w ∂w F = [ρA (x) + Mi δ (x − Lm )] − EI (x) 2 ∂t 2 ∂x2  1 − kf + ki δ (x − Ls ) w2 (x, t) + f (x, t) w (x, t) 2 1 − ko (w (x, t) − z (t))2 δ (x − Lo ) 2   1 ∂w 2 εo Vo2 − (p (x, t) + Po (w (t))) + Fr (w) 2 ∂x 2

(3.41)

and F(C1 ) is given by Eq. (3.36). It is noted that Eq. (3.41) is not a symmetric quadratic. However, when the displacement-induced stretching of the neutral axis can be ignored, that is, Po = 0, when there is no electrostatic force, that is Vo = 0, when there is no applied transverse force, that is, f = 0, then Eq. (3.41) becomes

3.2

Derivation of Governing Equation and Boundary Conditions

Fsq

 2 2  2 ∂ w ∂w 1 1 = [ρA (x) + Mi δ (x − Lm )] − EI (x) 2 ∂t 2 ∂x2 1 − ko (w (x, t) − z (t))2 δ (x − Lo ) 2  2  ∂w 1 1 − kf + ki δ (x − Ls ) w2 (x, t) − p (x, t) 2 2 ∂x

95

(3.42)

which is a symmetric quadratic provided that z (t) is independent of w (x,t).

3.2.2 Governing Equation To obtain the governing equations for the beam, we use Eq. (B.123) of Appendix B with u = w; that is, Fw −

∂Fwx ∂Fw˙ ∂ 2 Fwxx − + = 0. ∂x ∂t ∂x2

(3.43)

Using Eq. (3.41), it is found that   εo Vo2 bFˆ r (w) Fw = − kf + ki δ (x − Ls ) w (x, t) + f (x, t) + 2 (do − w)2 − ko (w (x, t) − z (t)) δ (x − Lo )   ∂ ∂w ∂ 2w ∂Fwx =− p (x, t) − Po (w (t)) 2 ∂x ∂x ∂x ∂x   2 2 2 ∂ Fwxx ∂ w ∂ = − 2 EI (x) 2 2 ∂x ∂x ∂x

(3.44)

∂ 2w ∂Fw˙ = (ρA (x) + Mi δ (x − Lm )) 2 ∂t ∂t where Fˆ r (w) = 1 + c2



do − w b

0.76 (3.45)

and c2 = 0.24c1 = 0.204 + 0.6 In arriving at Eq. (3.46), we have used Eq. (3.30).

 0.24 h . b

(3.46)

96

3

Thin Beams: Part I

Using Eq. (3.44) in Eq. (3.43), we obtain the following governing equation of motion for the beam   ∂w ∂ 2w ∂ p (x, t) − Po (w (t)) 2 ∂x ∂x ∂x   + kf + ki δ (x − Ls ) w + ko (w (x, t) − z (t)) δ (x − Lo )

∂2 ∂x2



EI (x)

∂ 2w ∂x2



−

+ [ρA (x) + Mi δ (x − Lm )]

(3.47)

εo Vo2 bFˆ r (w) ∂ 2w = + f (x, t) . ∂t2 2 (do − w)2

To obtain the governing equation of the single degree-of-freedom system, we use ¯ where F¯ is given by Eq. (3.35), and Case 5 of Table B.1 of Appendix B with F = F, u = z; that is, F¯ z −

∂ F¯ z˙ = 0. ∂t

(3.48)

Substituting Eq. (3.35) in Eq. (3.48), it is found that

mo

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

(3.49)

It is seen that Eq. (3.47) is, in general, a nonlinear equation due to the electrostatic term and the stretching of the neutral axis caused by the transverse displacement as indicated by Po (w (t)). It is pointed out that there are no restrictions on the relative locations of Lm , Ls , and Lo and they can include the case where Lm = Ls = Lo , which is a configuration of interest.

3.2.3 Boundary Conditions The boundary conditions are determined directly from Eqs. (B.124a,b) of Appendix B. Upon comparing Eq. (3.36) with the first equation of Eq. (B.36), it is found that a11 = ML

a21 = MR

a12 = JL

a22 = JR

A11 = kL

A21 = kR

A12 = ktL

A22 = ktR .

(3.50)

Then, using Eqs. (3.41) and (3.50) in Eqs. (B.124a,b), the following boundary conditions at each end of the beam are obtained.

3.2

Derivation of Governing Equation and Boundary Conditions

97

At x = 0 Either w (0, t) = 0 or 

∂ 2w ∂w − (p (x, t) + Po (w (t))) ∂t2 ∂x x=0   2 ∂ ∂ w + = 0 0 ≤ kL ≤ ∞ EI (x) 2 ∂x ∂x x=0

kL w + ML

(3.51a)

and either ∂w (0, t)/∂x = 0 or  ktL

∂w ∂ 3w ∂ 2w − EI (x) 2 + JL 2 ∂x ∂x∂t ∂x

=0

0 ≤ ktL ≤ ∞

(3.51b)

 ∂ 2w ∂w kR w + MR 2 + (p (x, t) + Po (w (t))) ∂x x=L ∂t    2 ∂ ∂ w − = 0 0 ≤ kR ≤ ∞ EI (x) 2 ∂x ∂x x=L

(3.52a)

x=0

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

and either ∂w (L, t)/∂x = 0 or  ∂w ∂ 3w ∂ 2w + JR ktR + EI (x) ∂x ∂x∂t2 ∂x2

=0

0 ≤ ktR ≤ ∞

(3.52b)

x=L

Each term of the boundary conditions has an interpretation. The displacement is w, the slope is ∂w/∂x, the moment M is EI∂ 2 w/∂x2 (recall Eq. (3.10)), and the shear  2 2 force V is ∂ EI (x) ∂ w/∂x /∂x (recall Eq. (3.11)). Consequently, it is seen that kα w and Mα w ¨ are shear forces and ktα wx and Jα w ¨ x are moments where α = L, R. In addition, the quantity (p (x, t) + Po (w (t))) ∂w/∂x is the vertical component of the sum of the axial forces at the end of the beam. Equations (3.51a,b) and (3.52a,b) have numerous special cases depending on whether the spring constants are 0 or ∞. For example, if kR → ∞ and ktR → ∞, then if we divide Eq. (3.52a) by kR and Eq. (3.52b) by ktR and take the limit as the respective quantities approach infinity, it is seen that the boundary conditions simplify to w (L, t) = 0 and ∂w (L, t)/∂x = 0. These are the boundary conditions for a clamped end. On the other hand, if kR = ktR = MR = JR = 0, then we have that V = M = 0. These are the boundary conditions for a free end. Several special cases of these boundary conditions are summarized in Table 3.1. It is pointed out that as a practical matter, Po (w) can only be generated if axial motion of the beam is restricted. Thus, when an end is free, Po (w) = 0 since there is no restriction of the neutral axis for this boundary condition.

Boundary condition

Clamped

Hinged (simply supported, pinned)

Pinned with torsion spring kt

Free

Free with axial force

Case

1

2

3

4

5

∂ 2y =0 ∂η2   ∂ ∂ 2y ∂y i (η) 2 = S (η, τ ) ∂η ∂η ∂η

∂ 2w =0 ∂x2   2 ∂ w ∂w ∂ EI (x) 2 = p (x, t) ∂x ∂x ∂x EI (x)

P is tensile Valid at either end of the beam PL2 S= EIo

∂ 2y =0 ∂η2   2 ∂ ∂ y i (η) 2 = 0 ∂η ∂η

∂2w =0 ∂x2   2 ∂ ∂ w EI (x) 2 = 0 ∂x ∂x EI (x)

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

y=0 ∂ 2y ∂y i (η) 2 = so Ktα ∂η ∂η

=0

w=0

∂η2

Remarks

∂w ∂ 2w EI (x) 2 = so ktα ∂x ∂x

∂x2

=0

y=0

w=0 ∂2y

y=0 ∂y =0 ∂η

w=0 ∂w =0 ∂x ∂ 2w

Non dimensional form

Dimensional form

Table 3.1 Dimensional and non dimensional form of the boundary conditions at each end of a beam

98 3 Thin Beams: Part I

∂ 2y ∂3y = so jα ∂η2 ∂η∂τ 2   ∂ ∂2y ∂ 2y i (η) 2 = s1 mα 2 ∂η ∂η ∂τ

∂ 2w ∂ 3w = so Jα ∂x2 ∂x∂t2   ∂ 2w ∂ ∂2w EI (x) 2 = s1 Mα 2 ∂x ∂x ∂t

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

7

so = +1 and α = L at η = 0

EI (x) i (η)

s1 = +1 and α = R at η = 1 Jα Mα , mα = jα = mb L2 mb mb = ρAo L

s1 = −1 and α = L at η = 0

so = −1 and α = R at η = 1

so = +1 and α = L at η = 0

s1 = +1 and α = R at η = 1 kα L3 ktα L Kα = , Ktα = EIo EIo Case 1: kα → ∞ & ktα → ∞ Case 2: kα → ∞ & ktα → 0

s1 = −1 and α = L at η = 0

so = −1 and α = R at η = 1

∂ 2y ∂y = so Ktα ∂η ∂η2   ∂ ∂ 2y i (η) 2 = s1 Kα y ∂η ∂η i (η)

∂w ∂2w = so ktα ∂x ∂x2   ∂ 2w ∂ EI (x) 2 = s1 kα w ∂x ∂x EI (x)

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

6

Remarks

Non dimensional form

Dimensional form

Boundary condition

Case

Table 3.1 (continued)

3.2 Derivation of Governing Equation and Boundary Conditions 99

100 Table 3.2 conditions

3

Thin Beams: Part I

Convention for indicating axial constrained and axial unconstrained boundary Boundary condition

Representation

Hinged, axial motion restrained

Hinged, axial motion permitted

Clamped, axial motion restrained

Clamped, axial motion permitted

It is also pointed out that the formal either/or statement of the boundary conditions is not necessary since w = 0 at either end of the beam can be obtained from Eqs. (3.51a) by taking the limit as kL → ∞ and from (3.52a) by taking the limit as kR → ∞. Similar reasoning can be applied to ∂w/∂x = 0 at either end of the beam by taking the respective limits on ktL and ktR . Furthermore, the axial loading is always present in the boundary conditions given by Eqs. (3.51a) and (3.52a) whenever in Eq. (3.51a) kL < ∞ and whenever in Eq. (3.52a) kR < ∞. Although p (x,t) does not appear in the boundary conditions when kL → ∞ and/or kR → ∞, it still appears in the governing equation. However, in order for the force to be transmitted to the beam one has to realize that the boundary constraint that causes w (0, t) = 0 and/or w (L, t) = 0 does so while permitting the neutral axis to change length. This is usually indicated pictorially by having rollers placed between the displacement and ground. The convention for illustrating these different types of end conditions is shown in Table 3.2. Orthogonal Functions In order for one to be able to generate orthogonal functions, it was shown in Appendix B that F and F (C1 ) had to be symmetric quadratics. It is seen from Eq. (3.36) that F (C1 ) is a symmetric quadratic and that when F can be reduced to Eq. (3.42) it too will be a symmetric quadratic. However, in order for Eq. (3.42) to be a symmetric quadratic, it must be assumed that z (t) is prescribed; that is, it is a known quantity. In the present case, z (t) is determined from Eq. (3.49). Therefore, because of this coupling of systems, z (t) is a function of w, and hence, Eq. (3.42) is no longer a symmetric quadratic. Consequently, in order for Eq. (3.42) to become a symmetric quadratic, the single degree-of-freedom system is removed; that is, ko = 0 in Eq. (3.42). In other words, orthogonal functions cannot be generated when a single degree-of-freedom system is attached to a beam. (See also Balachandran and Magrab, 2009, p. 595.)

3.2

Derivation of Governing Equation and Boundary Conditions

101

3.2.4 Non Dimensional Form of the Governing Equation and Boundary Conditions Governing Equations If we assume 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 shape functions, then Eq. (3.47) can be converted to a non dimensional form by introducing the following quantities w Mi x t z do , τ = , y = , zˆ = , d = , mi = L to L L L mb kf L4 kβ L3 Lm Kf = , Kβ = , β = i, o ηα = , α = m, s, o EIo EIo L  Ko mo Ao ρ , Mo = , to = L 2 s 4o = ωo2 to2 = Mo mb EIo  ko εo bL −2 2 mb = ρAo L kg, e1 = V , ωo = rad/s 2EIo mo η=

(3.53)

where to is a characteristic time of the beam. Using Eq. (3.53) in Eq. (3.47), the non dimensional form of Eq. (3.47) is     ∂y ∂ 2y ∂ 2y ∂ S (η, τ ) − P (y (τ )) 2 i (η) 2 − ∂η ∂η ∂η ∂η     + Kf + Ki δ (η − ηs ) y + Ko y − zˆ (τ ) δ (η − ηo )

∂2 ∂η2

−

e21 Vo2 Fˆ r

(y)

(d − y)2

+ [a (η) + mi δ (η − ηm )]

∂ 2y ∂τ 2

(3.54)

= fˆ (η, τ )

where we have used the relation δ (x) = δ (Lη) = δ (η)/ |L| and 1 P (y (τ )) = 2



Ao L Io

2 1  0

∂y ∂η

2

p (η, τ ) L2 S (η, τ ) = EIo f τ ) L3 (η, , fˆ (η, τ ) = EIo   d − y 0.76 Fˆ r (y) = 1 + c2 . b/L

dη,

(3.55)

102

3

Thin Beams: Part I

The equation for the single degree-of-freedom system given by Eq. (3.49) becomes 1 d 2 zˆ + zˆ = y (ηo , τ ) . 4o dτ 2

(3.56)

Boundary Conditions Using Eq. (3.53) in Eqs. (3.51) and (3.52), the boundary conditions in non dimensional form become as follows. At η = 0    ∂ 2y ∂y ∂ ∂ 2y + i (η) 2 KL y + mL 2 − (S (η, t) + P (y (t))) ∂η ∂η ∂τ ∂η

η=0

=0

(3.57a)

0 ≤ KL ≤ ∞ and  KtL

∂y ∂ 3y ∂ 2y + jL − i (η) 2 2 ∂η ∂η∂τ ∂η

η=0

=0

0 ≤ KtL ≤ ∞

(3.57b)

At η = 1    ∂y ∂ ∂ 2y ∂ 2y − i (η) 2 KR y + mR 2 + (S (η, t) + P (y (t))) ∂η ∂η ∂τ ∂η

η=1

=0

(3.58a)

0 ≤ KR ≤ ∞ and  ∂y ∂ 3y ∂ 2y + jR KtR + i (η) 2 2 ∂η ∂η∂τ ∂η

η=1

=0

0 ≤ KtR ≤ ∞

(3.58b)

In Eqs. (3.57) and (3.58), the following non dimensional quantities have been introduced jα = Ktα

Jα Mα , mα = mb L 2 mb

ktα L kα L3 = , Kα = , EIo EIo

(3.59) α = L, R.

Special cases of these boundary conditions in non dimensional form are given in Table 3.1.

3.3

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

103

3.3 Natural Frequencies and Mode Shapes of Beams with Constant Cross Section and with Attachments 3.3.1 Introduction In this chapter, we shall ignore the effects of electrostatic forces and the nonlinear force P (w (t)); that is, we set Vo = P (w (t)) = 0. Additionally, in this section, we shall consider only beams of constant cross section; therefore, i (η) = a (η) = 1. The effects of changes in the beam cross section are examined in Section 3.8. Furthermore, it is assumed that the axial tensile force is constant; that is, p = po . Finally, since we are only interested in determining the natural frequencies and mode shapes, we set fˆ = 0. With these assumptions, Eq. (3.54) reduces to    ∂ 2y  ∂ 4y − So 2 + Kf + Ki δ (η − ηs ) y + Ko y − zˆ (τ ) δ (η − ηo ) 4 ∂η ∂η ∂ 2y + [1 + mi δ (η − ηm )] 2 = 0 ∂τ

(3.60)

where, from Eq. (3.55), So =

po L2 . EIo

(3.61)

Equation (3.56) remains the same.

3.3.2 Solution for Very General Boundary Conditions We shall determine the natural frequencies and modes shapes for a very general set of boundary conditions and then reduce the results to several special cases of interest. Although this introduces additional algebraic complexity at the outset, it greatly reduces the need to solve repeatedly a class of very similar systems every time the boundary conditions or the in-span attachments change. This approach will also provide a clearer interpretation of the effects that the boundary conditions have on the natural frequencies of the systems. Consider a beam with the following boundary conditions. At the end η = 0, the beam is free and is restrained by a translation spring kL (N/m) and a torsion spring ktL (Nm). At the end η = 1, the beam is also free and is restrained by a translation spring kR (N/m) and a torsion spring ktR (Nm). In addition, this end also has attached to it a mass MR (kg) with mass moment of inertia JR (kgm2 ). Thus, at η = 0, we have from Eq. (3.57) that ∂y ∂ 3y = −KL y + So 0 ≤ KL ≤ ∞ ∂η ∂η3 ∂ 2y ∂y = KtL 0 ≤ KtL ≤ ∞. ∂η ∂η2

(3.62)

104

3

Thin Beams: Part I

∂ 2y ∂y ∂ 3y = KR y + mR 2 + So 0 ≤ KR ≤ ∞ 3 ∂η ∂τ ∂η ∂ 2y ∂y ∂ 3y − j = −K 0 ≤ KtR ≤ ∞. tR R ∂η ∂η2 ∂η∂τ 2

(3.63)

At η = 1, we have from Eq. (3.58) that

We assume a solution to Eq. (3.60) and (3.56) of the form y (η, τ ) = Y (η) ej zˆ (τ ) = Zo ej

2τ

(3.64)

2τ

where 2 = ωto

(3.65)

and ω (rad/s) is the frequency. Then Eqs. (3.60) and (3.56), respectively, become  ∂ 4Y ∂ 2Y  Y + Ko (Y − Zo ) δ (η − ηo ) − S + K + K δ − η ) (η o f i s ∂η4 ∂η2

(3.66)

− [1 + mi δ (η − ηm )] Y = 0 4

and  1−

4 4o

 Zo = Y (ηo ) .

(3.67)

Combining Eqs. (3.66) and (3.67), we arrive at  ∂ 2Y  ∂ 4Y Mo  4  Yδ (η − ηo ) − So 2 + Kf + Ki δ (η − ηs ) Y −  4 ∂η ∂η 1 − 4 /4o

(3.68)

−4 [1 + mi δ (η − ηm )] Y = 0. The boundary conditions at η = 0, which are given by Eqs. (3.62), become Y (0) = −KL Y (0) + So Y (0)

0 ≤ KL ≤ ∞

Y (0)

0 ≤ KtL ≤ ∞

= KtL

Y (0)

(3.69)

and those at η = 1, which are given by Eq. (3.63), become Y (1) = A2 Y (1) + So Y (1)

0 ≤ KR ≤ ∞

Y (1)

0 ≤ KtR ≤ ∞

= B2

Y (1)

(3.70)

3.3

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

105

where the prime denotes the derivative with respect to η and A2 = KR − mR 4 B2 = jR 4 − KtR .

(3.71)

As noted previously, Eqs. (3.69) and (3.70) contain numerous special cases. These cases can be obtained by setting Kα and Ktα , α = L, R, to one of its limiting values, either 0 or ∞. For example, if we were to consider a beam clamped at η = 0 and free at η = 1, this case is arrived at as follows. The first equation of Eq. (3.69) is divided by KL and the second equation of Eq. (3.69) is divided by KtL , and the respective limits as KL → ∞ and as KtL → ∞ are taken. In Eq. (3.70), we set A2 = B2 = 0. This procedure produces the boundary conditions that appear as Cases 1 and 4 in Table 3.1. We will use this limiting procedure to examine numerous special cases. There are several methods that can be used to solve Eq. (3.68) subject to the boundary conditions given by Eqs. (3.69) and (3.70). For this particular system, we shall use the Laplace transform with respect to the spatial variable η. The Laplace transform can handle easily the discontinuities in the governing equation and, as will become evident shortly, we will be able to reduce immediately the solution with four unknown solution constants to a solution with two unknown constants, which greatly simplifies the subsequent algebra. It is this immediate reduction in the number of unknown constants that permits us to examine the case of very general boundary conditions and then reduce the result to numerous special cases. Taking the Laplace transform of Eq. (3.68) with respect to η, we obtain from Eq. (C.12) of Appendix C with f = 0 and from pair 5 of Table C.1 of Appendix C that ,    3 1 s − So s y (0) + s2 − So y (0) + sy (0) + y (0) Y¯ (s) = D (s) 3  −sηi Gi Y (ηi ) e −

(3.72)

i=1

where G1 = Ki ,

G2 = −mi 4 ,

Mo  4  G3 = −  1 − 4 4o

η1 = η s ,

η2 = ηm ,

η3 = ηo .

(3.73)

In Eq. (3.72), s is the Laplace transform parameter, Y¯ (s) is the Laplace transform of Y (η), the prime indicates the derivative with respect to η, and

106

3

Thin Beams: Part I

   D (s) = s4 − So s2 + Kf − 4 = s2 − δ 2 s2 + ε 2   S So 2 o + 4 − Kf ε2 = − + 2 2   So 2 S o + δ2 = + 4 − Kf . 2 2

(3.74)

It is noted that in Eq. (3.72), Y (0) is the displacement at η = 0, Y (0) is the slope at η = 0, Y (0) is proportional to the moment at η = 0, and Y (0) is proportional to the shear force at η = 0. These four quantities and the value of  represent the five unknown quantities that have to be determined. The inverse Laplace transform of Eq. (3.72) can be obtained from Eq. (C.17) of Appendix C with f (η) = 0. Thus, ˆ (η) + Y (0) Rˆ (η) + Y (0) Sˆ (η) + Y (0) Tˆ (η) Y (η) = Y (0) Q −

3 

Gi Y (ηi ) Tˆ (η − ηi ) u (η − ηi )

(3.75)

i=1

where u (η) is the unit step function, Y (ηi ) is the displacement at η = ηi , i = 1, 2, 3, ˆ (η), etc. are given by Eq. (C.15) of Appendix C. and Q From the boundary condition at η = 0 given by Eq. (3.69), it is seen that we can replace in Eq. (3.75) Y (0) with KtL Y (0) and Y (0) with −KL Y (0) + So Y (0). Thus, Eq. (3.75) becomes Y (η) = Y (0) fˆ1 (η) + Y (0) fˆ2 (η) −

3 

Gi Y (ηi ) Tˆ (η − ηi ) u (η − ηi )

(3.76)

i=1

where ˆ (η) − KL Tˆ (η) fˆ1 (η) = Q fˆ2 (η) = Rˆ (η) + KtL Sˆ (η) + So Tˆ (η) .

(3.77)

The remaining two constants, Y (0) and Y (0), are determined by substituting Eq. (3.76) into the boundary conditions at η = 1, which are given by Eq. (3.70). Performing this substitution, we arrive at the following system of equations   ˆ {Y} = gˆ [C]

(3.78)

 cˆ () cˆ 12 () ˆ [C] = 11 cˆ 21 () cˆ 22 ()     gˆ 1 () Y (0) {Y} = , gˆ = . gˆ 2 () Y (0)

(3.79)

where

3.3

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

107

It is noted that if the Laplace transform and the operations used to arrive at Eq. (3.78) weren’t used, Eq. (3.78) would have been a 4×4 system of equations. ˆ are The coefficients of [C]  

 ˆ (1) − A2 Q ˆ (1) − KL Tˆ (1) cˆ 11 () = 4 − Kf Rˆ (1) − KL Q  

  cˆ 12 () = 4 − Kf Sˆ (1) + KtL Tˆ (1) − A2 Rˆ (1) + KtL Sˆ (1) + So Tˆ (1)   

cˆ 21 () = 4 − Kf Sˆ (1) − KL Rˆ (1) + So Tˆ (1)    − B2 4 − Kf Tˆ (1) − KL Sˆ (1)

     ˆ (1) + KtL Rˆ (1) + So Tˆ (1) + So Sˆ (1) cˆ 22 () = 4 − Kf Tˆ (1) − B2 Q     ˆ (1) + So Sˆ (1) + So Rˆ (1) + So Tˆ (1) + KtL Q (3.80)   and elements of the column vector gˆ are gˆ l () =

3 

Gi Y (ηi ) pˆ l (ηi )

l = 1, 2

(3.81)

i=1

where ˆ (1 − ξ ) − A2 Tˆ (1 − ξ ) pˆ 1 (ξ ) = Q pˆ 2 (ξ ) = Rˆ (1 − ξ ) + So Tˆ (1 − ξ ) − B2 Sˆ (1 − ξ ) .

(3.82)

In arriving at these results, we have used Eq. (3.74) to find that δ 2 − ε2 = So and δ 2 ε2 = 4 − Kf and we have used Eq. (C.16) of Appendix C. Solving Eq. (3.78) for the unknown quantities Y (0) and Y (0), it is found that Y (0) = Y (0) =

1

3 

ˆ () D

i=1

1

3 

ˆ () D

i=1

Gi Y (ηi ) hˆ 1i () (3.83) Gi Y (ηi ) hˆ 2i ()

where hˆ 1i () = cˆ 22 () pˆ 1 (ηi ) − cˆ 12 () pˆ 2 (ηi ) hˆ 2i () = cˆ 11 () pˆ 2 (ηi ) − cˆ 21 () pˆ 1 (ηi )

i = 1, 2, 3

(3.84)

and ˆ () = cˆ 11 () cˆ 22 () − cˆ 12 () cˆ 21 () . D

(3.85)

108

3

Thin Beams: Part I

Substituting Eq. (3.83) into Eq. (3.76) and collecting terms, we obtain

Y (η) =

1

3 

ˆ () D

i=1

ˆ i (, η, ηi ) Gi Y (ηi ) H

(3.86)

where ˆ i (, η, ηi ) = hˆ 1i () fˆ1 (η) + hˆ 2i () fˆ2 (η) H ˆ () Tˆ ( [η − ηi ]) u (η − ηi ) −D

i = 1, 2, 3.

(3.87)

It is noted that Eq. (3.86) is more general than it appears. By changing the definitions of G1 and/or G2 and/or G3 one can also use these results to consider a beam with, say, three in-span springs by setting Gj = Kj , j = 1, 2, 3 or a beam with three in-span masses by setting Gj = −mj 4 , j = 1, 2, 3. It is also noted that these results can be extended to include N attachments, N > 3, by simply changing the upper limit of the sum to N and assigning Gi , i > 3, the appropriate physical description of each additional attachment: either a spring, a mass, or a single degree-of-freedom system. It is also noted that in terms of the type of attachments represented by G1 and G2 , each of them is a limiting case of G3 . If, 4 in G3 , we letKo → ∞, then  o → ∞ and G3 → −Mo  . If G3 is rewritten as 4 4 G3 = Ko  /  − Ko /Mo , then when Mo → ∞, it is seen that G3 → Ko . It is also noted that when ηs = ηm = ηo , it is found from Eq. (3.84) that hˆ 11 () = hˆ 12 () = hˆ 13 () and hˆ 21 () = hˆ 22 () = hˆ 23 () and, therefore, ˆ 2 (, η, ηm ) = H ˆ 3 (, η, η0 ). ˆ 1 (, η, ηs ) = H H At this point, we shall consider the case of the beam without the single degreeof-freedom system. The case of the beam with the single degree-of-freedom system is considered in Section 3.5.1. Hence, we set Mo = 0 and, therefore, G3 = 0 and Eq. (3.86) becomes

Y (η) =

1

2 

ˆ () D

i=1

ˆ i (, η, ηi ). Gi Y (ηi ) H

(3.88)

To determine the characteristic equation, it is noted that Eq. (3.88) must be satisfied at η = η1 = ηs and at η = η2 = ηm . Therefore, setting η to each of these values in Eq. (3.88), we obtain ˆ () = Y (η1 ) D

2 

ˆ i (, η1 , ηi ) Gi Y (ηi ) H

i=1

ˆ () = Y (η2 ) D

2  i=1

(3.89) ˆ i (, η2 , ηi ) Gi Y (ηi ) H

3.3

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

109

which can be written in matrix form as   ˆ 1 (, η1 , η1 ) − D ˆ 2 (, η1 , η2 ) ˆ () G1 H G2 H Y (η1 ) = 0. (3.90) ˆ 1 (, η2 , η1 ) ˆ 2 (, η2 , η2 ) − D ˆ () Y (η2 ) G1 H G2 H It is seen from the procedure used to obtain Eq. (3.90), that the size of this system of equations is equal to the number of in-span attachments placed on the beam. The natural frequencies are those values of  = n for which the determinant of the coefficients of Eq. (3.90) equals zero; that is, those values of n that are solutions to

  ˆ 1 (n , η1 , η1 ) − D ˆ 2 (n , η2 , η2 ) − D ˆ (n ) G2 H ˆ (n ) G1 H ˆ 2 (n , η1 , η2 ) H ˆ 1 (n , η2 , η1 ) = 0. − G 1 G2 H

(3.91)

Equation (3.91) is valid for the general boundary conditions given by Eqs. (3.69) and (3.70). When there are no in-span attachments, that is, when mi = Ki = 0, the characteristic equation given by Eq. (3.91) reduces to ˆ (n ) = 0. D

(3.92)

From this point forward, we shall examine analytically and numerically only the case where So = Kf = 0. However, the qualitative effects of So and Kf are shown in Section 3.6 and in Section 3.8.5 the effects of So on tapered beams are considered.

3.3.3 General Solution in the Absence of an Axial Force and an Elastic Foundation To consider the case of beam without an elastic foundation and without an axial force, we set So = Kf = 0. Then, from Eq. (3.74) it is seen that δ = ε = . Under these conditions, the results of Section 3.3.2 simplify as follows. The boundary conditions at η = 0, which are given by Eq. (3.69), become Y (0) = −KL Y (0)

0 ≤ KL ≤ ∞

Y (0)

0 ≤ KtL ≤ ∞

= KtL

Y (0)

(3.93)

and those at η = 1, which are given by Eq. (3.70), become Y (1) = A2 Y (1)

0 ≤ KR ≤ ∞

Y (1)

0 ≤ KtR ≤ ∞

= B2

Y (1)

(3.94)

where the prime denotes the derivative with respect to η and A2 and B2 are given by Eq. (3.71).

110

3

Thin Beams: Part I

Equation (3.76) becomes Y (η) = Y

(0) f1 (η) + Y (0) f2 (η) −

3 

Gi Y (ηi ) T (η − ηi ) u (η − ηi )/3 (3.95)

i=1

where it is found that Eq. (3.77) can be written as f1 (η) = Q (η) − KL T (η)/3

(3.96)

f2 (η) = R (η)/ + KtL S (η)/2

and the non dimensional functions Q (η), R (η), S (η), and T (η) are given by Eq. (C.19) of Appendix C. Equation (3.78) becomes 

c11 () c21 ()



c12 () c22 ()

Y (0) Y (0)



=

g1 () g2 ()

 (3.97)

where   c11 () = 3 R () − KL Q () − A2 Q () − KL T ()/3   c12 () = 2 S () + KtL T () − A2 R ()/ + KtL S ()/2   c21 () = 2 S () − KL R ()/ − B2 T () − KL S ()/2   c22 () = T () + KtL Q () − B2 Q () + KtL R ()/

(3.98)

and gl () =

3 

Gi Y (ηi ) pl (, ηi ) l = 1, 2.

(3.99)

i=1

In Eq. (3.99), p1 (, ξ ) = Q ( [1 − ξ ]) − A2 T ( [1 − ξ ])/3 p2 (, ξ ) = R ( [1 − ξ ])/ − B2 S ( [1 − ξ ])/2 .

(3.100)

The solution given by Eq. (3.86) becomes 1  Gi Y (ηi ) Hi (, η, ηi ) D () 3

Y (η) =

(3.101)

i=1

where Hi (, η, ηi ) = h1i () f1 (η) + h2i () f2 (η) − D () T ( [η − ηi ]) u (η − ηi )/3

i = 1, 2, 3

(3.102)

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

3.3

111

and h1i () = c22 () p1 (, ηi ) − c12 () p2 (, ηi ) h2i () = c11 () p2 (, ηi ) − c21 () p1 (, ηi ) i = 1, 2, 3.

(3.103)

The quantity D () is given by D () = c11 () c22 () − c12 () c21 () .

(3.104)

In the remainder of this section, we shall consider the case of the beam without the single degree-of-freedom system; that is, when G3 = 0. Then the characteristic equation is obtained from Eq. (3.90), which becomes 

G1 H1 (, η1 , η1 ) − D () G1 H1 (, η2 , η1 )

G2 H2 (, η1 , η2 ) G2 H2 (, η2 , η2 ) − D ()



Y (η1 ) Y (η2 )

 = 0. (3.105)

The natural frequencies are those values of  = n for which the determinant of the coefficients of Eq. (3.105) equals zero; that is, those values of n that are solutions to (G1 H1 (n , η1 , η1 ) − D (n )) (G2 H2 (n , η2 , η2 ) − D (n )) − G1 G2 H2 (n , η1 , η2 ) H1 (n , η2 , η1 ) = 0.

(3.106)

When there are no in-span attachments, that is, when mi = Ki = 0 (G1 = G2 = 0), Eq. (3.106) reduces to D (n ) = 0.

(3.107)

Equations (3.106) and (3.107) are valid for the boundary conditions given by Eqs. (3.93) and (3.94). The limiting procedure3 discussed in Section 3.3.2 is used to obtain the characteristic equations for several sets of boundary conditions, the results of which are presented in Table 3.3. To this end, the following definitions are introduced D (n ) = D(p) (n )         Hj n , ηi , ηj = −T n ηi − ηj u ηi − ηj D(p) (n )   3 i, j = 1, 2 n + Hn(p) ηi , ηj

(3.108)

where p is the case number appearing in the leftmost column of Table 3.3 indicating (p) a specific set of boundary conditions and the expression for Hn (x, y) corresponding to the boundary conditions appear in the fourth column of that case’s row. It is noted 3

The MATLAB function limit from the Symbolic Math toolbox was used.

Clamped (KL → ∞) (KtL → ∞)

Clamped (KL → ∞) (KtL → ∞)

Free (KL = 0) (KtL = 0)

2

3

4

Free (A2n = 0) (B2n = 0)

Free (A2n = 0) (B2n = 0)

D(4) (n ) = R (n ) T (n ) − S2 (n ) = −0.5 (1 − cos n cosh n )        (4) Hn (x, y) = Q (n x) T (n ) Q n 1 − y − S (n ) R n 1 − y        + R (n x) R (n ) R n 1 − y − S (n ) Q n 1 − y

D(3) (n ) = R (n ) T (n ) − Q2 (n ) = −0.5 (1 + cos n cosh n )        Hn(3) (x, y) = T (n x) T (n ) R n 1 − y − Q (n ) Q n 1 − y        + S (n x) R (n ) Q n 1 − y − Q (n ) R n 1 − y

D(2) (n ) = S2 (n ) − R (n ) T (n ) = 0.5 (1 − cos n cosh n )        (2) Hn (x, y) = T (n x) S (n ) S n 1 − y − R (n ) T n 1 − y        + S (n x) S (n ) T n 1 − y − T (n ) S n 1 − y

D(1) (n ) = T 2 (n ) − R2 (n ) = − sin n sinh n        Hn(1) (x, y) = T (n x) T (n ) T n 1 − y − R (n ) R n 1 − y        + R (n x) T (n ) R n 1 − y − R (n ) T n 1 − y

Hinged (A2n → ∞) (B2n = 0)

Hinged (KL → ∞) (KtL = 0)

1

Clamped (A2n → ∞) (B2n → ∞)

Functions in the characteristic equation, Eq. (3.110), and the mode shapes, Eq. (3.120), when mi = 0 and/or Ki = 0

η=1 [Eq. (3.94)]

Case η = 0 (p) [Eq. (3.93)]

Boundary conditions

S (n ) T (n η) + S (n η) T (n )

T (n ) T (n η) + S (n η) Q (n )

S (n ) Q (n η) + R (n η) R (n )

Yn(2) (η) = −

Yn(3) (η) = −

Yn(4) (η) = −

Yn (η) = sin (n η)

(1)

Mode shape for mi = Ki = 0

Table 3.3 Functions comprising the characteristic equation and corresponding mode shape for a beam with and without in-span attachments for several boundary conditions. In all cases, So = Kf = 0. The quantities A2n and B2n are given in Eq. (3.109)

112 3 Thin Beams: Part I

6

5

∗

Free with translation spring (KtL = 0)

Clamped (KL → ∞) (KtL → ∞)

Case η = 0 (p) [Eq. (3.93)]

Free with translation spring (A2n = KR ) (B2n = 0)

Hinged (A2n → ∞) (B2n = 0)

η=1 [Eq. (3.94)]

Boundary conditions

Yn(5) (η) = −

S (n ) T (n η) + S (n η) T (n )

Mode shape for mi = Ki = 0

+ KL KR T (n )/6n

D(6) (n ) = D(4) (n ) − (KL + KR ) D(5) (n )/3n + KL KR D(1) (n )/6n Yn(6) (η) = Cn(6) [Q (n η)    (6) (4) (1) − KL T (n η) /3n Hn (x, y) = Hn (x, y) + KL KR 6n Hn (x, y)         + R (n η) + KL R (n x) R (n ) Q n 1 − y − Q (n ) R n 1 − y        3 KR R (n )/3n − S (n ) / + T (n x) S (n ) R n 1 − y − T (n ) Q n 1 − y (6) Cn =     n     Do + KR R (n x) S (n ) T n 1 − y − Q (n ) R n 1 − y        D = R ( ) 3 o n /n + Q (n x) R (n ) R n 1 − y − T (n ) T n 1 − y − (KL + KR ) Q (n )/3n

= 0.5 (cos n sinh n − sin n cosh n )        Hn(5) (x, y) = T (n x) Q (n ) T n 1 − y − S (n ) R n 1 − y        + S (n x) T (n ) R n 1 − y − R (n ) T n 1 − y

D(5) (n ) = Q (n ) T (n ) − R (n ) S (n )

Functions in the characteristic equation, Eq. (3.110), and the mode shapes, Eq. (3.120), when mi = 0 and/or Ki = 0

Table 3.3 (continued)

3.3 Natural Frequencies and Mode Shapes of Beams with Constant Cross Section. . . 113

Hinged with torsion spring (KL → ∞)

Clamped (KL → ∞) (KtL → ∞)

7

8

(n ) + (KtL + KtR

) D(5) ( n )/n

Yn(7) (η) = −Cn(7) T (n η)

Mode shape for mi = Ki = 0

+ KtL KtR D(2) (n )/2n + R (n η)   + KtL S (n η) /n (x, y) = Hn(1) (x, y) − KtL KtR 2n Hn(2) (x, y)         R (n ) + KtL S (n )/n (7) + KtL S (n x) T (n ) R n 1 − y − R (n ) T n 1 − y Cn =        T (n ) /n + T (n x) Q (n ) T n 1 − y − S (n ) R n 1 − y         + KtR T (n x) Q (n ) T n 1 − y − R (n ) S n 1 − y        + R (n x) T (n ) S n 1 − y − S (n ) T n 1 − y /n

=

D(1)

    Yn(8) (η) = Cn(8) T (n η) D(8) (n ) = D(3) (n ) + A2n 3n D(5) (n ) + (A2n B2n ) 4n D(2) (n )   +S (n η) + B2n n (Q (n ) R (n ) − S (n ) T (n ))  4  (2) (8) (3) T (n ) − A2n S (n )/3n Hn (x, y) = Hn (x, y) + (A2n B2n ) n Hn (x, y) Cn(8) =         3 A 2n T (n )/n − Q (n ) + B2n S (n x) Q (n ) S n 1 − y − S (n ) Q n 1 − y        + T (n x) R (n ) Q n 1 − y − T (n ) S n 1 − y /n         + A2n S (n x) T (n ) R n 1 − y − R (n ) T n 1 − y        /3n + T (n x) Q (n ) T n 1 − y − S (n ) R n 1 − y

Hn(7)

D(7) (n )

Functions in the characteristic equation, Eq. (3.110), and the mode shapes, Eq. (3.120), when mi = 0 and/or Ki = 0

The natural frequencies of a beam without in-span attachments that is clamped at one end and hinged at the other end equals those for a beam free at one end and hinged at the other end. However, the mode shapes are different.

Free with attachments

Hinged with torsion spring (A2n → ∞) (B2n = −KtR )

η=1 [Eq. (3.94)]

3

∗

η=0 [Eq. (3.93)]

Case (p)

Boundary conditions

Table 3.3 (continued)

114 Thin Beams: Part I

3.3

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

115

   that T n x − y u (x − y) = 0 when either x = y, since T (0) = 0, or when x < y, since u (x − y) = 0. For Case 8 in Table 3.3, A2n = KR − mR 4n

(3.109)

B2n = jR 4n − KtR .

Substituting Eq. (3.108) into Eq. (3.106), we obtain the following characteristic equation  

G1 Hn(p) (η1 , η1 )−3n D(p) (n ) G2 Hn(p) (η2 , η2 ) − 3n D(p) (n )   − G1 G2 −T (n [η1 − η2 ]) u (η1 − η2 ) D(p) (n ) + Hn(p) (η1 , η2 )   × −T (n [η2 − η1 ]) u (η2 − η1 ) D(p) (n ) + Hn(p) (η2 , η1 ) = 0

(3.110)

p = 1, 2, ..., 8. The applicability of Eq. (3.110) to a specific set of boundary conditions is determined by the selection of p. We now obtain the following special cases of Eq. (3.110) with respect to the in-span attachments. η1 = η2 (ηs = ηm ) When η1 = η2 , Eq. (3.110) becomes   1 D(p) (n ) D(p) (n ) − 3 (G2 + G1 ) Hn(p) (η1 , η1 ) = 0 n

p = 1, 2, ..., 8. (3.111)

As will be seen subsequently in Eq. (3.115), D(p) (n ) = 0 yields the natural frequency coefficients for a beam without in-span attachments; therefore, this solution is ignored and Eq. (3.111) becomes D(p) (n ) −

1 (G2 + G1 ) Hn(p) (η1 , η1 ) = 0 3n

p = 1, 2, ..., 8.

(3.112)

It is noted from Eq. (3.112) that when G2 = G1 and G2 is a mass, the masses simply add. When G2 is a spring, then the springs’ stiffness add; that is, they are two springs in parallel. Single Intermediate Rigid Support When G1 → ∞ (Ki → ∞), Eq. (3.112) becomes Hn(p) (η1 , η1 ) = 0

p = 1, 2, ..., 8.

(3.113)

Values of n that satisfy Eq. (3.113) give the natural frequency coefficients for a uniform beam with a rigid intermediate support where the displacement is zero; however, the slope is not necessarily zero.

116

3

Thin Beams: Part I

We now consider several special cases of Eq. (3.112). η1 = η2 (ηs = ηm ) and G1 = 0 (K i = 0) or G2 = 0 (mi = 0) When either G1 = 0 or G2 = 0, we use Eq. (3.112) to obtain D(p) (n ) −

 Gj (p)  H ηj , η j = 0 3n n

j = 1, 2

p = 1, 2, ..., 8.

(3.114)

η1 = η2 (ηs = ηm ) and G1 = G2 = 0 (mi = K i = 0) When G2 = G1 = 0, we use Eq. (3.112) to obtain D(p) (n ) = 0

p = 1, 2, ..., 8.

(3.115)

The values of n that satisfy Eq. (3.115) are the natural frequency coefficients for a beam without in-span attachments. Two Intermediate Rigid Supports If G1 = K1 and G2 = K2 in Eq. (3.110) and G1 → ∞ and G1 → ∞, then Eq. (3.110) becomes   −T (n [η1 − η2 ]) u (η1 − η2 ) D(p) (n ) + Hn(p) (η1 , η2 )   × −T (n [η2 − η1 ]) u (η2 − η1 ) D(p) (n ) + Hn(p) (η2 , η1 ) − Hn(p) (η1 , η1 ) Hn(p) (η2 , η2 )

(3.116)

= 0. p = 1, 2, ..., 8

Equation (3.116) is the characteristic equation for a beam with two arbitrarily positioned in-span rigid supports; however, the slopes at these locations are not necessarily zero. Mode Shapes for G1 = 0 (K i = 0) and G2 = 0 (mi = 0) To determine the mode shape for the case where G1 = 0 and G2 = 0 and for the boundary conditions given by Eqs. (3.93) and (3.94), it is found that one expression for the ratio of the coefficients of Eq. (3.105) at  = n is Cn =

Y (η1 ) −G2 H2 (n , η1 , η2 ) = . Y (η2 ) G1 H1 (n , η1 , η1 ) − D (n )

(3.117)

Therefore, from Eq. (3.101) with G3 = 0, the mode shape is 

G2 G1 Y (η1 ) H1 (n , η, ηs ) + H2 (n , η, η2 ) D (n ) Y (η2 ) D (n ) G2 G1 H1 (n , η, η1 ) + H2 (n , η, η2 ) = Cn D (n ) D (n )

Yn (η) = Y (η2 )

(3.118)

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

3.3

117

where we have set the scale factor Y (η2 ) = 1. It will be seen in at the end of Section 3.10.2 that the value of the scale factor does not matter. In the special case where η1 = η2 , H1 (n , η, η1 ) = H2 (n , η, η2 ) and Eq. (3.118) becomes Yn (η) = H1 (n , η, η1 ) .

(3.119)

When the special cases of the general boundary conditions as given in Table 3.3 are considered, we use Eq. (3.108) to write Eq. (3.118) as ,

Yn(p) (η)

(p) Hn (η, η1 ) −T (n [η − η1 ]) u (η − η1 ) + (p) D (n ) , (p) Hn (η, η2 ) + G2 −T (n [η − η2 ]) u (η − η2 ) + (p) D (n )

= Cn(p) G1

(3.120)

(p)

where Yn (η) is the mode shape for the boundary conditions given by the pth case in Table 3.3 and

Cn(p)

  (p) −G2 −T (n [η1 − η2 ]) u (η1 − η2 ) D(p) (n ) + Hn (η1 , η2 )   . = (p) G1 −T (n [η2 − η1 ]) u (η2 − η1 ) D(p) (n ) + Hn (η2 , η1 ) − 3n D (n ) (3.121)

When η1 = η2 , Eq. (3.120) can be written as (p)

Yn(p) (η) =

Hn (η, η1 ) − T (n [η − η1 ]) u (η − η1 ) . D(p) (n )

(3.122)

Mode Shapes for G1 = 0 (K i = 0) or G2 = 0 (mi = 0) To determine the mode shape for the case when G1 = 0 or G2 = 0 and for the boundary conditions given by Eqs. (3.93) and (3.94), we use Eq. (3.101) with G3 = 0 to obtain   Hj n , η, ηj Yn (η) = D (n )

j = 1, 2.

(3.123)

When the special cases of the general boundary conditions as given in Table 3.3 are considered, we use Eq. (3.108) to write Eq. (3.123) as Yn(p) (η)

 (p)       Hn η, ηj = (p) − T n η − ηj u η − ηj D (n )

j = 1, 2.

(3.124)

118

3

Thin Beams: Part I

This mode shape is also valid for the case when Gj → ∞. Mode Shapes for G1 = 0 (K i = 0) and G2 = 0 (mi = 0) When G1 = 0 and G2 = 0, g1 = g2 = 0 and Eq. (3.95) becomes Y (η) = Y (0) f1 (η) + Y (0) f2 (η)     = Y (0) Q (η) − KL T (η)/3 + Y (0) R (η)/ + K1L S (η)/2 (3.125) where we have used Eq. (3.96). Equation (3.97) becomes  [C] {Y} =

c11 () c12 () c21 () c22 ()



Y (0) Y (0)

 =

 0 0

(3.126)

where the coefficients of [C] are given by Eq. (3.98). For those values of  = n that satisfy det [C] = 0; that is, for those values for which D (n ) = 0, we obtain from Eq. (3.126) Y (0) c12 (n ) =− Y (0) c11 (n )   2 n S (n ) + KtL T (n ) − A2 R (n )/n + KtL S (n )/2n   =− 3n R (n ) − KL Q (n ) − A2 Q (n ) − KL T (n )/3n

Cn =

KtL < ∞ (3.127)

which is valid for KtL < ∞. Then, the mode shape becomes   Yn (η) = Cn Q (n η) − KL T (n η)/3n   + R (n η)/n + KtL S (n η)/2n

KtL < ∞

(3.128)

and we have set the scale factor Y (0) = 1. When KtL → ∞, we multiply Eq. (3.128) by 2n /KtL and take the limit as KtL → ∞, to obtain   Yn (η) = Cˆ n 2n Q (n η) − KL T (n η)/n + S (n η)

KtL → ∞ (3.129)

where Cˆ n = −

T (n ) − A2 S (n )/2n   3n R (n ) − KL Q (n ) − A2 Q (n ) − KL T (n )/3n

KtL → ∞. (3.130)

From Eq. (3.93), it is seen that the case of KtL → ∞ indicates that the slope is zero at η = 0; thus, Eqs. (3.129) and (3.130) are used when the end at η = 0 is clamped. When we consider the special cases of the boundary conditions given in Table 3.3 (p) for G1 = G2 = 0, we denote the mode shape Yn (η), where p is the case number

3.3

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

119

(p)

given in Table 3.3. The expressions for Yn (η) are given in Table 3.3 and correspond (p) to the values of n that satisfy Dn (n ) = 0. Node Points A node point is a stationary point of the system when it is vibrating at one of its natural frequencies. It is determined by finding those values of η = ηnode for which (p) Yn (ηnode ) = 0. A list of the node points corresponding to the first four mode shapes for cases 1 to 5 of Table 3.3 is given in Table 3.4. The location of the node points is important in the placement of sensors and in understanding the effects that in-span attachments have on the natural frequency. Strain Mode Shapes Displacement mode shapes are important in the design of many types of sensors and structures. In some types of sensors, strain is used as the sensing mechanism. In these cases, one should place the strain sensor where the strain is a maximum. From Eq. (3.8), it is seen that the strain is proportional to the second derivative of the displacement. Hence, the strain node points can be located by determining those (p) values of η = ηnode for which d2 Yn (ηnode )/dη2 = 0. In general, the strain and displacement nodes points are different. One exception is a beam hinged at both ends; for this case, the strain and displacement mode shapes are the same. A list of the strain node points corresponding to the first four mode shapes for cases 1 to 5 of Table 3.3 is given in Table 3.4. Free-Free Beam: Rigid Body Modes In addition to the mode shapes given by Eq. (3.120) for a beam free at both ends (p = 4), there exist two rigid body modes given by Yn (η) = Co

and

Yn (η) = Do + Eo η.

Both of these modes are associated with n = 0. The left hand expression represents a mode shape in which each point on the beam has the same displacement. The right hand expression represents a mode shape in which the beam undergoes a uniform translation and a rigid body rotation. In both cases, the strain mode shapes are zero. These rigid body modes will be ignored; however, they play an important role when the Rayleigh-Ritz method is used in Section 6.5.2 to determine the natural frequencies and mode shapes of rectangular plates with freely supported edges. Natural Frequency The values of the natural frequency coefficient are converted to the natural frequency fn in Hertz by using Eqs. (3.53) and (3.65). Thus, 2n 2n fn = = 2π to 2π L2

 EIo Hz. ρAo

(3.131)

Boundary conditions

Hinged-hinged

Clamped-clamped

Clamped-free

Free-free

Clamped-hinged

Case

1

2

3

4

5

Displacement Strain

Displacement Strain

Displacement Strain

Displacement Strain

Displacement Strain

Type

0, 1 0.264, 1

0.224, 0.776 0, 1

0 1

0, 1 0.224, 0.776

0, 1 0, 1

First mode

0, 0.557, 1 0.147, 0.554, 1

0.132, 0.5, 0.868 0, 0.5, 1

0, 0.783 0.217, 1

0, 0.5, 1 0.132, 0.5, 0.868

0, 0.5, 1 0, 0.5, 1

Second mode

Node points, ηnode

0, 0.386, 0.692, 1 0.102, 0.383, 0.692, 1

0.0944, 0.356, 0.644, 0.906 0, 0.358, 0.642, 1

0, 0.504, 0.868 0.132, 0.496, 1

0, 0.358, 0.642, 1 0.0944, 0.356, 0.644, 0.906

0, 0.333, 0.667, 1 0, 0.333, 0.667, 1

Third mode

0, 0.295, 0.529, 0.765, 1 0.0778, 0.293, 0.529, 0.765, 1

0.0735, 0.277, 0.5, 0.723, 0.927 0, 0.279, 0.5, 0.721, 1

0, 0.358, 0.644, 0.906 0.0944, 0.356, 0.642, 1

0, 0.279, 0.5, 0.721, 1 0.0735, 0.277, 0.5, 0.723, 0.927

0, 0.25, 0.5, 0.75, 1 0, 0.25, 0.5, 0.75, 1

Fourth mode

Table 3.4 Displacement node points and strain node points for the mode shapes corresponding to the four lowest natural frequencies for five sets of boundary conditions. Nodes points include the boundaries

120 3 Thin Beams: Part I

3.3

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

121

We see that the natural frequency is proportional to the square of the natural frequency coefficient. Thus, if the difference in two frequency coefficients is, say, 5%, then there is a 10% difference in the natural frequency. To illustrate one way to use Eq. (3.131), we shall compare two beams, each with boundary conditions, same material, and same geometric ratio  the same  2 Io / Ao L . In this case, Eq. (3.131) can be expressed as fn(j) =

cn Hz Lj

j = 1, 2

(3.132)

where the constant cn is 2 cn = n 2π



 Io Ao L2

E ρ

m/s.

(3.133)

 The ratio of the natural frequencies of the two beams, therefore, is fn(1) fn(2) = L2 /L1 . Thus, if a MEMS-scale element which has a length, say, L1 = 0.5 mm is compared to that of a machinery element where, say, L2 = 0.5 m, we should expect the MEMS-scale element to have a natural frequency 1000 times greater than that of the machine element.

3.3.4 Numerical Results The characteristic equation given by Eq. (3.110) has been evaluated for the eight sets of boundary conditions given in Table 3.3 for selected values of mi , Ki , ηs , and ηm and for selected values KL , KR , KtL , KtR , and mR . The values of 1 /π for Cases 1 to 5 are presented in Table 3.5 and those for Cases 6, 7, and 8, respectively, are presented in Tables 3.6, 3.7, and 3.8. In Table 3.5, we have also shown the effects of placing a very stiff spring (Ki → ∞) at the interior locations ηs = 0.5 and ηs = 0.7 when mi = 0. In examining the three cases of symmetric boundary conditions; that is, cases 1, 2, and 4, we note the following. For Cases 1 and 2, when Ki → ∞ we see that the first natural frequency coefficient is equal to that of the second natural frequency coefficient of the case where mi = Ki = 0. For Case 4, it is seen that when Ki → ∞ the first natural frequency coefficient is equal to twice the first natural frequency coefficient of a clamped free beam. In other words, we have created two cantilever beams, each half the length. The results presented in Table 3.6 clearly show the effects of the rigid body mode of a beam that has both of its ends free. √ The natural frequency of the beam as a single degree-of-freedom system is ωn = (kL +√kR + ki )/(mb + Mi ), which in non dimensional form can be expressed as 1 = 4 (KL + KR + Ki )/(1 + mi ). This approximation is valid for relatively small values of KL , KR , and Ki , where the effects of the stiffness of the springs on the bending of the beam is small. To verify this, consider the case where KL = KR = 10, Ki = 0 and mi = 0.5. Substituting

Hinged

Clamped

Clamped

Free

Clamped

1

2

3

4

5

Hinged

Free

Free

Clamped

Hinged

η=1

1.4228 1.2246 1.0943 0.77828 0.59014 0.56609 0.54124 0.43638 1.4586 1.3599 1.3076 1.2263 1.1917 1.0427 0.93875 0.67434

0.1 0.5 1 5 0.1 0.5 1 5 0.1 0.5 1 5 0.1 0.5 1 5

1 = 1.5056 2 = 2.4998 3 = 3.5 n = n + 0.5 (n ≥ 4) 1 = 0.5969 2 = 1.4942 3 = 2.5002 n = n − 0.5 (n ≥ 4) 1 = 1.5056 2 = 2.4998 3 = 3.5 n = n + 0.5 (n ≥ 4) 1 = 1.2499 2 = 2.25 3 = 3.25 n = n + 0.25 (n ≥ 4)

0.95534 0.84012 0.75859 0.54744

ηm = 0.5

0.1 0.5 1 5

mi

Ki = 0a

n = n (n ≥ 1)

n = n /π (Ki = 0) (mi = 0)

1.1969 1.0548 0.95218 0.68591

1.4958 1.4727 1.4591 1.4365

0.57765 0.52258 0.47921 0.35427

1.4621 1.3234 1.208 0.88059

0.96939 0.87832 0.80493 0.59299

ηm = 0.7 10 50 100 500 ∞ 10 50 100 500 ∞ 10 50 100 500 ∞ 10 50 100 500 ∞ 10 50 100 500 ∞

Ki 1.0477 1.1908 1.3151 1.7687 2.0000 1.5242 1.5919 1.6649 2.0298 2.4998 0.64418 0.75148 0.81774 0.94228 1.00000 0.56163 0.81384 0.93155 1.1232 1.1937 1.2764 1.3671 1.4571 1.8286 2.1602

ηs = 0.5

mi = 0a

1.0316 1.1296 1.2143 1.4645 1.6335 1.5145 1.5473 1.583 1.7519 2.0058 0.71969 0.94359 1.0785 1.3289 1.3922 0.62254 0.91965 1.0754 1.3635 1.427 1.2735 1.3542 1.433 1.7274 1.9753

ηs = 0.7

10 50 100 500

10 50 100 500

10 50 100 500

10 50 100 500

10 50 100 500

Ki

1.2173 1.3053 1.3931 1.7704

0.54914 0.79962 0.91965 1.121

0.63752 0.74586 0.81351 0.9415

1.4406 1.5055 1.5758 1.9333

1.0011 1.139 1.2593 1.7086

mi = 0.1

ηs = ηm = 0.5

1.0655 1.1442 1.2237 1.5841

0.50968 0.75089 0.87511 1.1116

0.61325 0.72425 0.79652 0.93826

1.2401 1.297 1.3588 1.6804

0.88066 1.0033 1.1112 1.529

mi = 0.5

0.95938 1.0307 1.1029 1.4352

0.47504 0.70438 0.82786 1.0975

0.58761 0.69952 0.77563 0.93383

1.1082 1.1593 1.2148 1.5053

0.79528 0.90642 1.0044 1.3884

mi = 1

0.68921 0.7407 0.79287 1.0351

0.36159 0.54021 0.64165 0.94627

0.47569 0.57579 0.652 0.88285

0.78822 0.82463 0.86427 1.0722

0.57396 0.65437 0.72539 1.006

mi = 5

a

3

Since the boundary conditions for Cases 1, 2, and 4 are symmetrical, the natural frequency coefficients at ηs = ηm = 0.7 will equal those at ηs = ηm = 0.3. This is not true for Cases 3 and 5.

η=0

Case

Boundary conditions

1 /π

Table 3.5 Values of the natural frequency coefficient 1 /π for a beam with and without in-span attachments and with boundary conditions corresponding to cases 1 to 5 of Table 3.3

122 Thin Beams: Part I

mi = 0 ηs = 0.3

Ki = 0 ηm = 0.3

10 100 1000

Ki = 0 mi = 0

10 100 1000

KL

Ki = 10

10 100 1000

KL

mi = 0.1

KR

Attachments

0.72065 0.7968 0.80141

KL = 10

0.63029 0.68042 0.68449

KL = 10

0.64691 0.70516 0.70976

KL = 10

0.73701 0.95433 0.98819

KL = 100

0.69634 0.88855 0.91602

KL = 100

0.70516 0.9157 0.9478

KL = 100

0.73845 0.97871 1.0227

KL = 1000

0.70184 0.92298 0.96003

KL = 1000

0.70976 0.9478 0.99028

KL = 1000

0.57755 0.61122 0.61421

KL = 10

10 100 1000

KR 0.8121 1.1309 1.1473

KL = 10

Ki = 100

10 100 1000

KR

mi = 0.5

0.84452 1.1416 1.2018

KL = 100

0.66364 0.80659 0.82449

KL = 100

KL = 1000

0.85122 1.1423 1.2068

KL = 1000

0.67242 0.84499 0.86992

KL = 10 0.53256 0.55819 0.56059

10 100 1000

KR

0.82317 1.1972 1.2932

KL = 10

Ki = 1000

10 100 1000

KR

mi = 1.0

0.90783 1.3119 1.4683

KL = 100

0.62913 0.73978 0.753

KL = 100

0.938 1.3455 1.5157

KL = 1000

0.64092 0.77871 0.79723

KL = 1000

Table 3.6 Values for the lowest natural frequency coefficient 1 /π of a beam free at each end and with translation springs at each end as a function of the stiffness of the boundary translation springs and the properties of the in-span attachments

3.3 Natural Frequencies and Mode Shapes of Beams with Constant Cross Section. . . 123

mi = 0 ηs = 0.3

Ki = 0 ηm = 0.3

0.1 1 10

Ki = 0 mi = 0

1.0099 1.0466 1.1708

KtL = 0.1

0.1 1 10

KtR

Ki = 1

0.1 1 10

KtR

1.0131 1.0497 1.1734

KtL = 0.1

0.97898 1.0132 1.127

KtL = 0.1

mi = 0.1

KtR

Attachments

1.0494 1.0845 1.2057

KtL = 1.0

1.016 1.0488 1.1598

KtL = 1.0

1.0466 1.0819 1.2034

KtL = 1.0

1.1723 1.2049 1.3242

KtL = 10

1.1435 1.1739 1.283

KtL = 10

1.1708 1.2034 1.3228

KtL = 10

0.1 1 10

KtR

Ki = 10

0.1 1 10

KtR

mi = 0.5

1.0406 1.0756 1.1957

KtL = 0.1

0.88694 0.91486 1.0038

KtL = 0.1

1.0731 1.107 1.2255

KtL = 1.0

0.92308 0.9496 1.0355

KtL = 1.0

1.1858 1.218 1.3365

KtL = 10

1.0538 1.0779 1.1599

KtL = 10

0.1 1 10

KtR

Ki = 100

0.1 1 10

KtR

mi = 1.0

1.2199 1.2491 1.3568

KtL = 0.1

0.81274 0.83656 0.91092

KtL = 0.1

1.2339 1.2633 1.3724

KtL = 1

0.84702 0.86946 0.9406

KtL = 1.0

1.2883 1.3186 1.4346

KtL = 10

0.97414 0.99389 1.0593

KtL = 10

Table 3.7 Values for the lowest natural frequency coefficient 1 /π of a beam hinged at each end and with torsion springs at each end as a function of the stiffness of the boundary torsion springs and the properties of the in-span attachments

124 3 Thin Beams: Part I

3.3

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

125

Table 3.8 Values for the lowest natural frequency coefficient 1 /π of a cantilever beam as a function of the boundary attachments and the the properties of the in-span attachments for jR = KtR = 0 Ki = mi = 0 KR = 0

mR = 0

1 /π

mR

1 /π

KR

1 /π

mR

KR = 1

KR = 10

KR = 100

0.1 0.5 1.0

0.54837 0.45199 0.39722

1 10 100

0.6398 0.84 1.1588

0.1 0.5 1.0

0.58853 0.48557 0.42681

0.78113 0.65041 0.57259

1.1421 1.0524 0.94992

Ki = 10 at ηs = 0.75 and mi = 0 KR = 0

mR = 0

1 /π

mR

1 /π

KR

1 /π

mR

KR = 1

KR = 10

KR = 100

0.1 0.5 1.0

0.68001 0.5571 0.48833

1 10 100

0.76646 0.90916 1.1874

0.1 0.5 1.0

0.70313 0.57659 0.50548

0.84346 0.69782 0.6126

1.1705 1.0741 0.96448

Ki = 100 at ηs = 0.75 and mi = 0 KR = 0

mR = 0

1 /π

mR

1 /π

KR

1 /π

mR

KR = 1

KR = 10

KR = 100

0.1 0.5 1.0

1.0292 0.80834 0.6992

1 10 100

1.1637 1.2263 1.3688

0.1 0.5 1.0

1.0383 0.81585 0.70573

1.1089 0.87552 0.75762

1.3534 1.1856 1.0325

mi = 0.1 at ηm = 0.75 and Ki = 0 KR = 0

mR = 0

1 /π

mR

1 /π

KR

1 /π

mR

KR = 1

KR = 10

KR = 100

0.1 0.5 1.0

0.53275 0.44592 0.39403

1 10 100

0.61463 0.80562 1.111

0.1 0.5 1.0

0.57164 0.47901 0.42336

0.75711 0.64108 0.56776

1.0972 1.0257 0.93727

mi = 1.0 at ηm = 0.75 and Ki = 0 KR = 0

mR = 0

1 /π

mR

1 /π

KR

1 /π

mR

KR = 1

KR = 10

KR = 100

0.1 0.5 1.0

0.44864 0.40438 0.36979

1 10 100

0.49697 0.64755 0.8882

0.1 0.5 1.0

0.48082 0.43403 0.39715

0.63058 0.57661 0.53048

0.88355 0.86262 0.83272

126

3

Thin Beams: Part I

these values into the previous expression, we obtain 1 /π = 0.6083, which compares reasonably well (∼5%) with the value of 0.5776 given in Table 3.6. For another example, we choose KL = KR = Ki = 10 and mi = 0.0 and find that 1 /π = 0.745, which compares reasonably well (∼3%) with the value of 0.7207 given in Table 3.6. When values of KL and KR become very large, the boundary conditions approach those of a beam hinged at each end. In this case, when Ki = mi = 0, 1 /π → 1. Detailed numerical results for two sets of boundary conditions of some practical interest are also presented. We shall use the results of these two cases as a basis of comparison when we remove, in Section 3.7, the assumption that the mass is a concentrated mass. The first set of boundary conditions is that of a cantilever beam with a mass and spring at its free end, which is Case 8 of Table 3.3. This configuration has application to atomic force microscopy, where a probe comes in contact with a sample. The probe is typically a vibrating cantilever beam with a small protuberance attached at its free end. The sample is often a deformable structure. The probe can be modeled as a cantilever beam with a mass attached to its free end and the sample can be modeled as an elastic spring (Melcher et al. 2007; Rabe et al. 1996). While the probe is in contact with the sample, the system can be considered that described by Case 8 of Table 3.3 with B2n = 0 and with mi = Ki = 0. Therefore, the natural frequency coefficients are determined from D(8) (n ) = D(3) (n ) +



 KR (5) − m  (n ) = 0 R n D 3n

(3.134a)

or  R (n ) T (n ) − Q2 (n ) +

 KR − m  R n (Q (n ) T (n ) − R (n ) S (n )) = 0 3n (3.134b)

where we have used the relations given in Table 3.3. The results obtained from determining 1 /π from Eq. (3.134b) for a range of values of mR and KR are shown in Fig. 3.4. We see from Eq. (3.134a) that when KR → 0 and mR → 0, D(8) (n ) → D(3) (n ), which is the solution for a cantilever beam with no attachments at its free end. On the other hand, when KR → ∞, D(8) (n ) → D(5) (n ), which is the solution for a beam clamped at one end and hinged at the other. Because of the range of values selected for KR , these limiting cases are closely approximated in Fig. 3.4. The second set of boundary conditions considered is that of a beam clamped at both ends, which is Case 2 of Table 3.3. The numerical results are presented in Figs. 3.5 to 3.10 for a wide range of parameter combinations to show the trends of the natural frequency coefficients. In Fig. 3.5, the effects on the natural frequency coefficient of the location and the magnitude of the stiffness of an in-span spring are shown. It is seen that the lowest natural frequency coefficient reaches a peak of 1 /π ∼ = 2.5 when the Ki > 2500 and ηs = 0.5. From Case 2 in Table 3.5, it is seen that a clamped-clamped beam without attachments has a second natural frequency of 1 /π = 2.4998. From Case 2 of Table 3.4 it is seen that at this second natural

3.3

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

m

127

Ω1/π = 1.25

R

KR

Ω1/π = 1.25

Ω1/π

1

0.5

0 1

Ω1/π = 0.234 0 log10(mR)

4

Ω1/π = 0.592

2 0

−1

log10(KR)

−2 −2

Fig. 3.4 Lowest natural frequency coefficient for a cantilever beam having a mass and translation spring at its free end

ηs

Ω1/π

2.5

Ki

2

1.5 4 3 log10(Ki)

2

1

0

0.2

0.4 ηs

0.6

0.8

1

Fig. 3.5 First natural frequency coefficient of a beam clamped at both ends and restrained by an in-span spring attached at ηs

128

3

Thin Beams: Part I

2 2.

log10(Ki)

2 1.9

1.

2.5

8

1.6

1.8 1.7

1.6

2.3

2.1

1. 8 1. 7

2

1.7

1.9

1.6

3

2.4

2.1

1.8

3.5

2.4 2.3 2.2 2.1 2 1.9

4

6 1.

1.7

2 1.6

1.5

1

0

0.2

0.4

0.6

0.8

1

ηs

Fig. 3.6 Contours of constant value of the lowest natural frequency coefficient for the surface in Fig. 3.5 mi η

1.6

m

1.4

Ω1/π

1.2 1 0.8

−1 −0.5 0 0.5 log10(mi)

1

1

0.6

0.8

0.4

0.2

0

η

m

Fig. 3.7 First natural frequency coefficient of a beam clamped at both ends and carrying a mass at ηm

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

0.7

0.8

1

log10(mi)

0.2

−0.2

1.3

1.4

1.5

0

1

1.

1. 2

1.1

1.5

1

0.9

1.2

0.4

8

0.

1.4

0.6

9 0.

1.1 1 .3 1.4

1.5

0.8

129

0.9 1 1 1.2 .1 1.3 1.4 1.5

1

1.3

3.3

1.2

−0.4 1.3 1.4

1.5

−0.8 −1

1.5

1.

4

−0.6

0

0.2

0.4

ηm

0.6

0.8

1

Fig. 3.8 Contours of constant value of the lowest natural frequency coefficient for the surface in Fig. 3.7 mi Ki

0.5

2.5 Ω1/π = 2.182 2 Ω1/π

Ω1/π = 1.042 1.5

1

0.5 3

Ω1/π = 1.441

Ω1/π = 0.669

2.5 0

1.5 log10(Ki)

1

0.5

2 −0.5 1

−1

log10(mi)

Fig. 3.9 Lowest natural frequency coefficient of a beam clamped at both ends as a function of mi and Ki for ηs = ηm = 0.5

130

3

Thin Beams: Part I

mi 0.3

Ki

2 Ω1/π = 1.827 Ω1/π = 1.066 Ω1/π

1.5

1 Ω1/π = 1.471 0.5 3

Ω1/π = 0.755 1

2.5 0.5

2

0

1.5 log10(Ki)

−0.5 1

−1

log10(mi)

Fig. 3.10 Lowest natural frequency coefficient of a beam clamped at both ends as a function of mi and Ki for ηs = ηm = 0.3

frequency a node point is at η = 0.5. Therefore, by placing a stiff spring at or near the node point of this second natural frequency, one effectively creates a system whose lowest natural frequency is now equal to that of the second natural frequency of a beam without attachments. This happens because the beam is forced to assume the mode shape corresponding to the second natural frequency of the beam without attachments. From the contour curves shown in Fig. 3.6, it is seen that certain natural frequencies can be obtained from a range of combinations of the values of Ki and ηs . It is also noted that as the spring is placed closer to the center of the beam, the magnitude of Ki that is required to maintain a constant natural frequency decreases. In Fig. 3.7, the effects on the natural frequency coefficient of the location and magnitude of an in-span mass are shown. It is seen that the effect of the mass is to lower the first natural frequency, which reaches its minimum value at ηm = 0.5 for all values of mi . From the contour curves shown in Fig. 3.8, it is seen that as with the beam restrained with an in-span spring, certain natural frequencies can be obtained from a range of combinations of the values of mi and ηm . In Figs. 3.9 and 3.10, respectively, the effects on the natural frequency coefficient as a function of the magnitude of an in-span mass and in-span spring each placed at ηs = ηm = 0.5 and at ηs = ηm = 0.3 are shown. It is seen that as the value of mi increases the natural frequency coefficient decreases and that as Ki increases, the natural frequency coefficient increases in a nonlinear manner.

3.3

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

131

To show the effects that the location of a rigid in-span support has on the lowest natural frequency coefficient, five sets of boundary conditions are examined. The rigid in-span support model is obtained by letting Ki → ∞ and the natural frequency coefficients for this case are determined from Eq. (3.113). The results are shown in Fig. 3.11. Also shown in the figure are the maximum values of the natural frequencies, denoted max , and the locations of the rigid support, denoted ηmax , at which these maxima occur. Referring to the location of the node points given in Table 3.4 to the values of ηmax listed in Fig. 3.11, it is seen for the cases of the hinged-hinged, clamped-clamped, clamped-free, and clamped-hinged beams that the locations of the rigid support at which the maximum values of 1 occur coincide with the locations of the node points of the second natural frequency. For the free-free beam, the maximum natural frequency occurs when the rigid support is at the location of the node point of the lowest natural frequency. It is noted that the case of a cantilever beam with an intermediate rigid support is the model of a one type of beam with an overhang. In Fig. 3.12, the effects on the lowest natural frequency coefficient of the location of two rigid in-span supports are shown for a beam clamped at each end and in Fig. 3.13 they are shown for a cantilever beam. It is seen in Fig. 3.12 that the maximum frequency occurs when the two supports are located in the vicinity of the node points of the third natural frequency of a beam without the rigid supports. From Table 3.4, the interior locations of these node points are 0.358 and 0.462; thus, when either η1 = 0.358 and η2 = 0.462 or η1 = 0.462 and η2 = 0.358, the maximum 2.5 c-c c-h 2 h-h

Ω1/π

1.5 f-f 1

BC

Ωmax/π

ηmax

0.5

h-h c-c c-f f-f c-h

2 2.5 1.494 1.506 2.25

0.5 0.5 0.783 0.224, 0.776 0.557

0

c-f

0

0.2

0.4

0.6

0.8

1

ηs

Fig. 3.11 Lowest natural frequency coefficients for a beam with an in-span rigid support as a function of its location for five sets of boundary conditions: c = clamped; f = free; and h = hinged

132

3 η1

Ω1

Fig. 3.12 Lowest natural frequency coefficient as a function of the location of two rigid supports for a beam clamped at each end

Thin Beams: Part I

η2

14 12 10 8 6 4 2 0.8

0.8 0.6 η1

η1

Fig. 3.13 Lowest natural frequency coefficient as a function of the location of two rigid supports for a cantilever beam

0.6 0.4

0.4 0.2

η2

0.2

η2

Ω1

8 6 4 2

0.8

0.8 0.6

0.6

η1 0.4

0.4 0.2

0.2

η2

value of the natural ferequency occurs. When one of the rigid supports is placed at either end of the beam, the resulting curve is the same as the curve identified as ‘c-c’ in Fig. 3.11. Turning our attention to the cantilever beam, it is seen in Fig. 3.13 that the maximum value of the frequency coefficient occurs in the vicinity of the node points of the third natural frequency of the beam without the rigid supports. From Table 3.4, the interior locations of these nodes points are 0.504 and 0.869;

3.3

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

133

thus, when either η1 = 0.504 and η2 = 0.869 or η1 = 0.869 and η2 = 0.504, the maximum value of the natural frequency occurs. When η2 = 0, the resulting curve is the same as that identified as ‘c-f’ in Fig. 3.11.

3.3.5 Cantilever Beam as a Biosensor Cantilever beams are used in many different ways as biosensors (Ziegler 2004). One such application is to use a material for the cantilever beam that absorbs a certain molecule. This absorption changes the mass of the beam so that there is a shift in the beam’s natural frequency. The sensitivity of this type of device is a function of the smallest frequency change that can be detected. To show how such a cantilever beam sensor works, we assume that the beam is vibrating at its lowest resonance 1 = 0.5969π . Then, from Eq. (3.131) 21 f1 = 2π L2

 EIo Hz ρAo

(3.135)

where f1 is the lowest natural frequency. If it can be assumed that the molecules are being absorbed uniformly along the length of the beam and the mass of these molecules per unit length is mm , then Eq. (3.135) becomes 21 fm = 2π L2

 EIo Hz. ρAo + mm

(3.136)

Therefore, the change (decrease) in the natural frequency is  21 EIo EIo − ρAo 2π L2 ρAo + mm ⎞ 1 ⎠ Hz. = f1 ⎝1 −  1 + mm ρAo

21 f = f1 − fm = 2π L2 ⎛



(3.137)

If mm /ρAo << 1, then Eq. (3.137) can be approximated by f ≈

f1 mm Hz 2ρAo

and, therefore,  mm ≈ 2ρAo

f f1

 kg/m.

(3.138)

Assuming that f1 can be measured accurately, we see that the overall accuracy of such a technique relies on the accuracy of the knowledge of ρ and Ao and the

134

3

Thin Beams: Part I

correctness of the assumption that the molecules are uniformly distributed over the surface of the beam.

3.4 Single Degree-of-Freedom Approximation of Beams with a Concentrated Mass In modeling a system as a single degree-of-freedom system, it is standard practice to ignore the mass of the spring. We shall show how one can incorporate the mass of the spring into the model by comparing the natural frequencies obtained from the continuum model given by Eq. (3.112) to that of a single degree-of-freedom spring-mass model. We shall consider the model shown in Fig. 3.14. Although the beam can have any boundary conditions, we shall limit our examples to three cases: clamped at each end, hinged at each end, and a clamped at one end and free at the other end (cantilever). For the cantilever, the mass can be located at the free end. To determine the equivalent mass Meq of a single degree-of-freedom of the system shown in Fig. 3.14, which is a system of two springs in parallel, the equivalent stiffness of the system is (Balachandran and Magrab 2009, p. 37) keq = ki + kbeam .

(3.139)

In Eq. (3.139), the stiffness kbeam of a uniform beam with a point load applied at ηm is of the form (Balachandran and Magrab 2009, p. 607) kbeam =

αEIo L3

(3.140)

where α is a constant that is a function of the boundary conditions and the position of the load, ηm . Expressions for α are listed in Table 3.9 for the three sets of boundary conditions that will be considered. It is noted that α is essentially the non dimensional spring constant for the beam for a given set of boundary conditions and for a given position of the mass, η m . The natural frequency for a single degree of freedom system formed by the parallel springs with constants kbeam and ki and the attached mass Mi is given by

η = ηm = ηs

Beam Mi

BC at η=0

ki

Meq BC at η=1

⇒ keq

Fig. 3.14 Equivalent single degree-of-freedom model for a beam with an attached spring and concentrated mass at the same in-span location

3.4

Single Degree-of-Freedom Approximation of Beams with a Concentrated Mass

135

Table 3.9 Constant c2 used in the determination of the equivalent mass of a single degree-offreedom system as a function of location, boundary conditions, and attached spring; the constant c1 ≈ 1 in all cases c2 in Eq. (3.147) Boundary conditions

ηm (= ηs )

α

Clamped/clamped

3 3 (1 − η )3 ηm m

Hinged/hinged

3 2 (1 − η )2 ηm m

Clamped/free

3 3 ηm

Ki = α/10

Ki = α

Ki = 5 α

0.5 (α = 192) 0.374 0.4 (α = 217) 0.409 0.3 (α = 323.9) 0.555

0.375 0.410 0.559

0.377 0.418 0.598

0.390 0.467 0.854

0.5 (α = 48) 0.4 (α = 52.1) 0.3 (α = 68)

0.488 0.523 0.659

0.488 0.523 0.661

0.490 0.528 0.678

0.498 0.550 0.774

1.0 (α = 3) 0.75 (α = 7.1) 0.5 (α = 24)

0.237 0.573 1.839

0.237 0.573 1.849

0.238 0.574 1.941

0.244 0.577 2.411

 ωsdof =

Ki = 0

 keq = Mi

ki + kbeam rad/s. Mi

(3.141)

The lowest natural frequency coefficient of a beam carrying a mass Mi is 1 , where 1 is a solution to Eq. (3.112) for a specific set of boundary conditions. Then, from Eq. (3.135), we have ωbeam

2 = 21 L



 EIo EIo 2 = 1 rad/s ρAo mb L3

(3.142)

where mb is the mass of the beam and is given by Eq. (3.53). We are interested in determining under what conditions ωsdof = ωbeam . Thus, equating Eqs. (3.141) and (3.142) and using Eq. (3.140), it is found that mi =

α 41

  Ki 1+ . α

(3.143)

where Ki is given by Eq. (3.53) and mi = Mi /mb . Equation (3.143) indicates that in order for these two natural frequencies to be equal, mi is not a constant but a function of the spring constant and the location of the mass. Thus, we seek an empirical relation such that Mi → Meq , where Meq is an equivalent mass. To find this equivalent mass, we examine Eq. (3.143) by first considering a cantilever beam with a mass at its free end (ηm = 1) and with Ki = 0. A numerical evaluation of Eq. (3.114) for this set of boundary conditions (Case 3 of Table 3.3) was conducted to determine the values of 1 for a wide range of values for mi . The results are shown in Fig. 3.15, where we have plotted it in the form given by Eq. (3.143). The value of α = 3 for this case is obtained from Table 3.9. It is seen

136

3

Thin Beams: Part I

120

100

α/Ω1

4

80

60

40

20

0

0

20

40

60

80

100

mi

Fig. 3.15 Plot of Eq. (3.143) with 1 determined from Eq. (3.114) and shown as squares, and a straight line fit, shown as a solid line

that the values of α 41 versus mi is a straight line. Thus, it reasonable to assume that the equivalent mass Meq of the single degree-of-freedom model can be expressed as Meq = c1 Mi + c2 mb .

(3.144)

Then, Eq. (3.143) with Ki = 0 becomes c1 mi + c2 =

α . 41

(3.145)

A numerical curve-fit to Eq. (3.145) for values of 0.1 ≤ mi ≤ 100 gives c1 = 1 and c2 = 0.237 and is shown as a solid line in Fig. 3.15. Then, the corrected version of Eq. (3.141) becomes  ωsdof, corr =

keq = ccorr ωsdof rad/s Meq

(3.146)

where the correction factor is  −1/2 ccorr = c1 + c2 mi .

(3.147)

The error using Eq. (3.146) instead of using Eq. (3.114) is typically less than 0.02% for mi > 0.5. When mi < 0.5 and Ki >> α, the error can be substantial:

3.5

Beams with In-Span Spring-Mass Systems

137

2% to 9%. In this case, it is better to use Eq. (3.114). For the case where ki = 0 and the mass is placed at the center for beams with clamped or hinged ends or placed at the free end of a cantilever beam, the results given in Table 3.9 agree with (Blevins 1979, p. 158–159). The other cases are not given in this reference. It is seen from Eq. (3.147) that as the ratio mi increases, the correction ccorr approaches unity. To see this, we refer to Table 3.9 and consider the example of a beam clamped at both ends. When mi = 0.1, ccorr = 0.544, when mi = 1, ccorr = 0.899, and when mi = 5, ccorr = 0.977. When this process is repeated for the three sets of boundary conditions and for several values of ηm and Ki /α one obtains the results given in Table 3.9. It has been found that in all cases c1 for all practical purposes equals unity. It is seen from the table that the value of c2 varies greatly as a function of the boundary conditions, the location of the mass ηm , and the ratio of the stiffness of the beam to the stiffness of the attached spring.

3.5 Beams with In-Span Spring-Mass Systems 3.5.1 Single Degree-of-Freedom System There are instances where an elastically mounted inertia element is attached to an elastic support beam. We shall examine such a system with the goal of determining if such a coupled system can be approximated by a two degree-of-freedom system that was described in Section 2.7. We start with Eq. (3.101), which is 1  Gi Y (ηi ) Hi (, η, ηi ). D () 3

Y (η) =

(3.148)

i=1

Equation (3.148) is valid for the general boundary conditions given by Eqs. (3.93) and (3.94). To determine the characteristic equation, it is noted that Eq. (3.148) must be satisfied at η = ηs = η1 , η = ηm = η2 , and η = ηo = η3 . Therefore, setting η to each of these values in Eq. (3.148), it is found that 1  Gi Y (ηi ) Hi (, η1 , ηi ) D () 3

Y (η1 ) =

i=1

1  Gi Y (ηi ) Hi (, η2 , ηi ) D () 3

Y (η2 ) =

i=1

1  Gi Y (ηi ) Hi (, η3 , ηi ) D () 3

Y (η3 ) =

i=1

(3.149)

138

3

Thin Beams: Part I

which can be written in matrix form as ⎫ ⎤⎧ ⎡ g11 g12 g13 ⎨ Y (ηs ) ⎬ [G] {Y} = ⎣ g21 g22 g23 ⎦ Y (ηm ) = 0 ⎩ ⎭ g31 g32 g33 Y (ηo )

(3.150)

where   gij = Gj Hj , ηi , ηj − δij D () .

(3.151)

and δ ij is the kronecker delta. We are interested in the particular case where ηs = ηm = ηo , which is shown Fig. 3.16. It is seen from Eq. (3.103) that when ηs = ηm = ηo , h11 () = h12 () = h13 () and h21 () = h22 () = h23 () and, therefore, from Eq. (3.102) it is found that H1 (, η, ηs ) = H2 (, η, ηm ) = H3 (, η, ηo ). Furthermore, since T (0) = 0 when η = ηs = ηm = ηo , Eq. (3.102) yields Hj (, ηs , ηs ) = h11 () f1 (ηs ) + h21 () f2 (ηs ) ηs = ηm = ηo ; j = 1, 2, 3.

(3.152)

Thus, the elements of [G] become   gij = Gj H1 , ηj , ηj − δij D ()

η1 = η2 = η3 .

(3.153)

The natural frequencies are those values of  = n for which det [G] = 0, where the elements of [G] are given by Eq. (3.153); that is, those values of n that are solutions to   Mo 4n 4  H1 (n , ηs , ηs ) = 0 D (n ) + mi n − Ki +  1 − 4n 4o (3.154) ηs = ηm = ηo . In arriving at Eq. (3.154), we have used Eq. (3.73). Using Eq. (3.108), Eq. (3.154) can be written as

mo

BC at x=0

Fig. 3.16 Beam with a concentrated mass, a spring, and a single degree-of-freedom system attached at the same in-span location for arbitrary boundary conditions

ko Mi ki

Lm = Ls = Lo L

BC at x=L

3.5

Beams with In-Span Spring-Mass Systems

139



(p)

D

 Ki Mo n  Hn(p) (ηs , ηs ) = 0 (n ) + mi n − 3 +  n 1 − 4n 4o

(3.155)

ηs = ηm = ηo (p)

where D(p) (n ) and Hn (ηs , ηs ) are given in Table 3.3 corresponding to the set of boundary conditions indicated by the superscript p. When Mo = 0, Eq. (3.155) reduces to Eq. (3.112). When 4n << 4o , Eq. (3.155) can be written as (p)

D

  Ki (n ) + (mi + Mo ) n − 3 Hn(p) (ηs , ηs ) = 0 n

4n << 4o

(3.156)

which is equivalent to replacing mi with mi + Mo in Eq. (3.112). The natural frequency coefficient associated with this equivalent mass is denoted  n . We shall use this special case when the numerical results are discussed. The corresponding mode shapes are determined from Eqs. (3.148) and (3.150) in the following manner. From Eq. (3.150) with  = n , it is determined that g23 g31 − g21 g33 Y (ηm ) = =1 Y (ηs ) g22 g33 − g23 g32 g21 g32 − g22 g31 Y (ηo ) = =1 Y (ηs ) g22 g33 − g23 g32

(3.157)

where we have used Eq. (3.154). Then, arbitrarily setting Y (ηs ) = 1 and using Eq. (3.157), we obtain the following mode shape from Eq. (3.148) Yn (η) = Cn H1 (n , η, ηs ) where

(3.158)

  1 Mo 4n 4  . Cn = Ki − mi n −  D (n ) 1 − 4n 4o

Again using Eq. (3.108) and, for convenience, setting Cn = 1, the mode shape given by Eq. (3.158) can be expressed as (p)

Yn(p) (η) =

Hn (η, ηs ) − T (n [η − ηs ]) u (η − ηs ) . D(p) (n )

(3.159)

It is seen that Eq. (3.159) is identical to Eq. (3.122). However, there is one important distinction: as discussed at the end of Section 3.2.1, the mode shape given by Eq. (3.159) is not an orthogonal function. From Eq. (3.67), the relative amplitude of the mass of the single degree-of-freedom system at  = n is

140

3

Thin Beams: Part I

 −1 4n = 1− 4 . (p) o Yn (ηo ) Zn,o

(3.160)

To compare the value of n obtained from Eq. (3.155) for the lowest natural frequency coefficient with that obtained for a two degree-of-freedom spring-mass model given by Eq. (2.208), we convert Eq. (2.208) to the notation of this section as follows. When k32 = 0, Eq. (2.208) becomes      ˆ 21,2 = 1 1 + ωr2 (1 + mr ) ∓ 1 + ωr2 (1 + mr ) 2 − 4ωr2  2

(3.161)

where ˆ 21,2 = 

2 ω1,2 2 ωn1

,

ωr2 =

2 ωn2 2 ωn1

,

mr =

mo . Mi

(3.162)

However, from Eqs. (3.53), we obtain 2 = ωn2 2 ωn1

ko 4 = 2o , mo to

mr =

Mo mi

keq ki + kbeam Ki + α 4 = = = 2 = 2s Mi Mi to mi to

(3.163)

where we have used Eqs. (3.139) and (3.140) and the expressions for α are given in Table 3.9. Substituting Eq. (3.163) into Eq. (3.162), we obtain ˆ 21,2 = 

2 t2 ω1,2 o

4s

=

˜4  1,2 4s

,

ωr2 =

4o 4s

(3.164)

and Eq. (3.161) becomes ˜ 21,2 

     2 1 4 4 4 4 4 4 s + o (1 + mr ) ∓ s + o (1 + mr ) − 4s o . (3.165) = 2

We shall consider the cases of a beam clamped at each end and a cantilever beam. Representative natural frequency coefficients and mode shapes for each of these systems have been obtained from Eqs. (3.155), (3.156), (3.159), and (3.160) and are plotted in Figs. 3.17 and 3.18. In Fig. 3.17, the beam is clamped at both ends and in Fig. 3.18 the beam is a cantilever. It is seen from these figures that the attached single degree-of-freedom system affects the natural frequencies and modes shapes of the beam in a complicated manner. From the mode shapes and numerical values presented in Figs. 3.17 and 3.18, the following observations are noted. To be able to identify which natural frequency coefficient is associated with a beam mode shape and which is associated with the

3.5

Beams with In-Span Spring-Mass Systems

141

Ω1/π = 1.165 Ω′1/π = 1.402 Zo,1/Y1(ηo) = 9.75

Ω2/π = 1.489 Ω′1/π = 1.402 Zo,2/Y2(ηo) = −0.718

Ω3/π = 2.468 Ω′2/π = 2.461 Zo,3/Y3(ηo) = −0.0586

Ω4/π = 3.311 Ω′3/π = 3.283 Zo,4/Y4(ηo) = −0.0174

(a) Ω1/π = 1.393 Ω′1/π = 1.402 Zo,1/Y1(ηo) = 1.22

Ω2/π = 2.201 Ω′2/π = 2.461 Zo,2/Y2(ηo) = −7.01

Ω3/π = 2.478 Ω′2/π = 2.461 Zo,3/Y3(ηo) = −1.2

Ω4/π = 3.317 Ω′3/π = 3.283 Zo,4/Y4(ηo) = −0.204

(b)

Fig. 3.17 Lowest four natural frequency coefficients n and corresponding modes shapes for a beam clamped at both ends for mi = 0.3, Ki = 100, ηs = ηm = ηo = 0.45, Mo = 0.1, and ˜ 1 /π = 1.143) and (b) Ko = 200 (o /π = 2.129,  ˜ 1 /π = (a) Ko = 20 (o /π = 1.197,  1.380). The dashed line is the mode shape for a beam without the single degree-of-freedom system but with the equivalent mass; that is, corresponding to  n , and the solid line is the mode shape for the beam with the single degree-of-freedom system

142

3

Thin Beams: Part I

Ω1/π = 0.8331 Ω′1/π = 1.076 Zo,1/Y1(ηo) = 16.3

Ω2/π = 1.112 Ω′1/π = 1.076 Zo,2/Y2(ηo) = −0.507

Ω3/π = 1.439 Ω′2/π = 1.388 Zo,3/Y3(ηo) = −0.136

Ω4/π = 2.315 Ω′3/π = 2.301 Zo,4/Y4(ηo) = −0.0182

(a)

Ω1/π = 1.06 Ω′1/π = 1.076 Zo,1/Y1(ηo) = 1.69

Ω2/π = 1.288 Ω′2/π = 1.388 Zo,2/Y2(ηo) = 9.34

Ω3/π = 1.527 Ω′2/π = 1.388 Zo,3/Y3(ηo) = −1.31

Ω4/π = 2.317 Ω′3/π = 2.301 Zo,4/Y4(ηo) = −0.12

(b)

Fig. 3.18 Lowest four natural frequency coefficients n and corresponding modes shapes for a cantilever beam for mi = 0.3, Ki = 100, ηs = ηm = ηo = 1.0, Mo = 0.1, and (a) Ko = 5 ˜ 1 /π = 0.835) and (b) Ko = 30 (o /π = 1.325,  ˜ 1 /π = 1.173). The (o /π = 0.846,  dashed line is the mode shape for a beam without the single degree-of-freedom system but with the equivalent mass; that is, corresponding to  n , and the solid line is the mode shape for the beam with the single degree-of-freedom system

3.5

Beams with In-Span Spring-Mass Systems

143

beam and the mass of the single degree-of-freedom system, the relative magnitudes of three quantities are needed:  n , o , and Zo,n , where  n , is determined from Eq. (3.156). For the results of a beam clamped at both ends given in Fig. 3.17, it is seen that when Ko = 20, 0 /π = 1.197 <  1 /π = 1.402 and the beam mode shapes corresponding to 1 and 2 are the same. However, for the mode shape corresponding to 1 /π = 1.165, the magnitude of Zo,n /Yn (ηo ) is very large, indicating that the amplitude of the mass with respect to Yn (ηo ) is very large and, from its sign, in phase with respect to it. The second lowest natural frequency coefficient of the system is 2 /π = 1.489, which corresponds fairly closely to the frequency coefficient of the beam and the mass of the single degree-of-freedom system as if this mass was attached directly to the beam. Furthermore, it is seen that 3 /π = 2.468 corresponds to the second mode shape of a beam without the single degree-of-freedom system but accounting for its mass; that is, the mode shape corresponding to  2 /π . On the other hand, when Ko = 200,  2 /π = 2.461 > o /π = 2.129 > 1 /π = 1.402 and the single degree-of-freedom system influences the second natural frequency of the beam. In this case, the second lowest natural frequency coefficient of the system is 2 /π = 2.201. Based on the magnitude of Zo,n /Yn (ηo ) and its sign, the amplitude of the mass with respect to Y (ηo ) is very large and out of phase with respect to it. The third lowest natural frequency coefficient of the system is 3 /π = 2.478, which corresponds fairly closely to the frequency coefficient of the beam and the mass of the single degree-of-freedom system as if this mass was attached directly to the beam. For this case, the mode shapes corresponding to these three natural frequency coefficients are different. From these two cases, it is seen that the single degree-of-freedom system does not substantially affect the ! ! natural frequencies and mode shapes of the beam except when !Zo,n /Yn (ηo )! is substantially greater than one. For all other natural frequencies, the beam behaves as a beam with an in-span spring and concentrated mass of magnitude mi + Mo . Similar conclusions are reached for the cantilever beam shown in Fig. 3.18. We now examine the validity of using Eq. (3.165) to predict the lowest natural frequency of the beam with a single degree-of-freedom system. From Fig. 3.17 and ˜ 1 /π = its legend, it is seen that for a beam clamped at both ends when Ko = 20,  ˜ 1.143 which compares well with 1 /π = 1.165. When Ko = 200, 1 /π = 1.380 which compares well with 1 /π = 1.393. It is seen that the numerical values are close to each other; however, there is no information regarding which mode shape ˜ 1 corresponds to the mass of the single degree-ofis involved. In the former case,  freedom system and in the latter case, the beam and the mass of the single degreeof-freedom systems as if it were attached directly to the beam. From Fig. 3.18 and its legend, it is seen that for a cantilever beam when Ko = 5, ˜ 1 /π = 0.835 which compares well with 1 /π = 0.833. When Ko = 30,  ˜ 1 /π =  1.173 which compares less well with 1 /π = 1.06. Again, these numerical values do not capture the corresponding mode. One can conclude from these observations that when a single degree-of-freedom system is attached to a beam, it is insufficient to just record the natural frequency coefficients. The values of  n , o , and Zo,n are also required, as is a display of the corresponding mode shapes.

144

3

Thin Beams: Part I

3.5.2 Two Degree-of-Freedom System with Translation and Rotation Consider a uniform beam of length L with the following boundary conditions. At the end x = 0 the beam is free and is restrained by a translation spring kL (N/m) and a torsion spring ktL (Nm). At the end x = L, the beam is also free and is restrained by a translation spring kR (N/m) and a torsion spring ktR (Nm). Attached to the interior of the beam is a two degree-of freedom system shown in the Fig. 3.19 that undergoes a translation z (t) and a rotation θ (t). For this system and boundary conditions, the characteristic equation and mode shapes will be determined. From a force balance and a moment balance on the mass shown in Fig. 3.20, the following equations of motion for the two degree-of-freedom ststem are obtained in terms of the parameters given in Eq. (3.53) as   d2 zˆ + 2K1 zˆ + (b1 − b2 ) K1 θ = K1 y (η1 , τ ) + y (η2 , τ ) 2 dτ    d2 θ 2 jo 2 + b1 + b22 K1 θ + (b1 − b2 ) K1 zˆ = K1 b1 y (η1 , τ ) − b2 y (η2 , τ ) dτ (3.166) mo

  where ηj = Lj /L, j = 1, 2, y ηj , τ is the displacement of the beam at ηj , and mo =

Mo , mb

jo =

Jo , mb L2

K1 =

k1 L 3 , EIo

bj =

aj L

j = 1, 2.

(3.167)

Since the spring forces from the two degree-of-freedom system act on the beam in the same direction as the beam’s inertia force, the equation of motion of the beam is

a1

ktL

k1

a2

θ(t) z(t)

Mo, Jo k1

Center of mass

ktR

E, A, I, ρ kL

Fig. 3.19 Nomenclature for a beam with a two degree-of-freedom system that rotates and translates

x w(x,t)

L1 L2 L

kR

3.5

Beams with In-Span Spring-Mass Systems

145 a1

Fig. 3.20 Forces on the mass of the two degree-of-freedom system given in Fig. 3.19

a2

Joθ Mo z k1(w(L1,t) −z−a1θ)

Center of mass k1(w(L2,t) −z+a2θ)

    ∂ 2y ∂ 4y + K θ δ − η θ δ − η = 0. y − z ˆ − b y − z ˆ + b + K + (η (η ) ) 1 1 1 1 2 2 ∂η4 ∂τ 2 (3.168) It is noted that η2 = η1 + b1 + b2 . The boundary conditions are those given by Eqs. (3.93) and (3.94) with mR = jR = 0. Thus, at η = 0 ∂ 3y = −KL y ∂η3

0 ≤ KL ≤ ∞

∂y ∂ 2y = KtL 2 ∂η ∂η

0 ≤ KtL ≤ ∞

(3.169)

and at η = 1 ∂ 3y = KR y ∂η3

0 ≤ KR ≤ ∞

∂y ∂ 2y = −KtR 2 ∂η ∂η

0 ≤ KtR ≤ ∞.

(3.170)

To determine the natural frequencies and mode shapes, we assume solutions of the form y (η, τ ) = Y (η) e j zˆ (τ ) = Zo e j θ (τ ) = o e

2τ

2τ

(3.171)

j2 τ

where 2 is given by Eq. (3.65). Upon substituting Eq. (3.171) into Eqs. (3.166) and (3.168), respectively, we arrive at a11 () Zo + a12 o = K1 [Y (η1 ) + Y (η2 )] a12 Zo + a22 () o = K1 [b1 Y (η1 ) − b2 Y (η2 )] and

(3.172)

146

3

Thin Beams: Part I

d4 Y + K1 (Y − Zo − b1 o ) δ (η − η1 ) dη4 + K1 (Y − Zo + b2 o ) δ (η − η2

) − 4 Y

(3.173) =0

where a11 () = 2K1 − mo 4 a12 = K1 (b1 − b2 )   a22 () = K1 b21 + b22 − jo 4 .

(3.174)

When b1 = b2 , it is seen from Eq. (3.174) that a12 = 0 and, therefore, from Eq. (3.172) it is found that the translation and rotation are independent of each other. Substituting Eq. (3.171) into the boundary conditions given by Eq. (3.169) and Eq. (3.170), it is found that at η = 0, Y (0) = −KL Y (0) Y (0) = KtL Y (0)

(3.175)

and at η = 1 Y (1) = KR Y (1) Y (1) = −KtR Y (1)

(3.176)

where the prime denotes the derivative with respect to η. Solving Eq. (3.172) for Zo and o , it is found that Zo = Z1 () Y (η1 ) + Z2 () Y (η2 ) o = 1 () Y (η1 ) − 2 () Y (η2 )

(3.177)

where K1 (a22 () − b1 a12 ) D2 () K1 Z2 () = (a22 () + b2 a12 ) D2 () K1 1 () = (b1 a11 () − a12 ) D2 () K1 2 () = (b2 a11 () + a12 ) D2 () Z1 () =

and

(3.178)

3.5

Beams with In-Span Spring-Mass Systems

147

D2 () = a11 () a22 () − a212 .

(3.179)

The values of  that satisfy D2 () = 0 are the natural frequency coefficients for the two degree-of-freedom system attached to a fix foundation; that is, with Y (η1 ) = Y (η2 ) = 0. Taking the Laplace transform of Eq. (3.173) with respect to η, using Eq. (C.12) of Appendix C with f = β = k = 0, and using pair 5 of Table C.1, we obtain Y¯ (s) =

s4

 1 Y (0) s3 + Y (0) s2 + Y (0) s + Y (0) 4 −

 −K1 (Y (η1 ) − Zo − b1 o ) e−η1 s − K1 (Y (η2 ) − Zo + b2 o ) e−η2 s . (3.180)

The inverse Laplace transform of Eq. (3.180) is given by Eq. (C.18) and pair 3 of Table C.1. Thus, Y (η) = Y (0) Q (η) + Y (0) R (η)/ + Y (0) S (η)/2 + Y (0) T (η)/3 −

2 

El T ( [η − ηl ]) u (η − η1 ) /3

(3.181)

l=1

where the Q (η), etc, are given by Eq. (C.19), E1 = K1 (Y (η1 ) − Zo − b1 o ) = e11 Y (η1 ) + e12 Y (η2 ) E2 = K1 (Y (η2 ) − Zo + b2 o ) = e21 Y (η1 ) + e22 Y (η2 )

(3.182)

and e11 = K1 [1 − Z1 () − b1 1 ()] e12 = K1 [b1 2 () − Z2 ()] e21 = K1 [b2 1 () − Z1 ()]

(3.183)

e22 = K1 [1 − Z2 () − b2 2 ()] . Employing the technique of Section 3.3.2, we use Eq. (3.175) to write Eq. (3.181) as Y (η) = Y (0) f1 (η) + Y (0) f2 (η) −

2 

El T ( [η − ηl ]) u (η − η1 ) /3

l=1

(3.184) where fl (η), l = 1, 2, is given by Eq. (3.96). The remaining two constants, Y (0) and Y (0), are determined by substituting Eq. (3.184) into the boundary conditions at η = 1, which are given by Eq. (3.176). Performing the substitutions, we arrive at the following system of equations

148

3



c11 () c21 ()



c12 () c22 ()

Y (0) Y (0)



=

Thin Beams: Part I



q1 () q2 ()

(3.185)

where cij () are given by Eq. (3.98) with A2 = KR and B2 = −KtR and qi () =

2 

El pi (, ηl )

i = 1, 2.

(3.186)

l=1

The quantity pl (, ηl ) is given by Eq. (3.100) with A2 = KR and B2 = −KtR . Solving Eq. (3.185) for Y (0) and Y (0) yields 1  El h1l () D () 2

Y (0) =

l=1

(3.187)

1  Y (0) = El h2l () D () 2

l=1

where hij () are given by Eq. (3.103) and D () is given by Eq. (3.104). Substituting Eq. (3.187) into Eq. (3.184) and using Eq. (3.182) results in 1  Bl (, η) Y (ηl ) Y (η) = D () 2

(3.188)

l=1

where B1 (, η) = e11 H1 (, η, η1 ) + e21 H2 (, η, η2 )

(3.189)

B2 (, η) = e12 H1 (, η, η1 ) + e22 H2 (, η, η2 )

and Hl (, η, ηl ) is given by Eq. (3.102). To determine the characteristic equation, it is noted that Eq. (3.188) must be satisfied at η = η1 and at η = η2 . Therefore, setting η to each of these values in Eq. (3.188), we obtain 1  Bl (, η1 ) Y (ηl ) D () 2

Y (η1 ) =

l=1

(3.190)

1  Y (η2 ) = Bl (, η2 ) Y (ηl ) D () 2

l=1

which in matrix form can be written as  B1 (, η1 ) − D () B2 (, η1 ) B1 (, η2 ) B2 (, η2 ) − D ()



Y (η1 ) Y (η2 )



=

0 0

 .

(3.191)

3.6

Effects of an Axial Force and an Elastic Foundation on the Natural Frequency

149

The natural frequencies are those values of  = n for which the determinant of the coefficients of Eq. (3.191) is zero; that is, those values of n that are solutions to [B1 (n , η1 ) − D (n )] [B2 (n , η2 ) − D (n )] − B2 (n , η1 ) B1 (n , η2 ) = 0. (3.192) To specialize Eq. (3.192) to the boundary conditions appearing in Table 3.3, one uses Eq. (3.108) in Eqs. (3.189) and Eq. (3.192) with p = 1, 2, . . . , 7. The mode shape is obtained from Eqs. (3.188) and (3.191) with  = n as Yn (η) =

1 {Cn B1 (n , η) + B2 (n , η)} D (n )

(3.193)

−B2 (n , η1 ) Y (η1 ) = Y (η2 ) B1 (n , η1 ) − D (n )

(3.194)

where Cn =

and for convenience we have set Y (η2 ) = 1. The corresponding mode shape of the rigid mass is obtained from Eq. (3.177) as Zo,n = Cn Z1 (n ) + Z2 (n ) o,n = Cn 1 (n ) − 2 (n )

(3.195)

and again we have again set Y (η2 ) = 1.

3.6 Effects of an Axial Force and an Elastic Foundation on the Natural Frequency We shall determine the effects that a constant axial force p (x, t) = po and an elastic foundation kf have on the natural frequency coefficient. Axial forces arise in beam models of turbine blades, where the centrifugal forces are the source of the axial forces, in vertical structures such as water towers, and in MEMS resonators (Singh et al. 2005). Elastic foundations arise when modeling railroad tracks. When the beam has a constant cross section and no in-span attachments, Eq. (3.60) reduces ∂ 4y ∂ 2y ∂ 2y − S + K y + =0 o f ∂η4 ∂η2 ∂τ 2

(3.196)

where So is given by Eq. (3.61) and Kf is given by Eq. (3.53). It is recalled that in Eq. (3.196) So represents a tensile axial force. For the case of harmonic oscillations of the form given by Eq. (3.64), Eq. (3.196) becomes  d4 Y d2 Y 4 Y = 0. − S + K −  o f dη4 dη2

(3.197)

150

3

Thin Beams: Part I

Rather than find a general solution to Eq. (3.197), we shall only obtain the solution to a beam that is hinged at each of its ends. This will be sufficient to illustrate the effects that po and kf have on the natural frequency coefficient. Thus, from Case 1 of Tables 3.3 and 3.5, the solution for a beam hinged at both ends is Yn (η) = sin (nπ η)

n = 1, 2, ... .

(3.198)

Substituting Eq. (3.198) into Eq. (3.197) leads to the characteristic equation n =

 4

(nπ )4 + So (nπ )2 + Kf

n = 1, 2, ...

(3.199)

where n is the nth natural frequency coefficient. Since Kf > 0, it is seen that the presence of the elastic foundation always increases the natural frequencies of the beam, and that a tensile axial force always increases the natural frequencies and a compressive axial force always decreases them. For compressive axial forces one must make sure that the buckling limits of the beam are not exceeded. The buckling limits are also a function of the boundary conditions. The effects of axial loads on the natural frequencies and mode shapes for a wide variety of boundary conditions are available (Shaker 1975; Karnovsky and Lebed 2000, Chapter 10).

3.7 Beams with a Rigid Extended Mass 3.7.1 Introduction A more realistic case of a mass attached to a beam will now be considered. When we considered the model of a point mass, it was tacitly assumed that the center of gravity of the attached mass was located at the point of attachment and that the mass was essentially hinged at this point. We shall lift these assumptions and consider two practical cases that closely approximate the case of a proof mass supported by MEMS-scale cantilever beams. In the first case, a rigid mass is attached to the free end of a cantilever beam and its center of mass is located a distance from the beam’s end. For the second case, the system consists of two beams each attached to an opposing side of a rigid body of finite length.

3.7.2 Cantilever Beam with a Rigid Extended Mass Referring to Fig. 3.21a, a mass ME is attached to the free end of a cantilever beam such that its center of mass is a distance do from the point of attachment. Referring to Fig. 3.21b and c, it is seen that the displacement of the center of mass is wCM (L) and, therefore, the translational inertia force of the extended mass is ME w ¨ CM (L). Thus, at the free end of the beam, the shear force is caused by the inertia force of

3.7

Beams with a Rigid Extended Mass

151

Center of mass: ME, JE

Fig. 3.21 (a) Equilibrium positions of an extended mass ME with mass moment of inertia JE that is attached to the free end of a cantilever beam (b) Displacement and rotation of the center of mass and (c) Free body diagram

Beam do

L

(a)

wCM(L) = w(L) + dow′(L) w(L)

w′(L)

(b)

V

MEwCM

V

M

JEw′CM do

(c)

the center of mass and the moment is caused by the rotational inertia of the center of mass about the attachment point (Zhou 1997; Seidel and Csepregi 1984; Kirk and Wiedemann 2002; Krylov and Maimon 2004). Hence, from a force balance and a moment balance about the attachment point, the boundary conditions at the free end are4 4 The force balance and moment balance method has been used here because it is also required when a beam with an in-span rigid mass is analyzed. However, these boundary conditions could have been determined from the energy approach as follows. The kinetic energy of the mass about the end of the beam is given by

F (C1 ) = Then,

1 ME 2



∂ 2 w (L, t) ∂w (L, t) + do ∂t ∂x∂t

2

 + JE

∂ 2 w (L, t) ∂x∂t

2 .

 2  ∂ w (L, t) ∂ 3 w (L, t) ∂ (C1 )  Fw(L,t) + d = −M E o ˙ ∂t ∂t2 ∂x∂t2  

 ∂ ∂ 2 w (L, t) ∂ 3 w (L, t) ∂ 3 w (L, t) 1) − − JE Fw(C M + d . = −d o E o ˙ x (L,t) ∂t ∂t2 ∂x∂t2 ∂x∂t2 −

152

3

!  2 ∂ 3w ∂ w ∂ 3 w !! EI 3 ! = ME + d o ∂x x=L ∂t2 ∂x∂t2 x=L !   ∂ 3w ∂ 2w ∂ 2w ! EI 2 !! = − do ME 2 + JE + do2 ME ∂x x=L ∂t ∂x∂t2

Thin Beams: Part I

(3.200) x=L

where JE is the mass moment of inertia about the center of mass. When do = 0, these boundary conditions reduce to that of a concentrated mass; that is, to those given by Case 7 in Table 3.1. The boundary conditions at the clamped end are w|x=0 = 0 ! ∂w !! = 0. ∂x !x=0

(3.201)

If the parameters given in Eq. (3.53) are introduced, then Eqs. (3.200) and (3.201), respectively, become !  2 ∂ 3 y !! ∂ 3y ∂ y = m + d E L ∂η3 !η=1 ∂τ 2 ∂η∂τ 2 η=1 !   ∂ 3y ∂ 2 y !! ∂ 2y 2 = −d m − j + d m L E E E L ∂η2 !η=1 ∂τ 2 ∂η∂τ 2

(3.202) η=1

and y=0 ∂y =0 ∂η

(3.203)

where mE =

ME , mb

jE =

JE , L 2 mb

dL =

do . L

(3.204)

It is assumed that there are no electrostatic forces and no in-plane forces on the beam, the beam has no in-span attachments, and there is no elastic foundation. In this case, the governing equation in terms of the non dimensional quantities is given by Eq. (3.60) with Ki = mi = Ko = Kf = So = 0; that is, ∂ 2y ∂ 4y + = 0. ∂η4 ∂τ 2

(3.205)

To determine the natural frequencies, a solution for y (η, τ ) that is given by Eq. (3.64) is assumed. Then Eq. (3.205) becomes ∂ 4Y − 4 Y = 0. ∂η4

(3.206)

3.7

Beams with a Rigid Extended Mass

153

The boundary conditions given by Eqs. (3.202) and (3.203), respectively, become   Y (1) = −mE 4 Y (1) + dL Y (1)     Y (1) = 4 dL mE Y (1) + jE + dL2 mE Y (1)

(3.207)

and Y (0) = 0

(3.208)

Y (0) = 0

where the prime denotes the derivative with respect to η. The solution to Eq. (3.206) is given by Eq. (C.18) of Appendix C with K = f = 0; thus, Y (η) = Y (0) Q (η) + Y (0) R (η)/ + Y (0) S (η)/2 + Y (0) T (η)/3 .

(3.209)

Following the procedure used in Section 3.3.2, we use Eq. (3.208) in Eq. (3.209) and find Y (η) = C1 S (η)/2 + C2 T (η)/3 .

(3.210)

For notational convenience, in Eq. (3.210) we have replaced Y (0) with C1 and Y (0) with C2 . To determine C1 and C2 , Eq. (3.210) is substituted into Eq. (3.207) to arrive at  [B ()] {C} =

b11 () b21 ()

b12 () b22 ()



C1 C2



=

0 0

 (3.211)

where b11 () = T () + a () S ()/2 + b () R ()/ b12 () = Q + a () T ()/3 + b () S ()/2 b21 () = Q − b () S ()/2 − e () R ()/

(3.212)

b22 () = R ()/ − b () T ()/3 − e () S ()/2 and a () = mE 4 b () = dL mE 4   e () = jE + dL2 mE 4 .

(3.213)

154

3

Thin Beams: Part I

0.6 dL = 0 0.55 0.5

Ω1/π

0.45 0.4 0.35

dL = 0.1 jE = 0

0.3

jE = 0.05

0.25

jE = 0.1

0.2 −3 10

10−2

10−1 mE

100

101

Fig. 3.22 Lowest natural frequency coefficients for a cantilever beam with an extended mass ratio mE as a function of mE for several values of jE and dL

A non-trivial solution to Eq. (3.211) exists only at  = n , where n are solutions to det [B ()] = 0; that is, b11 (n ) b22 (n ) − b12 (n ) b21 (n ) = 0.

(3.214)

When dL = 0, Eq. (3.214) reduces to D(8) (n ) of Case 8 of Table 3.3, where A2n = −a (n ) and B2n = e (n ) and a (n ) and e (n ) are given by Eq. (3.213) with dL = 0. The evaluation of Eq. (3.214) for the lowest natural frequency coefficient 1 /π is given in Fig. 3.22 as a function of mE for several combinations of values of jE and dL . It is seen from this figure that that both dL > 0 and jE > 0 have the effect of lowering the first natural coefficient, irrespective of the value of the value of mE . When mE > 3, jE has almost no effect on the first natural coefficient and it is primarily a function of dL .

3.7.3 Beam with an In-Span Rigid Extended Mass Consider the case of a beam of constant cross section that has a portion of its length replaced by a rigid body as shown in Fig. 3.23. This is a model of a proof mass suspended by two cantilevered elastic supports (Kopmaz and Telli 2002; Ilanko 2003; Zou et al. 2008; Urey et al. 2005). A solution to this system is obtained by considering the beam as two separate beams, one on each side of the extended mass. The system is made continuous by requiring the satisfaction of certain continuity conditions at their respective common interface. Since we are going to consider each

3.7

Beams with a Rigid Extended Mass

155 Center of mass: ME, JE

Fig. 3.23 (a) Two beams supporting a rigid mass (b) Displacement and rotation of the extended mass and (c) Free body diagram. The prime denotes the derivative of wl with respect to xl , l = 1, 2

E1, A1, I1, ρ1 x1

E2, A2, I2, ρ2

w1

w2 do

a

x2

b

c

L

(a) w1′(a)

w1(a)

wCM = w1(a) + dow1′(a)

bw1′(a)

w2(a)

w2′(a)

(b)

MEwCM V2

V1

JEwCM ′ x1 x1 = a

V2 M 2

M1 V 1 do

x2 x2 = c

b

(c)

beam separately, we shall also assume that they have different properties as indicated in the Fig. 3.23a. Referring to this figure and using the dimensional form of Eq. (3.205), the governing equation of each beam can be expressed as

El Il

∂ 4 wl ∂ 2 wl + ρ A =0 l l ∂t2 ∂xl4

l = 1, 2

(3.215)

where wl = wl (xl , t). The boundary conditions at xl = 0, l = 1, 2, are wl |xl = 0 = 0 ! ∂wl !! =0 ∂xl !xl = 0

l = 1, 2.

At the other end of each beam, we have the following continuity conditions

(3.216)

156

3

 w1 + b

∂w1 ∂x1

x1 =a

Thin Beams: Part I

= w2 |x2 =c

! ! ∂w1 !! ∂w2 !! = − ∂x1 !x1 =a ∂x2 !x2 =c ,

-

,

-

∂ 3 w1 ∂ 2 w1 ∂ 3 w1 + E1 I1 2 + do E1 I1 3 JE 2 ∂t ∂x1 ∂x1 ∂x1 

ME

∂ 3 w1 ∂ 2 w1 + do 2 2 ∂t ∂t ∂x1

 − E1 I1

∂ 3 w1 ∂x13

,

x1 =a

∂ 2 w2 ∂ 3 w2 = E2 I2 2 + (b − do ) E2 I2 3 ∂x2 ∂x2

x1 =a

! ∂ 3 w2 !! = E2 I2 3 ! ∂x2 !

x2 =c

. x2 =c

(3.217)

The last two continuity conditions in Eq. (3.217) were obtained from Fig. 3.23c by taking, respectively, the sum of moments about the center of mass and by summing the vertical forces. When the right hand beam, which is described by w2 , is removed, the first two continuity conditions of Eq. (3.217) do not apply. The last two continuity conditions simplify to become the boundary conditions expressed as ! ∂ 3 w1 !! E1 I1 3 ! ∂x1 ! ! ! 1! E1 I1 2 ! ∂x1 !

 = ME x1 =a

,

∂ 2w

=− x1 =a

∂ 3 w1 ∂ 2 w1 + d o ∂t2 ∂x1 ∂t2

∂ 3 w1 JE 2 ∂t ∂x1

+ do E1 I1

x1 =a

∂ 3w

1 3 ∂x1

(3.218)

. x1 =a

It is seen that the moment boundary condition given by the second equation of Eq. (3.218) is different from that given by the second equation of Eq. (3.200). They are, however, equivalent. Recall that the moment balance to arrive at Eq. (3.217) and, hence, Eq. (3.218) was taken about the center of mass whereas the moment balance to arrive at Eq. (3.200) was taken about the attachment point. Thus, referring to Fig. 3.23c, it is seen that Eq. (3.218) equals Eq. (3.200), since, when the moments are summed about the attachment point, JE → JE + do2 ME E1 I1

∂ 3 w1 ∂ 2w → M . E ∂t2 ∂x13

There are several ways to place the governing equations into non dimensional form. We choose the following parameters so that its special cases can be more easily compared to the previously obtained results

3.7

Beams with a Rigid Extended Mass

157

x2 x1 a c , ξ = , aL = , cL = , L L L L  t ρ2 A2 2 ρ1 A1 to = L s, τ = , α = , E1 I1 to ρ1 A1 η=

λ4 = αβ,

m E =

m = ρ1 A1 L kg,

ME , m

dL =

do , L

β=

E1 I1 , E2 I2

j E =

bL =

b L (3.219)

JE , mL2

where L = a + b + c. Notice that m is the mass of a fictitious beam of length L having the properties of beam 1. This quantity appears when this non dimensional conversion is performed. However, in order to compare beams with different attachment ratios, it is better to consider a mass ratio that is proportional to the mass ratio of the actual beam. The total mass of the two beams is mb = ρ1 A1 a + ρ2 A2 c = ρA1 L (aL + αcL ) = ρA1 L (aL + α [1 − aL − bL ]). Therefore, we define the mass ratio and rotational inertia ratio, respectively, as m E =

ME ME = mb mcE (3.220)

JE j E = mcE L2 where cE = aL + α (1 − aL − bL ) . Using Eq. (3.219) in Eq. (3.215), we obtain ∂ 2 w1 ∂ 4 w1 + =0 ∂η4 ∂τ 2 2 ∂ 4 w2 4 ∂ w2 + λ = 0. ∂ξ 4 ∂τ 2

(3.221)

The boundary conditions given by Eq. (3.216) become w1 |η=0 = 0

and

w2 |ξ =0 = 0

and

! ∂w1 !! =0 ∂η !η=0 ! ∂w2 !! =0 ∂ξ !ξ =0

and the continuity conditions given by Eq. (3.217) become

(3.222)

158

3

 ∂w1 w1 + bL ∂η

η=aL

Thin Beams: Part I

= w2 |ξ =cL

! ! ∂w1 !! ∂w2 !! = − ∂η !η=aL ∂ξ !ξ =cL  ∂ 2 w1 ∂ 3 w1 ∂ 3 w1 β j E cE 2 + + d L ∂τ ∂η ∂η2 ∂η3   2  ∂ 3 w1 ∂ 3 w1 ∂ w1 − β mE cE + d L ∂τ 2 ∂τ 2 ∂η ∂η3

 η=aL

η=aL

∂ 2 w2 ∂ 3 w2 + − d (b ) L L ∂ξ 2 ∂ξ 3

=

! ∂ 3 w2 !! = . ∂ξ 3 !ξ =cL

ξ =cL

(3.223)

The system is assumed to undergo harmonic oscillations at a frequency ω of the form w1 (η, τ ) = W1 (η) ej

2τ

(3.224) w2 (ξ , τ ) = W2 (ξ ) ej

2τ

where 2 = ωto . Substituting the first equation of Eq. (3.224) into the first equation of Eq. (3.221) and the second of Eq. (3.224) into the second equation of Eq. (3.221), we obtain d 4 W1 − 4 W1 = 0 dη4 (3.225) d 4 W2 − λ4 4 W2 = 0. dξ 4 The boundary conditions given by Eq. (3.222) become W1 |η=0 = 0 W2 |ξ =0 = 0

and

! dW1 !! =0 dη !η=0

and

! dW2 !! =0 dξ !ξ =0

and the continuity conditions given by Eq. (3.223) become

(3.226)

3.7

Beams with a Rigid Extended Mass

 w1 + b L

159

dW1 dη

η=aL

= W2 |ξ =cL

! ! dW1 !! dW2 !! = − dη !η=aL dξ !ξ =cL  dW1 d 3 W1 d 2 W1 β −j E cE 4 + d + L ∂η dη2 dη3  β

−m E cE 4

  dW1 d 3 W1 W1 + dL − dη dη3

 η=aL

η=aL

=

d 2 W2 d 3 W2 + − d (b ) L L dξ 2 dξ 3

! d3 W2 !! = . dξ 3 !ξ =cL

ξ =cL

(3.227)

From Eqs. (3.206) and (3.209), it is seen that the solutions to Eq. (3.225) are W1 (η) = W1 (0) Q (η) + W1 (0) R (η)/ + W1 (0) S (η)/2 + W1 (0) T (η)/3

0 ≤ η ≤ aL

W2 (ξ ) = W2 (0) Q (λξ ) + W2 (0) R (λξ )/(λ) + W2 (0) S (λξ )/(λ)2 + W2 (0) T (λξ )/(λ)3 0 ≤ ξ ≤ cL (3.228) where Q (η), etc. are given by Eq. (C.19) of Appendix C. Using Eq. (3.226) in Eq. (3.228), we obtain W1 (η) = C1 S (η)/2 + C2 T (η)/3

0 ≤ η ≤ aL

W2 (ξ ) = C3 S (λξ )/(λ)2 + C4 T (λξ )/(λ)3

0 ≤ ξ ≤ cL .

(3.229)

For convenience, in Eq. (3.229) we have replaced W1 (0) with C1 , etc. The four unknown quantities are determined by substituting Eq. (3.229) into the continuity conditions given by Eq. (3.227). When this substitution is performed, the following system of equations is obtained ⎡

e11 () ⎢ e21 () [E ()] {C} = ⎢ ⎣ e31 () e41 () where

e12 () e22 () e32 () e42 ()

e13 () e23 () e33 () e43 ()

⎤⎧ ⎫ e14 () ⎪ C1 ⎪ ⎪ ⎬ ⎨ ⎪ e24 () ⎥ ⎥ C2 = 0 e34 () ⎦ ⎪ C3 ⎪ ⎪ ⎭ ⎩ ⎪ C4 e44 ()

(3.230)

160

3

Thin Beams: Part I

e11 () = S (aL )/2 + bL R (aL )/ e12 () = T (aL )/3 + bL S (aL )/2 e13 () = −S (λcL )/(λ)2 e14 () = −T (λcL )/(λ)3 e21 () = R (aL )/ e22 () = S (aL )/2 e23 () = R (λcL )/(λ) e24 () = S (λcL )/(λ)2 e31 () = −j E cE β3 R (aL ) + βQ (aL ) + dL βT (aL )

(3.231)

e32 () = −j E cE β2 S (aL ) + βR (aL )/ + dL βQ (aL ) e33 () = −Q (λcL ) − (bL − dL ) (λ) T (λcL ) e34 () = −R (λcL )/(λ) − (bL − dL ) Q (λcL ) e41 () = −m E cE β3 (S (aL )/ + dL R (aL )) − βT (aL )

e42 () = −m E cE β2 (T (aL )/ + dL S (aL )) − βQ (aL ) e43 () = − (λ) T (λcL ) e44 () = −Q (λcL ) .

The natural frequency coefficients are the values of  = n for which det [E (n )] = 0. The mode shapes corresponding to these natural frequency coefficients are W1n (η) = S (n η) /2n + C2n T (n η) /3n W2n (ξ ) = C3n S (λn ξ )/(λn )2 + C4n T (λn ξ )/(λn )3

(3.232)

where C2n , C3n , and C4n are determined from ⎫ ⎫ ⎧ ⎤⎧ e22 (n ) e23 (n ) e24 (n ) ⎨ C2n ⎬ ⎨ −e21 (n ) ⎬ ⎣ e32 (n ) e33 (n ) e34 (n ) ⎦ C3n = −e31 (n ) ⎩ ⎭ ⎭ ⎩ e42 (n ) e43 (n ) e44 (n ) −e41 (n ) C4n ⎡

(3.233)

and, for convenience, we have set C1n = 1. Numerical values for the lowest natural frequency coefficient 1 /π are given in Figs. 3.24 and 3.25 for m E = 0.5 and α = β = 1; that is, for beams with the same geometrical and physical properties, as a function of aL and bL for several combinations of values of j E and dL . It is seen that the values aL , bL , dL , and j E , affect 1 such that its maximum value varies in a complicated manner as a function of these quantities. In other words, without graphs like these, it is difficult to predict whether 1 increases or decreases as one or more of these parameters changes.

3.7

Beams with a Rigid Extended Mass

161

(a) Ω1/π = 1.55

Ω1/π = 1.69

1.8

Ω1/π

1.6 1.4 1.2

Ω1/π = 1.23

Ω1/π = 1.43

1 0.5 0.45 0.4 0.35 0.3 0.25 0.2 aL

0.3 0.2 0.1

0

bL

(b) Ω1/π = 1.62 Ω1/π = 1.62 1.8

Ω1/π

1.6 1.4 1.2

Ω1/π = 1.23

Ω1/π = 1.43

1 0.5 0.45 0.4 0.35 0.3 0.25 0 aL 0.2

0.3 0.2 0.1

bL

Ω1/π = 1.69

(c)

Ω1/π = 1.55

1.8

Ω1/π

1.6 1.4 1.2

Ω1/π = 1.23

Ω1/π = 1.43

1 0.5 0.45 0.4 0.35 0.3 0.25 0 0.2 aL

0.3 0.2 0.1

bL

Fig. 3.24 Lowest natural frequency coefficient for a beam clamped at each end with an extended rigid mass interior to the beam for m E = 0.5 and j E = 0 as a function of aL and bL (a) dL = 0.25bL (b) dL = 0.5bL and (c) dL = 0.75bL

162

3

Thin Beams: Part I

(a) Ω1/π = 1.43

1.8

Ω1/π = 1.53

Ω1/π

1.6 1.4 1.2 Ω1/π = 1.23 1 0.5 0.45 Ω1/π = 1.15 0.4 0.35 0.3 0.25 0 0.2 aL

0.3 0.2 0.1 bL

(b) Ω1/π = 1.48 Ω1/π = 1.48

1.8

Ω1/π

1.6 1.4 1.2 Ω1/π = 1.23 1 0.5 0.45 Ω1/π = 1.15 0.4 0.35 0.3 0.25 0 0.2 aL

0.3 0.2 0.1 bL

(c) Ω1/π = 1.53

1.8

Ω1/π = 1.43

Ω1/π

1.6 1.4 1.2 Ω1/π = 1.23 1 0.5 0.45 Ω1/π = 1.15 0.4 0.35 0.3 0.25 0 aL 0.2

0.3 0.2 0.1 bL

Fig. 3.25 Lowest natural frequency coefficient for a beam clamped at each end with an extended rigid mass interior to the beam for m E = 0.5 and j E = 0.05 as a function of aL and bL (a) dL = 0.25bL (b) dL = 0.5bL and (c) dL = 0.75bL

3.7

Beams with a Rigid Extended Mass

163

(a) Extended mass: Ω1/π = 1.6907 Concentrated mass: Ω1/π = 1.3904 Center of mass: dL = 0.25bL

(b) Extended mass: Ω1/π = 1.6216 Concentrated mass: Ω1/π = 1.3323 Center of mass: dL = 0.5bL

(c) Extended mass: Ω1/π = 1.5488 Concentrated mass: Ω1/π = 1.2953 Center of mass: dL = 0.75bL

Fig. 3.26 Mode shapes corresponding to the lowest natural frequency of a beam clamped at each end with an extended rigid mass interior to the beam for m E = 0.5, j E = 0, bL = 0.3, and aL = 0.2 (a) dL = 0.25bL (b) dL = 0.5bL and (c) dL = 0.75bL . For the case of the concentrated mass, the mass is located at aL + dL

In Figs. 3.26 and 3.27, representative mode shapes for the lowest natural frequencies have been plotted for m E = 0.5, j E = 0, bL = 0.3, aL = 0.2 and 0.35, and for three values of dL . For the case of the concentrated mass, the mass is located at aL + dL ; that is, at the location of the center of mass of the extended mass. It is seen that upon comparing the extended mass model to the concentrated mass model, there are substantial differences in the values of the natural frequencies and in the corresponding mode shapes.

164

3

Thin Beams: Part I

(a) Extended mass: Ω1/π = 1.7413 Concentrated mass: Ω1/π = 1.2953 Center of mass: dL = 0.25bL

(b) Extended mass: Ω1/π = 1.7494 Concentrated mass: Ω1/π = 1.2829 Center of mass: dL = 0.5bL

(c) Extended mass: Ω1/π = 1.7413 Concentrated mass: Ω1/π = 1.2953 Center of mass: dL = 0.75bL

Fig. 3.27 Mode shapes corresponding to the lowest natural frequency of a beam clamped at each end with an extended rigid mass interior to the beam for m E = 0.5, j E = 0, bL = 0.3, and aL = 0.35 (a) dL = 0.25bL (b) dL = 0.5bL and (c) dL = 0.75bL . For the case of the concentrated mass, the mass is located at aL + dL

3.8 Beams with Variable Cross Section 3.8.1 Introduction We shall now consider beams whose cross-sectional area varies with position along the length of the beam. Two cases will be considered. The first case will be a beam whose geometric properties vary continuously along the length of the beam. The second case will be a beam whose cross section is constant for portion of the beam and then changes abruptly to a different constant cross section for the remainder of the beam. The solution method for each of these types of beams is different and, therefore, they will be analyzed separately. We shall be concerned with determining the natural frequencies and mode shapes and we shall limit our discussion to a cantilever beam with and without a concentrated mass at its free end.

3.8

Beams with Variable Cross Section

165

3.8.2 Continuously Changing Cross Section We start with the non dimensional form of the governing equation given by Eq. (3.54), set Kf = Ki = Ko = mi = Vo = fˆ = S = P = 0, and assume a solution of the form y (η, τ ) = W (η) ej

2τ

(3.234)

to obtain d2 dη2

  d2 W i (η) − a (η) 4 W = 0 dη2

(3.235)

where 2 = ωto , ω is the frequency of oscillation, and to is given in Eq. (3.53). The switch from Y to W is to avoid confusion with the symbol for the Bessel function of the second kind, which is needed when one form of this equation is solved. If one were to examine a table of properties of plane areas (Tuma 1979, pp. 125–127) it would be found that the moment of inertia of several different beam cross sections can be expressed in the form I = ci bh3 where b is the width of the cross section, h is the height of the cross section, and ci is a constant that is a function of the shape of the cross section. For example, for a beam with a rectangular cross section, ci = 1/12; for a circular cross section ci = π/64 and b = h is the diameter of the circle; and for an elliptical cross section ci = π/64 and b and h are the major and minor axes, respectively, or the minor and major axes, respectively. However, circular and elliptical cross sections will only be considered when the tapers are equal in the y and z directions. In the notation used to obtain Eq. (3.235) and referring to Fig. 3.28, it is seen that I = Io i (η)

(3.236)

where Io = ci bo h3o i (η) = y (η) z3 (η)

(3.237)

and y (η) and z (η) are non dimensional quantities that, in general, are independent of each other. Such a beam is referred to as a double-tapered beam. In the analyses that follow, profiles that vary linearly and exponentially with respect to η will be considered. The area of the cross section can be written in the form A = ca bh

166

3 bo

bo ho

h1

Thin Beams: Part I bo

ho

ho h1

ho

bo

b1 b1 bo

bo ho

h1

bo ho h1

ho

b1

ho

bo b1 (a)

(b)

(c)

Fig. 3.28 Geometry and notation for linear- and exponential-tapered beams (a) Double-tapered (b) Tapered in the depth only and (c) Tapered in the height only

where ca is a constant that is a function of the shape of cross section. For example, for a beam with a rectangular or square cross section, c = 1; for a circular cross section c = π/4 and b = h is the diameter of the circle; and for an elliptical cross section c = π/4 and b and h are the major and minor axes, respectively. In the current notation, the area can be written as A = Ao a (η)

(3.238)

where Ao = ca bo ho a (η) = y (η) z (η) .

(3.239)

The taper functions y (η) and z (η), respectively, for linearly varying edges with the tapered end is at η = 0 are y (η) = (1 − β) η + β z (η) = (1 − α) η + α

(3.240)

where α = h1 /ho ≤ 1 and β = b1 /bo ≤ 1. For exponentially tapered edges, y (η) and z (η), respectively, vary as y (η) = eβe (1−η) z (η) = eαe (1−η) where the tapered end is at η = 0,

(3.241)

3.8

Beams with Variable Cross Section

167

αe = ln α βe = ln β. and, since α ≤ 1 and β ≤ 1, αe ≤ 0 and βe ≤ 0. For a cantilever beam with a mass at its free end, the boundary conditions are given by Cases 1 and 7 of Table 3.1. Thus, for harmonic oscillations given by Eq. (3.234), the boundary conditions at η = 0 are ! d2 W !! i (η) =0 dη2 !η=0 !  ! d2 W !! d 4 ! = m  W i (η) ! L η=0 dη dη2 !η=0

(3.242)

W|η=1 = 0 ! dW !! = 0. dη !η=1

(3.243)

and at η = 1 they are

The second boundary condition of Eq. (3.242) can be simplified as follows. If the indicated differentiation is performed, we obtain 

di (η) d2 W d3 W + i (η) dη dη2 dη3

η=0

! ! = mL 4 W !

η=0

.

From the first equation of Eq. (3.242), it is seen that d 2 W/dη2 = 0. Therefore, the boundary conditions given by Eq. (3.242) become ! d2 W !! =0 i (η) dη2 !η=0 ! ! d3 W !! 4 ! = m  W . i (η) ! L η=0 dη3 !η=0

(3.244)

In Eq. (3.244), mL = ML /mb , where mb = ρAo L is the mass of a beam of constant cross section of that at the end η = 1. Rather than use mb , we shall use the actual mass of the beam ma . For the frustum of a right pyramid, the mass is (Tuma 1979, p. 25) ma = where

  ρL ho bo + h1 b1 + ho bo h1 b1 = mb cαβ 3

(3.245)

168

3

cαβ =

  1 1 + αβ + αβ . 3

Thin Beams: Part I

(3.246)

Then, Eq. (3.244) becomes ! d2 W !! =0 i (η) dη2 !η=0 ! ! d3 W !! 4 ! = mc  W i (η) ! αβ η=0 dη3 !η=0

(3.247)

  where m = ML / mb cαβ . The mass ratio for a beam with an exponential taper is given in Section 3.8.4. The effects of a linear and an exponential taper on the natural frequencies will now be determined. However, to get a closed form solution to Eq. (3.235) we will be limited to the case where α = β for the linear double-tapered beam and to the case where βe = 0 for the exponential tapered beam.5 Three cases are examined: (1) a double tapered beam shown in Fig. 3.28a, (2) a beam in which z (η) = 1 and the taper is only in the y-direction as shown in Fig. 3.28b, and (3) a beam in which y (η) = 1 and the taper is only in the z-direction as shown in Fig. 3.28c.

3.8.3 Linear Taper For the linear tapered beam, we use Eqs. (3.237), (3.239) and (3.240) to obtain a (η) = y (η) z (η) = [(1 − β) η + β] [(1 − α) η + α] i (η) = y (η) z3 (η) = [(1 − β) η + β] [(1 − α) η + α]3 .

(3.248)

Double-Taper Beam with α = β When α = β, Eq. (3.248) becomes a (η) = y (η) z (η) = [(1 − α) η + α]2 i (η) = y (η) z3 (η) = [(1 − α) η + α]4 .

(3.249)

For this case, the cross sections can also be circular and elliptical.

5 These restrictions will be removed when the Rayleigh-Ritz solution method is used subsequently. The Rayleigh-Ritz method is an approximate solution method that often yields results very close to those obtained from known analytical solutions.

3.8

Beams with Variable Cross Section

169

Single-Taper Beam with z (η) = 1 For this case, we set α = 1 in Eq. (3.248) to obtain a (η) = (1 − β) η + β i (η) = (1 − β) η + β.

(3.250)

Single-Taper Beam with y (η) = 1 For this case, we set β = 1 in Eq. (3.248) to obtain a (η) = (1 − α) η + α i (η) = [(1 − α) η + α]3 .

(3.251)

If it is assumed that α = β, then it is seen from Eqs. (3.249) to (3.251) that a (η) and i (η) can be expressed as a (η) = [(1 − α) η + α]p i (η) = [(1 − α) η + α]q

(3.252)

where p and q are integers. Then the governing equation, Eq. (3.235), can be written as d2 dη2

  2 q d W [(1 − α) η + α] − [(1 − α) η + α]p 4 W = 0. dη2

(3.253)

To place Eq. (3.253) into a more useful form, we make the following transformation ϕ = (1 − α) η + α.

(3.254)

Thus, n dn n d = − α) (1 dηn dϕ n

n = 1, 2, ... .

(3.255)

Using Eqs. (3.254) and (3.255) in Eq. (3.253), we obtain d2 dϕ 2

  2 qd W ϕ − ϕ p λ4 W = 0 dϕ 2

(3.256)

 . 1−α

(3.257)

where λ=

170

3

Thin Beams: Part I

To transform the boundary conditions, it is noted that η = 0 corresponds to ϕ = α and that η = 1 corresponds to ϕ = 1. Therefore, from Eq. (3.243), the boundary conditions at ϕ = 1 are W|ϕ=1 = 0 ! dW !! =0 dϕ !ϕ=1

(3.258)

and from Eq. (3.247), those at ϕ = α are ! d2 W !! =0 dϕ 2 !ϕ=α ! d3 W !! = mcαβ (1 − α) α −q λ4 W|ϕ=α . dϕ 3 !ϕ=α

(3.259)

Solution It is seen from Eqs. (3.249) and (3.251) that q = p + 2. Thus, if we let p = n, where n = 1 or 2, then Eq. (3.256) is of the form d2 dϕ 2

  d2 W ϕ n+2 − ϕ n λ4 W = 0 dϕ 2

n = 1, 2.

(3.260)

Equation (3.260) can be factored into the following two operators 

1 d ϕ n dϕ

     d 1 d n+1 d ϕ n+1 + λ2 ϕ − λ2 W = 0. dϕ ϕ n dϕ dϕ

(3.261)

When, p = q = 1, the above factorization is not possible and the solution method is fairly complicated (Wang 1967; Banks and Kurowski 1977; Lee and Kuo 1992). Hence, this case will not be considered analytically and only an approximate method will be used to obtain the natural frequencies for this case. Hence, a solution is sought to the two second-order equations d dϕ

  dW ϕ n+1 ± λ2 ϕ n W = 0 dϕ

n = 1, 2.

(3.262)

The solutions to Eq. (3.262) are (Weisstein 2003, p. 197)   √   √  W = ϕ −n/2 C1 Jn 2λ ϕ + C2 Yn 2λ ϕ  √   √  + C3 In 2λ ϕ + C4 Kn 2λ ϕ

(3.263)

where Jn (x) and Yn (x), respectively, are Bessel functions of the first and second kind of order n and correspond to +λ2 and In (x), and Kn (x), respectively, are the modified Bessel functions of the first and second kind of order n and correspond to −λ2 .

3.8

Beams with Variable Cross Section

171

Before proceeding, the following identities are noted: d  −n/2  √  Jn 2λ ϕ ϕ dϕ d  −n/2  √  Yn 2λ ϕ ϕ dϕ d  −n/2  √  In 2λ ϕ ϕ dϕ d  −n/2  √  ϕ Kn 2λ ϕ dϕ

 √  = −λϕ −(n+1)/2 Jn+1 2λ ϕ  √  = −λϕ −(n+1)/2 Yn+1 2λ ϕ = λϕ

−(n+1)/2

(3.264)

 √  In+1 2λ ϕ

 √  = −λϕ −(n+1)/2 Kn+1 2λ ϕ .

Therefore,   √   √  dW = λϕ −(n+1)/2 −C1 Jn+1 2λ ϕ − C2 Yn+1 2λ ϕ dϕ  √   √  + C3 In+1 2λ ϕ − C4 Kn+1 2λ ϕ   √   √  d2 W 2 −(n+2)/2 C 2λ 2λ ϕ = λ ϕ J ϕ + C Y 1 n+2 2 n+2 dϕ 2  √   √  + C3 In+2 2λ ϕ + C4 Kn+2 2λ ϕ

(3.265)

  √   √  d3 W = λ3 ϕ −(n+3)/2 −C1 Jn+3 2λ ϕ − C2 Yn+3 2λ ϕ dϕ 3  √   √  + C3 In+3 2λ ϕ − C4 Kn+3 2λ ϕ . Using Eqs. (3.263) and (3.265) in the boundary conditions given by Eqs. (3.258) and (3.259), we obtain the following system of equations ⎡ ⎢ ⎢ ⎢ ⎣

Yn (2λ) In (2λ) Kn (2λ) Jn (2λ) Yn+1 (2λ) −In+1 (2λ) Kn+1 (2λ) Jn+1 (2λ)  √   √   √   √  Jn+2 2λ α Yn+2 2λ α In+2 2λ α Kn+2 2λ α a42,n (λ) a43,n (λ) a44,n (λ) a41,n (λ)

⎤⎧ ⎫ C1 ⎪ ⎪ ⎪ ⎪ ⎥⎨ ⎬ ⎥ C2 =0 ⎥ C3 ⎪ ⎦⎪ ⎪ ⎪ ⎩ ⎭ C3 (3.266)

where  √   √  a41,n (λ) = Jn+3 2λ α + mcαβ (1 − α) λα −(n+1)/2 Jn 2λ α  √   √  a42,n (λ) = Yn+3 2λ α + mcαβ (1 − α) λα −(n+1)/2 Yn 2λ α  √   √  a43,n (λ) = −In+3 2λ α + mcαβ (1 − α) λα −(n+1)/2 In 2λ α  √   √  a44,n (λ) = Kn+3 2λ α + mcαβ (1 − α) λα −(n+1)/2 Kn 2λ α .

(3.267)

The natural frequency coefficients are those values of λ = λl for which the determinant of coefficients of Eq. (3.266) vanishes for a given α. Once λl is determined, Eq. (3.257) is used to obtain l .

172

3

Thin Beams: Part I

The corresponding mode shapes are   √   √  Wl (ϕ) = ϕ −n/2 Jn 2λl ϕ + C2,l Yn 2λl ϕ  √   √  + C3,l In 2λl ϕ + C4,l Kn 2λl ϕ

(3.268)

where Cj,l , j = 2, 3, 4, are determined from the solution to ⎫ ⎧ ⎫ ⎤⎧ −J K Yn+1 C ) (2λ ) (2λ ) ⎬ ⎨ ⎨ l n+1 l n+1 l 2,l     (2λ√l )  −In+1  (2λ   √ √ √ ⎬ ⎣ Yn+2 2λl α In+2 2λl α Kn+2 2λl α ⎦ C3,l = −Jn+2 2λl α ⎭ ⎩ ⎩ ⎭ C4,l a43,n (λl ) a44,n (λl ) a42,n (λl ) −a41,n (λl ) (3.269) ⎡

and we have arbitrarily set C1,l = 1. The lowest natural frequency coefficient 1 for n = 2 and n = 1, respectively, are given in Figs. 3.29 and 3.30 as a function of the taper ratio α for several values of m. It is seen from these figures that, as with beams of constant cross section, the addition of the mass at the free end reduces the natural frequencies. However, for a double tapered beam there is an increase in the first natural frequency coefficient as the taper increases (the value of α decreases). For a single tapered beam, the changes in the natural frequency coefficients are similar to those of a double taper beam, but the magnitudes of the changes are not as large. From these results, it is 3 2.8 2.6

m=0

2.4 Ω1

m = 0.1 2.2 2 m = 0.2 1.8 m = 0.5 1.6 1.4

0

0.2

0.4

0.6

0.8

1

α

Fig. 3.29 Lowest natural frequency coefficient of a linear double-tapered cantilever beam (n = 2) carrying a mass at its free end as a function of the taper ratio α for several values of the mass ratio m

3.8

Beams with Variable Cross Section

173

2.6

2.4

Ω1

2.2

m=0 m =0.1

2

1.8

m = 0.2

1.6

m = 0.5

1.4

0

0.2

0.4

α

0.6

0.8

1

Fig. 3.30 Lowest natural frequency coefficient of a linear single-tapered cantilever beam (n = 1; y (η) = 1) carrying a mass at its free end as a function of the taper ratio α for several values of the mass ratio m

seen that for some combinations of α and m, the reduction in natural frequency due to the addition of the end mass can be nullified by choosing an appropriate taper.

3.8.4 Exponential Taper For a double exponentially tapered beam, Eqs. (3.237), (3.239), and (3.241) are used to obtain, a (η) = e(1−η)(βe +αe ) i (η) = e(1−η)(βe +3αe ) .

(3.270)

Then, Eq. (3.235) can be written as d2 dη2



e(1−η)(βe +3αe )

d2 W dη2



− 4 e(1−η)(αe +βe ) W = 0.

(3.271)

An analytical solution in terms of standard functions can be obtained if αe = 0; that is, when α = 1 and, therefore, z (η) = 1. In this case, Eq. (3.271) becomes

174

3

d2 dη2



e−βe η

d2 W dη2



Thin Beams: Part I

− 4 e−βe η W = 0.

(3.272)

The solution to Eq. (3.272) is (Cranch and Adler 1956)      W (η) = eδη C1 cosh εp η + C2 sinh εp η  + C3 cos (εm η) + C4 sin (εm η)

(3.273)

where δ = βe /2 and εp = εm =

√ √

2 + δ 2 2 − δ 2

2 > δ 2 .

(3.274)

Substituting Eq. (3.273) into the boundary conditions given by Eqs. (3.243) and (3.244), we obtain ⎡

cosh εp ⎢ ε sinh ε ⎢ p p ⎢ 2 ⎣ δ + εp2 c41

sinh εp εp cosh εp 2δεp c42

cos εm −εm sin εm 2 δ 2 − εm c43

⎤⎧ ⎫ sin εm C1 ⎪ ⎪ ⎪ ⎬ ⎨ ⎪ εm cos εm ⎥ ⎥ C2 =0 ⎥ 2δεm ⎦ ⎪ ⎪ ⎪ C3 ⎪ ⎭ ⎩ C4 c44

(3.275)

where me = ML /ma , ma = mb ce is the mass of the exponentially single-tapered beam, mb = ρAo L, and ce = −

 1  1 − eβe βe

c41 = δ 3 + 3δεp2 − me ce e−βe 4 c42 = 3δ 2 εp + εp3

(3.276)

2 − m c e−βe 4 c43 = δ 3 − 3δεm e e 3. c44 = 3δ 2 εm − εm

It is noted that since βe ≤ 0, ce > 0. The values of natural frequency coefficients n are those values for which the determinant of the coefficients in Eq. (3.275) is zero. The corresponding mode shapes are      Wn (η) = e−δη cosh εp,n η + C2,n sinh εp,n η     + C3,n cos εm,n η + C4,n sin εm,n η where

(3.277)

3.8

Beams with Variable Cross Section

175

2.8 me = 0

2.6

Ω1

2.4

2.2

me = 0.1

2

1.8 me = 0.4 1.6

1.4

0

0.2

0.4

β

0.6

0.8

1

Fig. 3.31 Lowest natural frequency coefficient of a cantilever beam with an exponential taper in the y-direction and a constant thickness in the z-direction as a function of the taper ratio β for several values of the attached mass ratio me

εp,n = εm,n =

 

2n + δ 2 2n − δ 2

2n > δ 2

(3.278)

and Cj,n , j = 2, 3, 4, are determined from the solution to ⎫ ⎧ ⎫ ⎤⎧ εp cosh εp −εm,n sin εm,n εm,n cos εm,n ⎪ ⎨ C2,n ⎪ ⎨ −εp.n sinh εp,n ⎪ ⎬ ⎪ ⎬ ⎢ ⎥ 2 2 −δ 2 − εp,n δ 2 − εm,n 2δεm,n ⎦ C3,n = . ⎣ 2δεp ⎪ ⎪ ⎪ ⎪ ⎩ ⎩ ⎭ ⎭ C4,n c42 c43 c44 −c41 (3.279) ⎡

The values 1 of as a function of β for several value of me are shown in Fig. 3.31. The values of 1 at β = 1 are those of a cantilever beam of constant cross section with a mass at its free end.

3.8.5 Approximate Solutions to Tapered Beams: Rayleigh-Ritz Method The previous class of beams with continuously variable cross sections are revisited, except this time we do not require that α = β in beams with rectangular cross sections and we do not require that αe = 0 in exponentially tapered beams. Again,

176

3

Thin Beams: Part I

we shall consider these beams without in-span attachments. The restrictions on α, β, and β e can be removed if we use an approximate solution method known as the Rayleigh-Ritz method (Meirovitch 2001, pp. 499–507). Since an approximate solution method is going to be used, we shall also consider the effects of an axial force and for a cantilever beam, we shall include the effects of an attached mass ML . To employ the Rayleigh-Ritz method for the determination of the natural frequencies and mode shapes, one assumes harmonic excitation of frequency ω and magnitude W(x) and forms a quantity , which is the difference between the maximum kinetic energy and the maximum potential energy. We start with Eqs. (3.37) to (3.39), set f = Po = Vo = kf = Mi = ki = ko = 0, and assume a constant axial force: that is, p (x, y) = po . The maximum potential energy is obtained by replacing w (x, t) with W (x) and the maximum kinetic energy is obtained by replacing ∂w (x, t) /∂t with ωW (x), which is the maximum velocity. Then,  = Tmax − Vmax

(3.280)

where

Tmax =

ω2 2

⎡ L ⎤

⎣ ρA (x) W 2 (x) dx + ML W 2 (0)⎦ 0

Vmax =

1 2

L "

 EI (x)

2 d2 W dx2

 + po

dW dx

(3.281)

2 # dx.

0

Introducing the quantities defined in Eqs. (3.53) and (3.61), Eqs. (3.280) and (3.281) can be written as ⎫ ⎡ ⎧ 1 ⎨

⎬ EI ⎣4 = a (η) Y 2 (η) dη + mL Y 2 (0) ⎩ ⎭ 2L 0

1 −

"



d2 Y i (η) dη2

2

 + So

dY dη

2 #

⎤

(3.282)

dη⎦

0

where mL = ML /(ρho bo L) , 2 = ωto , and Y = W/L. The Rayleigh-Ritz procedure is used to determine the natural frequency coefficient l and the corresponding mode shapes as follows. We assume a solution of the form Y (η) =

N  n=1

cn ϕn (η)

(3.283)

3.8

Beams with Variable Cross Section

177

where N is an appropriately chosen integer, cn are unknown coefficients, and ϕ n is a member of a set of N independent trial functions. The quantity N is selected as that integer for which  converges to within a specified value; that is, in going from a solution using N terms to a solution using N + 1 terms,  changes by less than a specified amount. There is some ‘art’ to using the Rayleigh-Ritz method, for the selection of ϕ n is not straightforward. There often exist many sets of trial functions. One may select one set over another set because one set converges faster than the other; that is, the value of N is smaller. Also, it is not necessary that ϕ n satisfy all of the boundary conditions, but they must satisfy at least the geometric boundary conditions, which in the case of the beam are Y and Y . In many cases, ϕ n are chosen as those functions that are the solution to a similar geometry with the same boundary conditions. We shall discuss our choice of ϕ n after the formulation of the Rayleigh-Ritz method is completed. Upon substituting Eq. (3.283) into Eq. (3.282), we obtain6 EI   = cp cl Gpl 2L

(3.284)

Gpl = 4 I1pl − I2pl

(3.285)

N

N

p=1 l=1

where

and

1 I1,pl =

a (η) ϕp (η) ϕl (η) dη + mL ϕp (0) ϕl (0) 0

1 I2,pl =

i (η) ϕp (η) ϕl (η) dη + So

0

1

(3.286) ϕp (η) ϕl (η) dη

0

and the prime denotes the derivative with respect to η. It is seen from Eq. (3.286) that I1,pl = I1,lp and I2,pl = I2,lp ; that is, these quantities are symmetric. The Rayleigh-Ritz method states that a necessary condition for a stationary value of  is attained when ∂ =0 ∂cn 6

Note that

n = 1, 2, ...N.

⎞2 ⎛ ⎞ ⎛  N N N N  N     ⎠ ⎠ ⎝ ⎝ cp ϕp = cp ϕp cl ϕl = cp cl ϕp ϕl . p=1

p=1

l=1

p=1 l=1

(3.287)

178

3

Thin Beams: Part I

Substituting Eq. (3.284) into Eq. (3.287) and performing the indicated operation,7 the following system of N equations in terms of N unknown constants cl is obtained N 

cl Gnl = −

l=1

N 

cl I2,nl + 4

l=1

N 

cl I1,nl = 0 n = 1, 2, ..., N

(3.288)

l=1

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

(3.289)

where ⎡   ⎢ ⎢ Ij = ⎢ ⎣

···

Ij,11 Ij,21 .. .

Ij,12 Ij,22

Ij,N1

Ij,N2

···

Ij,1N Ij,2N .. .

⎤ ⎥ ⎥ ⎥ j = 1, 2 ⎦

Ij,NN

⎧ c1 ⎪ ⎪ ⎪ ⎨ c2 {c} = . ⎪ ⎪ .. ⎪ ⎩ cN

⎫ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎭

.

It is seen that Eq. (3.289) is a standard eigenvalue formulation from which l and the corresponding mode shapes can be determined, once I1,nl and I2,nl have been determined. We shall now consider a tapered cantilever beam with an axial load So and with a mass attached to its free end. There are several candidates for the choice of the trail function. Depending on the taper one could, for example, use the mode shape Yn(3) (η) given in Table 3.2, or the mode shapes for a linear tapered beam given by Eq. (3.268), or the mode shapes for the exponentially tapered beam given by Eq. (3.277). These functions satisfy the boundary conditions and they are orthogonal functions so that I1 nl = 0, n = l. We choose instead a trial function that takes advantage of readily available computational programs.8 We shall obtain numerically the solution to the static version of the system under consideration and assume that it is subject to a static spatially varying load. The spatially varying load is selected so that we can generate N independent numerically obtained trial functions Y(n) whose magnitude varies with respect to η such that it simulates the nth mode shape of a cantilever beam. Thus, we start with Eq. (3.54), set P = Vo = Kf = mi = Ki = Ko = 0, 7

Note that

⎛ ⎞ ⎤ ⎡ , N N N N N N     1 1 1 ∂ ⎝  cp cl Gpl ⎠ = ⎣ cl Gnl + cp Gpn ⎦ = cl Gnl + cl Gln 2 ∂cn 2 2 p=1 l=1

=

1 2

,

l=1

p=1

N 

N 

l=1

cl Gnl +

l=1

cl Gnl =

l=1

where we have used the fact that Gpl is a symmetric quantity. 8 In our implementation, the MATLAB function bvp4c is used.

N  l=1

cl Gnl

l=1

3.8

Beams with Variable Cross Section

179

remove the time dependent term, set S = So , where So is a constant, and replace f with a static spatially varying loading to obtain d2 dη2



d2 Y (n) i (η) dη2

 − So

d 2 Y (n) = sin ((n − 0.5) π (1 − η)) dη2

n = 1, 2, ...N. (3.290)

In Eq. (3.290), i (η) is given by Eq. (3.248) for a beam with a linear taper and by Eq. (3.270) for an exponential taper. We require that the solution to Eq. (3.290) satisfy the boundary conditions for a cantilever beam with an axial tensile load. Thus, at η = 1, ! Y (n) !η=1 = 0 ! (3.291) dY (n) !! =0 ! dη η=1 and at η = 0



! d2 Y (n) !! =0 dη2 !η=0 dY (n) d3 Y (n) − So i (η) 3 dη dη

η=0

(3.292)

= 0.

The solution to Eqs. (3.290) to (3.292) is our trial function; that is, ϕn (η) = Y (n) (η). Since Y (n) (η) is in the form of an array of numerical values, we use standard numerical integration techniques to obtain I1nl and I2nl . This procedure for obtaining the trial function has been verified by comparing the results from this procedure with the results from the analytical solutions obtained previously. The results of such a comparison are shown in Table 3.10. It is seen from these results that there is excellent agreement between the exact solution and those obtained from the Rayleigh-Ritz solution. However, it is noted that to obtain this type of agreement for the case when mL > 0, one must use twice the number of terms in the RayleighRitz solution than is used for the case when mL = 0. Consequently, we can expect that the implementation of the Rayleigh-Ritz procedure as described here will yield values very close to their true values. It is noted that obtaining the trial functions Y (n) (η) in the manner indicated has several advantages. These functions satisfy the boundary conditions when mL = 0, they satisfy the ordinary differential equation without the inertia term but with relations i (η) and a (η) that are under consideration, and there are no restrictions on the form of i (η) and a (η). There is one caveat: one may not always be able to obtain a numerical solution to the static equations; that is, a converged solution cannot be obtained, or the computation time needed to obtain one solution may be too long. We have plotted the approximate results for a linear double-tapered beam in Fig. 3.32 and those for an exponentially double-tapered beam in Fig. 3.33 using

180

3

Thin Beams: Part I

Table 3.10 Comparison of the lowest natural frequency coefficients of a cantilever beam obtained from the Rayleigh-Ritz procedure using the solutions to Eqs. (3.390) to (3.392) as the trial functions to those obtained from the analytical solutions: Eq. (3.266) for a linear taper and Eq. (3.275) for an exponential taper Taper

mL

α

β

1

Source

Linear, single

0.0

0.1

1.0

2.1522 2.1519

Eq. (3.289), N = 3 Eq. (3.266)

Linear, single

0.5

0.1

1.0

1.0881 1.0851

Eq. (3.289), N = 6 Eq. (3.266)

Linear, double

0.0

0.1

0.1

2.6846 2.6842

Eq. (3.289), N = 3 Eq. (3.266)

Linear, double

0.5

0.1

0.1

0.88517 0.87866

Eq. (3.289), N = 6 Eq. (3.266)

Exponential, single

0.0

1.0

0.1

2.6060 2.6061

Eq. (3.289), N = 3 Eq. (3.275)

Exponential, single

0.5

1.0

0.2

1.3601 1.3598

Eq. (3.289), N = 6 Eq. (3.275)

N = 6 in both cases. It is seen from these figures that the variation of the natural frequency coefficient of the linear and exponentially double-tapered beams are fundamentally different with respect to the change in 1 as a function of α and β. In Fig. 3.34, the lowest natural frequency coefficient of a cantilever beam with a single and double linear tapered as a function of a tensile axial force is

3 2.8

Ω1

2.6 2.4 2.2 2 1.8 0 0.2 0.4 0.6 0.8 β

1

1

0.6

0.8

0.4

0.2

0

α

Fig. 3.32 Lowest natural frequency coefficient of a double-tapered cantilever beam with unequal linear tapers. The dotted line indicates the values of the 1 when α = β, which is also shown in Fig. 3.29 for the curve labeled m = 0

3.8

Beams with Variable Cross Section

181

3

Ω1

2.5

2

1.5 0.2

0.2 0.4

0.4 0.6

0.6 0.8

0.8 1

1

β

α

Fig. 3.33 Lowest natural frequency coefficient of a double-tapered cantilever beam with unequal exponential tapers. The dotted line indicates the values of the 1 when α = β and the edge of the surface at α = 1 is given in Fig. 3.31 for the curve labeled me = 0

6 5.5 5

α = 0.1 β = 0.1

4.5 α = 1 β = 0.1

Ω1

4

α = 0.1 β = 1

3.5

α=1 β=1

3 2.5 2

α 0.1 0.1 1 1

β 0.1 1 0.1 1

Sb −1.36 −0.321 −1.62 −2.47

1.5 1

0

10

20

30

40

50

So

Fig. 3.34 Lowest natural frequency coefficient of a cantilever beam with single and double linear tapers as a function of a tensile axial force

182

3

Thin Beams: Part I

presented. The values at which buckling occurs, denoted of Sb , are listed in Fig. 3.34 for the corresponding values of α and β. For a beam of constant cross section (α = β = 1) Sb = −π 2 /4. Also, as So → Sb , 1 → 0.

3.8.6 Triangular Taper: Application to Atomic Force Microscopy A geometry that is sometimes used in commercial atomic force microscopy cantilevers is the triangular shape shown in Fig. 3.35 (Chen et al. 1994; Abramovitch et al. 2007). This geometry is used because of its high lateral stiffness. The thickness of the beam is constant; that is, z (η) = 1. The width of the shape shown in Fig. 3.35 is given by y (η) = η = αo

0 ≤ η ≤ η1

(3.293)

η1 < η ≤ 1

where αo = 2co /bo ≤ 1 and η1 = L1 /L ≥ αo . We have a solid triangle when η1 = 1, which is equivalent to having co = bo /2. Thus, from Eq. (3.248) a (η) = y (η)

(3.294)

i (η) = y (η) .

Using the Rayleigh-Ritz procedure described in Section 3.8.5 for N = 5, the results shown in Fig. 3.36 are obtained. It is seen that changes in the first natural frequency coefficient vary in a complicated manner as a function of α o and η1 . We also note that for αo = 0.1 and αo = 0.25, 1 has a minimum value around η1 = 0.55.

co

bo

L1

Fig. 3.35 Geometry of a constant height triangular beam

co L

3.8

Beams with Variable Cross Section

183

2.8

αo = 0.8

2.6 αo = 0.5

Ω1

2.4

2.2

αo = 0.25

2

αo = 0.1

1.8

1.6 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

η1

Fig. 3.36 Lowest natural frequency coefficients for a triangular beam of constant thickness. The triangles are the beam shapes at η1 = αo

3.8.7 Constant Cross Section with a Step Change in Properties The geometry of a beam with a step change in its cross section occurs in certain types of piezoelectric beam energy harvesters and for certain configurations of AFM probes (Salehi-Khojin et al. 2008) and fiber optic probes (Yang et al. 1997). Since the cross sections are constant in each portion of the beam, we shall consider the case of two independent constant cross section beams. Referring to Fig. 3.37, one portion of the beam is clamped at its left end and the other portion of the beam is free and is carrying a concentrated mass ML at this free end. The common boundary of the two portions is subject to a set of continuity conditions. We shall see that the formulation and solution to these systems of beams is similar to that of a beam carrying a rigid extended mass at an in-span location. In fact, the continuity conditions

E1, A1, I1, ρ1

E2, A2, I2, ρ 2 w1

x1

Fig. 3.37 Geometry and notation for a beam with an abruptly changing constant cross section

ML

w2 x2

a

c L

184

3

Thin Beams: Part I

can be obtained directly as a special case of a beam carrying an extended rigid mass at an in-span location. For the geometry described in Fig. 3.37, we use the dimensional form of the governing equations given by Eq. (3.215); that is, El Il

∂ 4 wl ∂ 2 wl + ρ A = 0 l = 1, 2 l l ∂t2 ∂xl4

(3.295)

where wl = wl (xl , t). The boundary conditions at x1 = 0 are w1 |x1 =0 = 0 ! ∂w1 !! =0 ∂x1 !x1 =0

(3.296)

and those at x2 = 0 are obtained from Eq. (3.51a) as ! ∂ 2 w2 !! =0 ! ∂x22 !x =0 2 ! ! ! 3 ∂ w2 ! ∂ 2 w2 !! = −ML 2 ! . E2 I2 3 ! ∂t x2 =0 ∂x2 !x =0

(3.297)

2

The continuity conditions can be obtained from Eq. (3.217) by setting ME = JE = b = do = 0 to arrive at w1 |x1 =a ! ∂w1 !! ∂x1 !x1 =a ! ∂ 2 w1 !! E1 I1 2 ! ∂x1 !x =a 1 ! ∂ 3 w1 !! −E1 I1 3 ! ∂x1 ! x1 =a

= w2 |x2 =c ! ∂w2 !! = − ∂x2 !x2 =c ! ∂ 2 w2 !! = E2 I2 2 ! ∂x2 !x =c 2 ! ∂ 3 w2 !! = E2 I2 3 ! . ∂x2 !

(3.298)

x2 =c

The following parameters are introduced x2 x1 a c , ξ = , aL = , cL = L L L L t ρ 2 A2 ρ1 A1 to = L 2 s, τ = , α = , E1 I 1 to ρ 1 A1 ML λ4 = αβ, m = ρ1 A1 L kg, mL = m η=

β=

E1 I1 E2 I2

(3.299)

3.8

Beams with Variable Cross Section

185

where L = a+c and, therefore, cL = 1−aL . The quantity m is the mass of a fictitious beam of length L that has the properties of beam 1. This quantity appears when the non dimensional conversion is performed. However, in order to compare beams with different attachment ratios, it is better to consider a mass ratio that is proportional to the mass ratio of the actual beam. The mass of the beam is mb = ρ1 A1 a + ρ2 A2 c = ρA1 L (aL + αcL ) = ρA1 L (aL + α [1 − cL ]). Therefore, we define the mass ratio as mL =

ML mco

(3.300)

where co = aL + α (1 − aL ) . Before proceeding, it is noted that for a beam in which the cross section of each beam is rectangular ρ2 b2 h2 ρ 1 b1 h1      3 b1 h1 E1 β= . E2 b2 h2 α=

(3.301)

If the two sections of the beams are made of the same material, then b2 h2 b1 h1    3 h1 b1 β= . b2 h2 α=

(3.302)

We now examine several special cases of Eq. (3.302). If b2 /b1 = h2 /h1 , then Eq. (3.302) becomes  α1 = 

h2 h1

h1 β1 = h2

2 4

(3.303) 1 = 2. α1

This case is shown in Fig. 3.38a. On the other hand, if b2 /b1 = 1, that is, only the height of the beam sections changes, then α2 = β2 =

h2 h1 1 α23

(3.304) .

186 Fig. 3.38 Three different stepped beams (a) b2 /b1 = h2 /h1 and α1 = (h2 /h1 )2 and β1 = 1/α12 (b) b2 /b1 = 1 and α2 = h2 /h1 and β2 = 1/α23 and (c) h2 /h1 = 1 and α3 = b2 /b1 and β3 = 1/α3

3

b1

b1

b1

h1

h1

h1

Thin Beams: Part I

b2

b1

b2 (a)

h1

h2

h2 (b)

(c)

This case is shown in Fig. 3.38b. Finally, if h2 /h1 = 1, that is, only the width of the beam sections changes, b2 b1 b1 1 β3 = = . b2 α3

α3 =

(3.305)

This case is shown in Fig. 3.38c. If the two portions of the beam are circular cylinders of radius r1 , 0 ≤ x1 ≤ L1 and r2 , 0 ≤ x2 ≤ L2 , then, when ρ1 = ρ2 and E1 = E2 ,  αc =

r2 r1

βc =

1 . αc2

2

Thus, the case of beams with circular cylindrical portions can be represented by Eq. (3.303). Substituting Eq. (3.299) into Eqs. (3.295), we obtain the governing equations ∂ 2 w1 ∂ 4 w1 + =0 ∂η4 ∂τ 2 2 ∂ 4 w2 4 ∂ w2 + λ = 0. ∂ξ 4 ∂τ 2

(3.306)

Substituting Eq. (3.299) into the boundary conditions given by Eqs. (3.296) and (3.297), the boundary conditions at η = 0 become w1 |η=0 = 0 ! ∂w1 !! =0 ∂η !η=0 and those at ξ = 0 become

(3.307)

3.8

Beams with Variable Cross Section

187

! ∂ 2 w2 !! =0 ∂ξ 2 !ξ =0 ! ! ∂ 2 w2 !! ∂ 3 w2 !! = − m βc . L o ∂ξ 3 !ξ =0 ∂τ 2 !ξ =0

(3.308)

Substituting Eq. (3.299) into Eqs. (3.298), the continuity conditions become w1 |η=aL = w2 |ξ =cL ! ! ∂w1 !! ∂w2 !! = − ∂η !η=aL ∂ξ !ξ =cL  2  2 ∂ w1 ∂ w2 β = 2 ∂η η=aL ∂ξ 2 ξ =cL !  3 ∂ w1 ∂ 3 w2 !! −β = . ∂η3 η=aL ∂ξ 3 !ξ =cL

(3.309)

We assume that the system is undergoing harmonic oscillations at a frequency ω of the form w1 (η, τ ) = W1 (η) ej w2 (ξ , τ ) = W2 (ξ ) ej

2τ

2τ

(3.310)

where 2 = ωto . Substituting the first of Eq. (3.310) into the first equation of Eq. (3.306) and the second of Eq. (3.310) into the second equation of Eq. (3.306), we obtain ∂ 4 W1 − 4 W1 = 0 ∂η4 ∂ 4 W2 − λ4 4 W2 = 0. ∂ξ 4

(3.311)

The substitution of Eq. (3.310) into the boundary conditions given by Eq. (3.307) yields W1 |η=0 = 0 ! ∂W1 !! =0 ∂η !η=0

(3.312)

and substitution of Eq. (3.310) into the boundary conditions given by Eq. (3.308) gives

188

3

Thin Beams: Part I

! ∂ 2 W2 !! =0 ∂ξ 2 !ξ =0 ! ! ∂ 3 W2 !! = mL βco 4 W2 !ξ =0 . ! 3 ∂ξ ξ =0

(3.313)

The continuity conditions given by Eq. (3.309) become W1 |η=aL = W2 |ξ =cL ! ! ∂W2 !! ∂W1 !! = − ∂η !η=aL ∂ξ !ξ =cL ! ! ∂ 2 W1 !! ∂ 2 W2 !! β = ∂η2 !η=aL ∂ξ 2 !ξ =cL ! ! ∂ 3 W1 !! ∂ 3 W2 !! β =− . ∂η3 !η=aL ∂ξ 3 !ξ =cL

(3.314)

From Eq. (3.228), the solutions to Eq. (3.311) are W1 (η) = W1 (0) Q (η) + W1 (0) R (η)/ + W1 (0) S (η)/2 + W1 (0) T (η)/3

0 ≤ η ≤ aL

W2 (ξ ) = W2 (0) Q (λξ ) + W2 (0) R (λξ )/(λ) + W2 (0) S (λξ )/(λ)2 + W2 (0) T (λξ )/(λ)3

0 ≤ ξ ≤ cL (3.315)

where Q (η), etc. are given by Eq. (C.19) of Appendix C. Using Eq. (3.312) in the first equation of Eq. (3.315) and Eq. (3.313) in the second equation of Eq. (3.315), we obtain W1 (η) = C1 S (η)/2 + C2 T (η)/3 0 ≤ η ≤ aL  mL co λ T (λξ ) W2 (ξ ) = D1 Q (λξ ) + α + D2 R (λξ )/(λ)

(3.316)

0 ≤ ξ ≤ cL

where, for convenience, we have replaced in Eq. (3.316) W1 (0) with C1 , etc. The unknown quantities C1 , C2 , D1 , and D2 , are determined by substituting Eq. (3.316) into Eq. (3.314). When this substitution is performed, we arrive at the following system of equations [G ()] {V} = 0 where

(3.317)

3.8

Beams with Variable Cross Section

⎡

g11 (, aL ) ⎢ g21 (, aL ) [G ()] = ⎢ ⎣ g31 (, aL ) g41 (, aL ) ⎧ ⎫ ⎪ ⎪ C1 ⎪ ⎪ ⎨ ⎬ C2 {V} = D1 ⎪ ⎪ ⎪ ⎩ ⎪ ⎭ D2

189

g12 (, aL ) g22 (, aL ) g32 (, aL ) g42 (, aL )

g13 (λ, cL ) g23 (λ, cL ) g33 (λ, cL ) g43 (λ, cL )

⎤ g14 (λ, cL ) g24 (λ, cL ) ⎥ ⎥ g34 (λ, cL ) ⎦ g44 (λ, cL )

(3.318)

and g11 (, aL ) = S (aL )/2 g12 (, aL ) = T (aL )/3 mL co λ T (λcL ) α g14 (λ, cL ) = − R (λcL )/(λ) g21 (, aL ) = R (aL )/ g13 (λ, cL ) = − Q (λcL ) −

g22 (, aL ) = S (aL )/2 g23 (λ, cL ) = λT (λcL ) + g24 (λ, cL ) = Q (λcL ) g31 (, aL ) = βQ (aL )

mL co (λ)2 S (λcL ) α (3.319)

g32 (, aL ) = βR (aL )/ g33 (λ, cL ) = − (λ)2 S (λcL ) − g34 (λ, cL ) = − λT (λcL ) g41 (, aL ) = βT (aL ) g42 (, aL ) = βQ (aL ) g43 (λ, cL ) = (λ)3 R (λcL ) +

mL co (λ)3 R (λcL ) α

mL co (λ)4 Q (λcL ) α

g44 (λ, cL ) = (λ)2 S (λcL ) . The values of  = n for which det [G (n )] = 0 are the natural frequency coefficients. The mode shapes corresponding to these natural frequency coefficients are 0 ≤ η ≤ aL W1n (η) = S (n η) /2n + C2n T (n η) /3n  mL co λn W2n (ξ ) = D1n Q (λn ξ ) + T (λn ξ ) α + D2n R (λn ξ )/(λn ) where C2n , D1n , and D2n are determined from

0 ≤ ξ ≤ cL

(3.320)

190

3

Thin Beams: Part I

⎫ ⎧ ⎫ ⎤⎧ g22 (n , aL ) g23 (λn , cL ) g24 (λn , cL ) ⎨ C2n ⎬ ⎨ −g21 (n , aL ) ⎬ ⎣ g32 (n , aL ) g33 (λn , cL ) g34 (λn , cL ) ⎦ D1n = −g31 (n , aL ) ⎭ ⎩ ⎩ ⎭ g42 (n , aL ) g43 (λn , cL ) g44 (λn , cL ) −g41 (n , aL ) D2n (3.321) ⎡

and we have arbitrarily set C1n = 1. These results reduce to the case of a uniform cantilever beam with a mass at its free end in either of two ways. For the first way, we set aL = 1 (cL = 0) and Eq. (3.318) yields ! ! g11 (, 1) ! ! g (, 1) |G| = !! 21 ! g31 (, 1) ! g41 (, 1) ! ! g11 (, 1) ! ! g (, 1) = !! 21 ! g31 (, 1) ! g41 (, 1)

g12 (, 1) g22 (, 1) g32 (, 1) g42 (, 1) g12 (, 1) g22 (, 1) g32 (, 1) g42 (, 1)

g13 (λ, 0) g23 (λ, 0) g33 (λ, 0) g43 (λ, 0) ! −1 0 !! 0 1 !! 0 0 !! mL β4 0 !

! g14 (λ, 0) !! g24 (λ, 0) !! g34 (λ, 0) !! g44 (λ, 0) ! (3.322)

= g32 (, 1) g41 (, 1) − g31 (, 1) g42 (, 1) + mL β4 (g11 (, 1) g32 (, 1) − g12 (, 1) g31 (, 1))   = β 2 −Q2 () + R () T () + mL  (R () S () − Q () T ()) where mL = ML /ρ1 A1 L. In arriving at Eq. (3.322), we have used the fact that co = 1, R (0) = S (0) = T (0) = 0, and Q (0) = 1. It is seen that Eq. (3.322) equals, within a multiplicative constant, D(8) (n ) when B2n = 0 and A2n = −mL 4n . For comparison purposes in what follows, we denote these frequency coefficients (n) aL =1 , which are the natural frequency coefficients of a uniform beam having the properties of beam 1. For the second way, we set aL = 0 (cL = 1) and Eq. (3.318) yields ! ! g11 (, 0) g12 (, 0) g13 (λ, 1) ! ! g (, 0) g22 (, 0) g23 (λ, 1) |G| = !! 21 ! g31 (, 0) g32 (, 0) g33 (λ, 1) ! g41 (, 0) g42 (, 0) g43 (λ, 1) ! ! ! 0 0 g13 (λ, 1) g14 (λ, 1) ! ! ! ! 0 0 g23 (λ, 1) g24 (λ, 1) ! ! ! =! ! ! 1 0 g33 (λ, 1) g34 (λ, 1) ! ! 0 1 g43 (λ, 1) g44 (λ, 1) !

! g14 (λ, 1) !! g24 (λ, 1) !! g34 (λ, 1) !! g44 (λ, 1) !

= g13 (λ, 1) g24 (λ, 1) − g14 (λ, 1) g23 (λ, 1) = −Q2 (λ) + R (λ) T (λ) + mL λ (R (λ) S (λ) − Q (λ) T (λ)) . (3.323)

3.8

Beams with Variable Cross Section

191

In arriving at Eq. (3.323), we have used the fact that co = α, R (0) = S (0) = T (0) = 0, and Q (0) = 1. We see that Eq. (3.323) equals D(8) (n ) when B2n = 0 and A2n = −mL 4n and we replace n with λn . However, it is noted that at the limit when aL = 0, mL = ML /ρ2 A2 L. In addition, when Eq. (3.323) is solved, the solution gives the value of λn = rn , where rn is the nth root. Consequently, at (n) aL = 0, n = rn /λ = rn /(αβ)0.25 . This solution is denoted aL =0 , which is the natural frequency coefficient of a uniform beam having the properties of beam 2. (n) (n) 0.25 However, since rn = (n) . This relation aL =1 , we have that aL =0 = aL =1 (αβ) seems to indicate that the natural frequency coefficients of a stepped beam vary from that of a uniform beam with the properties of beam 2 to that of a uniform beam with properties of beam 1. However, this turns out not to be the case; it will be seen that for certain combinations of values for aL , α, and mL , the natural frequency coefficient can be larger than that obtained for either of these values. The determinant of [G ()] has been evaluated to determine the lowest natural frequency coefficient as a function aL for the case where both portions of the beam are the same material; that is, where α and β are given by Eq. (3.302). They are shown in Figs. 3.39 to 3.41 for the mass ratios mL = 0, 0.2, and 1.0 and for the  2 following three sets of α and β: (1) α = h2 h1 , β = 1 α 2 ; (2) α = h2 h1 , β = 1 α 3 ; and (3) α = b2 b1 , β = 1 α. For these nine combinations of parameters, we have plotted in Fig. 3.39 the case where α = 0.4, in Fig. 3.40 the case where α = 0.6, and in Fig. 3.41 the case where α = 0.8. There are several interesting features appearing in these figures. First, in all cases, there is a maximum value of 1 that occurs at value of aL other than at aL = 0 or 2.5

mL = 0 2

Ω1

mL = 0.2 1.5 mL = 1

β = 1/α, α = b2/b1

1

β = 1/α2, α = (h2/h1)2 β = 1/α3, α = h2/h1 0.5

0

0.2

0.4

0.6

0.8

1

aL

Fig. 3.39 Lowest natural frequency coefficient of a stepped cantilever beam as a function of aL for α = 0.4, mL = 0, 0.2, and 1, and the three α and β combinations indicated

192

3

Thin Beams: Part I

2.5

mL = 0 2

Ω1

mL = 0.2 1.5 mL = 1

β = 1/α, α = b2/b1

1

β = 1/α2, α = (h2/h1)2 β = 1/α3, α = h2/h1 0.5

0

0.2

0.4

0.6

0.8

1

aL

Fig. 3.40 Lowest natural frequency coefficient of a stepped cantilever beam as a function of aL for α = 0.6, mL = 0, 0.2, and 1, and the three α and β combinations indicated

2.5

mL = 0 2

Ω1

mL = 0.2 1.5 mL = 1

β = 1/α, α = b2/b1

1

β = 1/α2, α = (h2/h1)2 β = 1/α3, α = h2/h1 0.5

0

0.2

0.4

0.6

0.8

1

aL

Fig. 3.41 Lowest natural frequency coefficient of a stepped cantilever beam as a function of aL for α = 0.8, mR = 0, 0.2, and 1, and the three α and β combinations indicated

3.8

Beams with Variable Cross Section

193

Table 3.11 Values of the geometric parameters that produce the maximum first natural frequency coefficient for a stepped beam for the three different beam geometries shown in Fig. 3.38 and for several values of mL Stepped beam geometry α2 = h2 h1 , β2 = 1 α23 (Fig. 3.38b)

 2 α1 = h2 h1 , β1 = 1 α12 (Fig. 3.38a)

α3 = b2 b1 , β3 = 1 α3 (Fig. 3.38c)

mL

1, max

α 2, max

aL, max

1, max

α 1, max

aL, max

1, max

α 3, max

aL, max

0.0 0.1 0.2 0.5 1.0

2.279 1.887 1.725 1.476 1.282

0.162 0.346 0.395 0.452 0.486

0.740 0.751 0.765 0.786 0.799

2.527 1.962 1.774 1.503 1.300

0.063 0.242 0.289 0.350 0.388

0.682 0.702 0.716 0.735 0.746

3.560 2.137 1.891 1.571 1.347

0.0035 0.116 0.155 0.212 0.252

0.510 0.590 0.601 0.605 0.601

aL = 1. Therefore, for a given value of mL and a given stepped beam geometry, the values of α and α L that result in a maximum value of 1 have been determined and tabulated in Table 3.11. Second, irrespective of the value of the attached mass or the value of aL , the value for 1 for the case of a beam where h2 /h1 = 1, that is, Case 3 above, is always equal to or larger than the values of 1 for the other two cases. This is to be expected because, of the three configurations considered, this is the stiffest one. It is also remarked that for Case 3 the value of the natural frequency coefficient is symmetrical about aL = 0.5 for all three values of mL since, for this case, αβ = 1. In Fig. 3.42, the mode shapes for the lowest three natural frequencies have been plotted for α = 0.4, aL = 0.4 and 0.8, and β = 1/α, β = 1/α 2 , and β = 1/α 3 . For comparison purposes, the mode shapes for a cantilever beam of constant cross section have also been plotted. It is seen that except for the mode shape corresponding to 1 for aL = 0.8 there are substantial differences between the mode shapes for all three types of stepped beams and those for beams with constant cross section.

3.8.8 Stepped Beam with an In-Span Rigid Support Consider the case of a stepped cantilever beam in which a rigid support is placed at the location where the abrupt change in properties occurs; that is, at aL . For this configuration, the continuity conditions given by Eq. (3.314) become W1 |η=aL = 0 W2 |ξ =cL = 0 ! ! ∂W2 !! ∂W1 !! = − ∂η !η=aL ∂ξ !ξ =cL ! ! ∂ 2 W1 !! ∂ 2 W2 !! β = . ∂η2 !η=aL ∂ξ 2 !ξ =cL Substituting Eq. (3.316) into Eq. (3.324) results in

(3.324)

194 Fig. 3.42 Mode shapes of cantilever stepped beams corresponding to the lowest three natural frequencies for aL = 0.4 and 0.8, α = 0.4 and for (a) β = 1/α, (b) β = 1/α 2 , and (c) β = 1/α 3 . The shapes shown with dashed lines are the mode shapes for a cantilever beam of constant cross section

3

Thin Beams: Part I

aL = 0.4

aL = 0.8

Ω1 = 2.253

Ω1 = 2.094

Ω2 = 4.784

Ω2 = 5.021

Ω3 = 7.846

(a)

Ω3 = 8.054

aL = 0.4

aL = 0.8

Ω1 = 2.08

Ω1 = 2.094

Ω2 = 4.201

Ω2 = 4.973

Ω3 = 7.638

Ω3 = 6.745

(b) aL = 0.4

aL = 0.8

Ω1 = 1.818

Ω1 = 2.093

Ω2 = 3.784

Ω2 = 4.84

Ω3 = 5.655

Ω3 = 6.894

(c)

3.8

Beams with Variable Cross Section

195

[G ()] {V} = 0 where

(3.325)

⎡

⎤ 0 0 g11 (, aL ) g12 (, aL ) ⎢ 0 0 g13 (λ, cL ) g14 (λ, cL ) ⎥ ⎢ ⎥ [G ()] = ⎢ ⎥ ⎣ g21 (, aL ) g22 (, aL ) g23 (λ, cL ) g24 (λ, cL ) ⎦ g31 (, aL ) g32 (, aL ) g33 (λ, cL ) g34 (λ, cL )

(3.326)

and {V} and gij , respectively, are given by Eqs. (3.318) and Eq. (2.319). When other boundary conditions are considered, one starts with Eq. (3.315) and uses the boundary conditions at η = 0 and at ξ = 0 to reduce the equations to the form given by Eq. (3.316). Then the continuity conditions given by Eq. (3.324) are used to obtain the characteristic determinant. The values of  = n for which det [G (n )] = 0 are the natural frequency coefficents. This determinant has been evaluated to obtain the lowest natural frequency coefficient 1 as a function aL for the case where both portions of the beam are the same material; that is, where α and β are given by Eq. (3.302). The quantity aL is both the location of the abrupt change in the properties of the two portions of the beam and the location of the rigid support. The results are shown in Figs. 3.43 to 3.45 for the mass ratios mL = 0 and 0.5 and for the following three 1.6 1.4 mL = 0

1.2

Ω1/π

1 mL = 0.5

0.8 0.6 β = 1/α, α = b2/b1

0.4

β = 1/α2, α = (h2/h1)2 0.2 0

β = 1/α3, α = h2/h1 0

0.2

0.4

0.6

0.8

1

aL

Fig. 3.43 Lowest natural frequency coefficient of a stepped cantilever beam with an in-span rigid support at aL as a function of aL for α = 0.4, mL = 0 and 0.5, and the three α and β combinations indicated

196

3

Thin Beams: Part I

1.6 1.4 mL = 0

1.2

Ω1/π

1 mL = 0.5

0.8 0.6

β = 1/α, α = b2/b1

0.4

β = 1/α2, α = (h2/h1)2 0.2 0

β = 1/α3, α = h2/h1 0

0.2

0.4

aL

0.6

0.8

1

Fig. 3.44 Lowest natural frequency coefficient of a stepped cantilever beam with an in-span rigid support at aL as a function of aL for α = 0.6, mL = 0 and 0.5, and the three α and β combinations indicated 1.6 1.4 mL = 0

1.2

Ω1/π

1 mL = 0.5

0.8 0.6

β = 1/α, α = b2/b1

0.4

β = 1/α2, α = (h2/h1)2 0.2 0

β = 1/α3, α = h2/h1 0

0.2

0.4

0.6

0.8

1

aL

Fig. 3.45 Lowest natural frequency coefficient of a stepped cantilever beam with an in-span rigid support at aL as a function of aL for α = 0.8, mL = 0 and 0.5, and the three α and β combinations indicated

3.9

Elastically Connected Beams

197

sets of α and β: (1) α = (h2 /h1 )2 , β = 1/α 2 ; (2) α = h2 /h1 , β = 1/α 3 ; and (3) α = b2 /b1 , β = 1/α. For these nine combinations of parameters, we have plotted in Fig. 3.43 the case where α = 0.4, in Fig. 3.44 the case where α = 0.6, and in Fig. 3.45 the case where α = 0.8. From the curves presented in these figures, we make the following observations. For the case of mL = 0, the maximum values of the frequency coefficient occur in the vicinity of the node point of the second natural of a cantilever beam with a uniform cross section, which from Case 3 of Table 3.4 is ηnode = 0.783. The values of the frequency coefficients at aL = 0 are the same values as those given in Figs. 3.39 to 3.41 at aL = 0 and with the corresponding parameters, since this end is the clamped end. When the support is located at the free end of the beam; that is, at aL = 1, the beam becomes a beam clamped at one end and hinged at the other. In this case, the mass has no effect since the displacement is zero and from Case 5 of Table 3.5, the lowest natural frequency coefficient is 1 /π = 1.2499.

3.9 Elastically Connected Beams 3.9.1 Introduction Consider the system composed of two beams shown in Fig. 3.46, where each of the beams has a constant cross section and a length L. The beams have different cross section properties and different physical properties. The two beams are connected by a continuous elastic spring of constant kf . In addition, each of the beams has a mass Mi attached at x = Lm with a spring of constant ki connecting them. The boundary conditions are the same for each beam and are given by Eqs. (3.51) and (3.52) with ML = JL = Po = p = 0. The foundation spring and the spring attached to the masses are compressed or stretched based on the relative displacements of the respective beams. The governing equation for each beam is obtained from Eq. (3.47) with Vo = p = Po = f = ko = 0. Thus,

ktL

MR, JR

Mi

ktR

E1, A1, I1, ρ1 kL

ki

ktL

x

kf

...

...

w1

MR, JR

Mi

kR ktR

E2, A2, I2, ρ2 x

Fig. 3.46 Two different beams of the same length connected by elastic springs and subject to the same boundary conditions

kL

w2 Lm

L

kR

198

3

Thin Beams: Part I

 ∂ 2 w1 ∂ 4 w1  [ρ + + k + k δ − L ) (w − w ) A + M δ − L )] =0 (x (x f i m 1 2 1 1 i m ∂x4 ∂t2  ∂ 4 w2  ∂ 2 w2 =0 E2 I2 4 + kf + ki δ (x − Lm ) (w2 − w1 ) + [ρ2 A2 + Mi δ (x − Lm )] ∂x ∂t2 (3.327) E1 I1

where wj = wj (x, t) , j = 1, 2, and we have assumed that w1 > w2 so that beam 1 experiences a force in a direction opposite to that of w1 and beam 2 experiences a force in the direction of w2 . Equation (3.327) is converted into non dimensional form by introducing the following parameters  x η= , L α=

Lm ηm = , L

ρ 2 A2 , ρ 1 A1

β=

mb = ρ1 A1 L kg,

to = L

2

ρ 1 A1 s, E1 I1

τ=

E1 I1 , λ4 = αβ, E2 I2 Mi ki L 3 mi = , Ki = , mb E1 I1

t to (3.328)

Kf =

kf L 4 . E1 I 1

Using Eq. (3.328) in Eq. (3.327), we obtain  ∂ 4 w1  ∂ 2 w1 [1 + K + K δ − η − w δ − η =0 + + m (η (η ) (w ) )] f i m 1 2 i m ∂η4 ∂τ 2   ∂ 2w   ∂ 4 w2 mi 2 4 1 + + β K + K δ − η − w =0 δ − η + λ (η (η ) (w ) ) f i m 2 1 m ∂η4 α ∂τ 2 (3.329) where wl = wl (η, τ ), l = 1, 2 and we have used the relation δ (x) = δ (Lη) = δ (η)/|L|. It is assumed that the system of beams is undergoing harmonic oscillations of the form w1 (η, τ ) = W1 (η) e j

2τ

w2 (η, τ ) = W2 (η) e j

2τ

(3.330)

where 2 = ωto . Substituting Eq. (3.330) into Eq. (3.329), we arrive at  d4 W1  + K + K δ − η (η ) (W1 − W2 ) − 4 [1 + mi δ (η − ηm )] W1 = 0 f i m dη4     d 4 W2 mi 4 4 1 + δ − η − λ + β K + K δ − η − W  W2 = 0. (η ) (W ) ) (η f i m 2 1 m α dη4 (3.331)

3.9

Elastically Connected Beams

199

We now consider two special cases of Eq. (3.331). The first case is when kf = 0 and Mi = ki = 0, and the second case is when Mi = 0 and ki = 0 and kf = 0. For the latter case, it will be assumed that α = β = 1. For each case, a different solution method will be used.

3.9.2 Beams Connected by a Continuous Elastic Spring For the case of two beams connected by a continuous elastic spring, we set Mi = ki = 0 and Eq. (3.331) becomes ∂ 4 W1 + Kf (W1 − W2 ) − 4 W1 = 0 ∂η4 ∂ 4 W2 + βKf (W2 − W1 ) − λ4 4 W2 = 0. ∂η4

(3.332)

Equation (3.332) has been used as a relatively simple approximation to determine the natural frequencies of double-wall carbon nanotubes (Yoon et al. 2003; Zhang et al. 2007; Oniszczuk 2000). The case of elastically connected stretched beams has also been investigated (Kelly and Srinivas 2009). A solution to Eq. (3.322) shall be obtained for α = β = λ = 1. For this case, we assume a solution of the form Wj (η) =

∞ 

Ajn Yn(p) (η)

j = 1, 2

(3.333)

n=1 (p)

where Yn (η) is given in the last column of Table 3.3 and corresponds to the pth set coefficient for of boundary conditions. The quantity n(p) is the natural   frequency  which D(p) n(p) = 0, where the expression for D(p) n(p) is given in the fourth column of Table 3.3. Substituting Eq. (3.333) into Eq. (3.332) with β = λ = 1, we obtain ∞  

  4n(p) + Kf − 4 A1n − Kf A2n Yn(p) (η) = 0

n=1 ∞   

 −Kf A1n + 4n(p) + Kf − 4 A2n Yn(p) (η) = 0

(3.334)

n=1 (p)

where we have used the fact that Yn (η) is a solution to (p)

∂ 4 Yn − 4n(p) Yn(p) = 0. ∂η4

(3.335)

200

3

Thin Beams: Part I

Hence, for any value of n, Eq. (3.334) will be satisfied if Ajn , j = 1, 2, is a solution to , -  −Kf 4n(p) + Kf − 4 A1n = 0. (3.336) [K ()] {A} = A2n −Kf 4n(p) + Kf − 4 The natural frequency coefficients  for the elastically coupled beams are those values of n for which det [K (n )] = 0. Thus, n is determined from 2

4n(p) + Kf − 4n − Kf2 = 0

(3.337)

4n = 4n(p) + Kf ∓ Kf .

(3.338)

or

In Eqs. (3.337) and (3.338), the natural frequency coefficient is denoted n since it directly corresponds to the nth frequency coefficient of the uncoupled beam and (p) the corresponding mode shape Yn (η); that is, the solution is only applicable at a specific value of n. Therefore, there are two natural frequency coefficients, 4n = 4n(+) = 4n(p) + 2Kf 4n = 4n(−) = 4n(p) .

(3.339)

It is seen that 4n(−) corresponds to the case where kf = 0. In our case, it represents the vibration of the coupled beams whose displacements are in phase; that is, W1 (η) = W2 (η). We shall show subsequently that the natural frequency coefficient 4n(+) is the frequency coefficient when the displacements are out of phase; that is, W1 (η) = −W2 (η). Finally, since 2 = ωto , we obtain the natural frequency from the square root of the expressions given in Eq. (3.339). Regarding 4n(+) , the following observation is made. If, in Eq. (3.335), one were to consider a single beam on an elastic foundation, then 4n(p) would be replaced by 4n −Kf . Recall Eq. (3.197) with So = 0. Therefore, the natural frequency coefficient for one beam on an elastic foundation is 4n = 4n(p) + Kf . Returning the case of two elastically coupled beams, it is seen from Eq. (2.339) that when α = β = λ = 1, 4n = 4n(p) + 2Kf . Consequently, the effects of the out phase motion of the two elastically connected beams is equivalent to a single beam on an elastic foundation with twice the displacement of the foundation of one beam. This factor of two is because, for the elastically coupled beams, both ends of the foundation undergo an extension/compression; for one beam, only one end of the foundation is extended or compressed. Corresponding to n , it is found from Eq. (3.333) that the mode shapes are (p)

W1n (η) = A1n Yn (η) (p)

W2n (η) = A2n Yn (η)

(3.340)

3.9

Elastically Connected Beams

201

since n exists only for a specific value of n. From Eq. (3.336), it is seen that at n , 4n(p) + Kf − 4n A2n = . A1n Kf

(3.341)

Then, the corresponding modes shapes are (p)

W1n (η) = Yn (η) W2n (η) =

   A2n (p) (p) Yn (η) = 4n(p) + Kf − 4n Kf Yn (η) . A1n

(3.342)

If, in Eq. (3.342), we set 4n = 4n(−) = 4n(p) , then A2n /A1n = 1, confirming that the displacements of the beams are in phase. On the other hand, when 4n = 4n(+) it is found that A2n /A1n = −1, indicating that the displacements of the beams are out of phase. Thus, using the results appearing in Tables 3.5 to 3.8 to obtain n(p) , specifying kf , and using Eq. (3.339) the natural frequency coefficients for the p sets of boundary conditions in Table 3.3 can be determined. The corresponding mode shapes are determined from the last column in Table 3.3 and Eq. (3.342).

3.9.3 Beams with Concentrated Masses Connected by an Elastic Spring The case of two uniform beams, each carrying a mass at the same in-span location and interconnected by a spring attached to the masses, is analyzed. When the beams are cantilever beams, this configuration has been used to model bimorph beams in energy harvesters (Yang and Yang 2009), grippers (Chonan et al. 1996), and as a means to determine the properties of nanofibers (Yuya et al. 2007). For this configuration, we set kf = 0 and α = β = 1 and Eq. (3.331) becomes, ∂ 4 W1 + Ki δ (η − ηm ) (W1 − W2 ) − 4 [1 + mi δ (η − ηm )] W1 = 0 ∂η4 ∂ 4 W2 + Ki δ (η − ηm ) (W2 − W1 ) − 4 [1 + mi δ (η − ηm )] W2 = 0. ∂η4

(3.343)

The boundary conditions for each beam are assumed to be those given by Eqs. (3.93) and (3.94). We shall obtain a solution to Eq. (3.343) subject to the boundary given by Eqs. (3.93) and (3.94) with Y replaced by W by taking the Laplace transform of Eq. (3.343) with respect to η. Thus, from Eq. (C.12) of Appendix C with f = β = 0 and using the appropriate definition for K, we obtain

202

3

1 s2 − 4 1 ¯ 2 (s) = W 2 s − 4 ¯ 1 (s) = W

Thin Beams: Part I

  W1 (0) s3 + W1 (0) s2 + W1 (0) s + W1 (0) + P12 e−ηm s   W2 (0) s3 + W2 (0) s2 + W2 (0) s + W2 (0) + P21 e−ηm s (3.344)

where   P12 = mi 4 − Ki W1 (ηm ) + Ki W2 (ηm )   P21 = mi 4 − Ki W2 (ηm ) + Ki W1 (ηm ) .

(3.345)

The inverse Laplace transform of Eq. (3.344) is obtained from Eq. (C.18) with f = 0 and using the appropriate values for K as Wj (η) = Wj (0) Q (η) + Wj (0) R (η)/ + Wj (0) S (η)/2 + Wj (0) T (η)/3 + Gj (η, ηm )

j = 1, 2

(3.346)

where G1 (η, ηm ) = P12 T ( [η − ηm ]) u (η − ηm )/3 G2 (η, ηm ) = P21 T ( [η − ηm ]) u (η − ηm )/3 .

(3.347)

The procedure that was used in Section 3.3.2 is again employed. Hence, using Eq. (3.93) with γ replaced by W in Eq. (3.346), we obtain Wj (η) = Wj (0) f1 (η) + Wj (0) f2 (η) + Gj (η, ηm )

j = 1, 2

(3.348)

where fl (η) , l = 1, 2, are given by Eq. (3.96). Substituting Eq. (3.348) into Eq. (3.94) with Y replaced by W and solving for Wj (0) and Wj (0), j = 1, 2, respectively, and substituting these expressions for Wj (0) and Wj (0) into Eq. (3.348) yields P12 H2 (, η, ηm ) D ()  H (, η, η )   2 m = − mi 4 − Ki W1 (ηm ) + Ki W2 (ηm ) D () P21 W2 (η) = − H2 (, η, ηm ) D ()   H (, η, η )  2 m = − mi 4 − Ki W2 (ηm ) + Ki W1 (ηm ) D ()

W1 (η) = −

(3.349)

where H2 (, η, ηm ) is given by Eq. (3.102) and D() is given by Eq. (3.104). To determine the characteristic equation, it is noted that Eq. (3.349) must be satisfied at η = ηm . Therefore, setting η = ηm in each of the equations in Eq. (3.349), the following system of equations is obtained

3.10

Forced Excitation

203



h¯ () h¯ 12 () [H ()] {W} = ¯ 11 h21 () h¯ 22 () where



W1 (ηm ) W2 (ηm )

 =0

(3.350)

 h¯ 11 () = D () + mi 4 − Ki H2 (, ηm , ηm ) h¯ 12 () = Ki H2 (, ηm , ηm ) h¯ 21 () = h¯ 12 ()

(3.351)

h¯ 22 () = h¯ 11 () . The natural frequency coefficients are those values of  = n for which det [H (n )] = 0; that is, those values of n that are solutions to      D (n ) + mi 4n − 2Ki H2 (n , ηm , ηm ) D (n ) + mi 4n H2 (n , ηm , ηm ) = 0. (3.352) To convert Eq. (3.352) into a form that represents the boundary conditions in Table 3.3, we use Eq. (3.108) in Eq. (3.352) to obtain a (n ) b (n ) = 0

(3.353)

where

 a (n ) = 3n D(p) (n ) + mi 4n − 2Ki Hn(p) (ηm , ηm ) b (n ) = D(p) (n ) + mi n Hn(p) (ηm , ηm ).

(3.354)

It is seen from Eq. (3.114) that b (n ) = 0 is the characteristic equation from which the natural frequency coefficients for a beam with a mass located at ηm are obtained. On the other hand, from Eq. (3.112) it is seen that a (n ) = 0 is the characteristic equation for a beam with a mass Mi and spring with stiffness 2ki that are each located at ηm . Thus, b (n ) = 0 is the case when the beams vibrate in phase and a (n ) = 0 is the case when the beams vibrate out of phase. The factor of two for the stiffness is due to the fact that the spring simultaneously undergoes an extension or compression at each of its ends. This results in twice the extension/compression of a spring that is only clamped at one of its ends. It is emphasized that these conclusions are valid for the eight sets of boundary conditions given in Table 3.3.

3.10 Forced Excitation 3.10.1 Boundary Conditions and the Generation of Orthogonal Functions To determine the response of a beam subjected to forced excitation, we shall use the separation of variables and the generation and application of orthogonal functions.

204

3

Thin Beams: Part I

From the discussion following Eq. (B.92) in Section B.2.1 of Appendix B, the boundary conditions given by Eqs. (3.51) and (3.52) will allow one to generate orthogonal functions provided that F is a symmetric quadratic; that is, when F = Fsq , where Fsq is given by Eq. (3.42). Equation (3.42) was obtained by removing the nonlinear terms; that is, by setting Po = Vo = 0. In addition, from the discussion at the end of Section 3.2.3 is it also necessary to set ko = 0; that is, the single degree-of-freedom system must be removed. Thus, when Po = 0 in the boundary conditions described by Eqs. (3.51) and (3.52) and their special cases given in Table 3.1, we will be able to generate orthogonal functions. Consequently, in terms of the non dimensional parameters, one can obtain a solution to ∂2 ∂η2

      ∂ 2y ∂ ∂y i (η) 2 − S (η, τ ) + Kf + Ki δ (η − ηs ) y ∂η ∂η ∂η

∂ 2y + [a (η) + mi δ (η − ηm )] 2 = fˆ (η, τ ) ∂τ

(3.355)

and the boundary conditions given in non dimensional form by Eqs. (3.57) and (3.58) with P (y (t)) = 0 by using orthogonal functions. To determine the orthogonality condition, we note from comparing Eq. (3.42) to the first equation of Eq. (B.70) of Appendix B that p (x) = ρA (x) + Mi δ (x − Lm ). From comparing Eq. (3.36) with the second equation of Eq. (B.70) and using Eq. (3.50), it is seen that a11 = ML , a12 = JL , a21 = MR , and a22 = JR . Therefore, the orthogonality condition given by Eqs. (B.90) to (B.92) in terms of the non dimensional quantities given by Eq. (3.53) is

1 a (η) Yn (η) Yj (η) dη + mi a (ηm ) Yn (ηm ) Yj (ηm ) + mL Yn (0) Yj (0) 0

(3.356)

+ jL Yn (0) Yj (0) + mR Yn (1) Yj (1) + jR Yn (1) Yj (1) = δnj Nn

where δ nj is the kronecker delta, the prime denotes the derivative with respect to η, and

1 Nn = 0

+

a (η) Yn2 (η) dη + mi a (ηm ) Yn2 (ηm ) + mL Yn2 (0) + jL Yn 2 (0)

(3.357)

mR Yn2 (1) + jR Yn 2 (1) .

It is seen from Eq. (3.356) that the orthogonality condition is valid for beams with variable cross sections. When the beam is of constant cross section; that is, a (η) = i (η) = 1, and when (p) (p) ML = JL = S = Kf = 0, then Yn (η) = Yn (η), where Yn (η) is given by Eq. (3.120) and its special cases tabulated in Table 3.3. When S and/or Kf are not equal to zero, the orthogonal functions can be obtained from Eqs. (3.88), (3.90), and (3.91) and their special cases.

3.10

Forced Excitation

205

3.10.2 General Solution The general solution of a beam of continuously variable cross section shall be obtained when the axial tensile force is a constant; that is, S (η, τ ) = So . Then Eq. (3.355) becomes ∂2 ∂η2

   ∂ 2y  ∂ 2y i (η) 2 − So 2 + Kf + Ki δ (η − ηs ) y ∂η ∂η

∂ 2y + [a (η) + mi δ (η − ηm )] 2 = fˆ (η, τ ) . ∂τ

(3.358)

We assume that the boundary conditions are those given by Eqs. (3.57) and (3.58) with P (y (t)) = ML = JL = 0. Then, the boundary conditions at η = 0 can be written as   i (0) y (0, τ ) = −KL y (0, τ ) + So y (0, τ ) i (0) y (0, τ ) = KtL y (0, τ )

0 ≤ KL ≤ ∞ 0 ≤ KtL ≤ ∞

(3.359)

and those at η = 1 can be written as   i (1) y (1, τ ) = KR y (1, τ ) + mR y¨ (1, τ ) + So y (1, τ ) i (1) y (1, τ )

= −KtR

y (1, τ ) − j

R

y¨ (1, τ )

0 ≤ KR ≤ ∞ 0 ≤ KtR ≤ ∞

(3.360)

where the prime denotes the derivative with respect to η and the over dot indicates the derivative with respect to the non dimensional time τ . The technique that is used to solve Eqs. (3.358) to (3.360) is as follows. We assume a solution of the form y (η, τ ) =

∞ 

ϕn (τ ) Yn (η)

(3.361)

n=1

where Yn (η) is a solution to ∂2 ∂η2

   ∂ 2 Yn  ∂ 2 Yn i (η) − S + Kf + Ki δ (η − ηs ) Yn o 2 2 ∂η ∂η

−4n [a (η) + mi δ (η

(3.362)

− ηm )] Yn = 0

and the boundary conditions at η = 0   i (0) Yn (0) = −KL Yn (0) + So Yn (0) i (0) Yn (0)

=

KtL Yn (0)

0 ≤ KL ≤ ∞ 0 ≤ KtL ≤ ∞

(3.363)

206

3

Thin Beams: Part I

and those at η = 1     i (1) Yn (1) = KR − mR 4n Yn (1) + So Yn (1)   i (1) Yn (1) = jR 4n − KtR Yn (1, τ )

0 ≤ KR ≤ ∞ 0 ≤ KtR ≤ ∞.

(3.364)

From the discussion in Section 3.10.1 and the discussion following Eq. (B.82) in Appendix B, it is known that Yn (η) is an orthogonal function. The modal amplitudes ϕn (τ ) are unknown quantities that are to be determined. Substituting Eq. (3.361) into Eq. (3.358) and using Eq. (3.362) results in ∞ 

 4n ϕn + ϕ¨n [a (η) + mi δ (η − ηm )] Yn (η) = fˆ (η, τ ) .

(3.365)

n=1

Upon substituting Eq. (3.361) into Eq. (3.359), the boundary conditions at η = 0, it is found that ∞     i (0) Yn (0) + KL Yn (0) − So Yn (0) ϕ (τ ) = 0 n=1 ∞    i (0) Yn (0) − KtL Yn (0) ϕ (τ ) = 0.

(3.366)

n=1

Substituting Eq. (3.361) into Eq. (3.360), the boundary conditions at η = 1, we find ∞       i (1) Yn (1) − KR Yn (1) − So Yn (1) ϕn (τ ) − mR Yn (1) ϕ¨n (τ ) = 0 n=1 ∞     i (1) Yn (1) + KtR Yn (1) ϕn (τ ) + jR Yn (1) ϕ¨n (τ ) = 0. n=1

(3.367) From Eq. (3.363), it is seen that Eq. (3.366) is satisfied. Using Eq. (3.364) in Eq. (3.367), we obtain ∞    4n ϕn (τ ) + ϕ¨n (τ ) mR Yn (1) = 0 n=1 ∞  



(3.368)

4n ϕn (τ ) + ϕ¨n (τ ) jR Yn (1) = 0.

n=1

The following operations are now performed. First, since Yn (η) is an orthogonal function, Eq. (3.365) is multiplied by Yl (η) and integrated with respect to η over the range 0 to 1 to obtain

3.10

Forced Excitation

207

∞ 

 4 n ϕn (τ ) + ϕ¨n (τ ) [a (η) + mi δ (η − ηm )] Yn (η)Yl (η) dη 1

n=1

1 =

0

(3.369)

fˆ (η, τ ) Yl (η) dη.

0

Next, the first equation of Eq. (3.368) is multiplied by Yl (1) and the second equation of Eq. (3.368) is multiplied by Yl (1) to obtain ∞     4n ϕn (τ ) + ϕ¨n (τ ) mR Yn (1) Yl (1) = 0 n=1 ∞  

4n ϕn (τ ) + ϕ¨n (τ )



jR Yn (1) Yl (1)



(3.370) = 0.

n=1

Upon adding Eqs. (3.369) and (3.370) and collecting terms, we arrive at ⎧ ∞  ⎨ 1  [a (η) + mi δ (η − ηm )] Yn (η) Yl (η) dη 4n ϕn (τ ) + ϕ¨n (τ ) ⎩ n=1 0 ⎫ ⎬ 1 + mR Yn (1) Yl (1) + jR Yn (1) Yl (1) = fˆ (η, τ ) Yl (η) dη. ⎭

(3.371)

0

From Eqs. (3.356) and (3.357), it is seen that when mL = jL = 0

1 Yn (η) Yl (η) dη + mi a (ηm ) Yn (ηm ) Yl (ηm ) + mR Yn (1) Yl (1) 0

+

jR Yn (1) Yl (1)

(3.372)

= δnl Nn

where δ nl is the Kronecker delta, and

1 Nn =

Yn2 (η) dη + mi a (ηm ) Yn2 (ηm ) + mR Yn2 (1) + jR Yn 2 (1) .

(3.373)

0

Then, Eq. (3.371) becomes ∂ 2 ϕn + 4n ϕn = Gn (τ ) ∂τ 2

(3.374)

208

3

Thin Beams: Part I

where 1 Gn (τ ) = Nn

1

fˆ (η, τ ) Yn (η) dη

(3.375)

0

and Nn is given by Eq. (3.373). The solution to Eq. (3.374) is given by Eq. (C.6) of Appendix C. When the initial conditions are zero, we obtain 1 ϕn (τ ) = 2 n Nn

τ 1 0

 sin 2n ξ fˆ (η, τ − ξ ) Yn (η) dηdξ

(3.376)

0

where we have used Eq. (3.375). Substituting Eq. (3.376) into Eq. (3.361), the final result is y (η, τ ) =

τ 1 ∞  Yn (η) n=1

2n Nn

0

 sin 2n ξ fˆ (η, τ − ξ ) Yn (η) dηdξ .

(3.377)

0

It is pointed out that Eq. (3.377) clearly shows that we were justified in arbitrarily setting the mode shape scale factor of Yn to unity. In the numerator of Eq. (3.377), we have the (indirect) product of Yn and Yn and in the denominator we have Nn , which is a function of the square of Yn . Consequently, whatever scale factor had been selected, it would have canceled.

3.10.3 Impulse Response Consider an impulse force of magnitude Fo that is applied at η = η1 and at τ = 0. This type of applied force is represented as fˆ (η, τ ) = Fˆ o δ (η − η1 ) δ (τ )

(3.378)

where, from Eq. (3.55), Fˆ o = Fo L3 /(EIo ). Substituting Eq. (3.378) into Eq. (3.377) gives y (η, τ ) =

∞  Yn (η) Yn (η1 ) n=1

2n Nn

 sin 2n τ .

(3.379)

It is seen from Eq. (3.379) that if the location of the load coincides with a node point of a particular mode shape; that is, when Yn (η1 ) = 0, then that mode is not excited and will not contribute to the response y (η, τ ).

3.10

Forced Excitation

209

To illustrate Eq. (3.379), we select a cantilever beam of constant cross section with a mass at its free end and with a spring located at an in-span location. We assume that the axial force and the elastic foundation are absent; that is, S = Kf = 0. This configuration corresponds to Case 8 of Table 3.3 with B2 = 0 and A2 = −mR 4 . Hence, from Table 3.3 and Eq. (3.124), when mi = 0 (G2 = 0) and Ki = 0 (G1 = 0) (8)

Hn (η, ηs ) − T (n [η − ηs ]) u (η − ηs ) D(8) (n )

Yn (η) = Yn(8) (η) =

(3.380)

and n are the solutions to Eq. (3.114); that is, D(8) (n ) −

Ki (8) H (ηs , ηs ) = 0. 3n n

(3.381)

In Eqs. (3.380) and (3.381), the following expressions from Table 3.3 are repeated here for convenience Hn(8) (η, ηs ) = Hn(3) (η, ηs ) − mR n {S (n η) [T (n ) R (n [1 − ηs ]) − R (n ) T (n [1 − ηs ])] + T (n η) [Q (n ) T (n [1 − ηs ]) − S (n ) R (n [1 − ηs ])]} Hn(3) (η, ηs )

= T (n η) [T (n ) R (n [1 − ηs ]) − Q (n ) Q (n [1 − ηs ])] + S (n η) [R (n ) Q (n [1 − ηs ]) − Q (n ) R (n [1 − ηs ])]

D(8) (n ) = R (n ) T (n ) − Q2 (n ) − mR n [Q (n ) T (n ) − R (n ) S (n )] . (3.382) When Ki = 0, n are the solutions to D(8) (n ) = 0 and, from Case 8 of Table 3.3, the mode shape is Yn (η) = Yn(8) (η) = −

T (n ) + mR n S (n ) T (n η) + S (n η) . mR n T (n ) + Q (n )

(3.383)

From Eq. (3.373), the norm is

1   2 2 Nn = Yn(8) (η) dη + mR Yn(8) (1) .

(3.384)

0

After some numerical experimentation, it is found that good convergence of Eq. (3.379) is obtained by using the lowest 10 natural frequencies and their corresponding mode shapes. The results are shown in Figs. 3.47 and 3.48. In both figures, the parameters are ηs = 0.75, Ki = 0 and 100, and mR = 0 and 0.4. In Fig. 3.47, the load is applied at η1 = 0.5 and in Fig. 3.48, at η1 = 1.0. The duration of the non

210

3

(a)

Thin Beams: Part I

(b) 0.4

0.5 y(η,τ)

y(η,τ)

0.2 0 −0.5 −1 1 0.8 0.6 η 0.4 0.2 0

(c)

0

0.5

1.5

1

−0.4 1 0.8 0.6 0.4 η 0.2

2

0

τ

(d)

0.3 0.2

0

0.5

1

1.5

2

2.5

3

τ

0.2 0.1

0.1

y(η,τ)

y(η,τ)

0 −0.2

0

0

−0.1

−0.1

−0.2 1 0.8 0.6 0.4 0.2 η 0

−0.2 1 0.8 0.6 0.4 0.2 η 0

0

0.1

0.2 τ

0.3

0.4

0.5

0

0.2

0.4 τ

0.6

0.8

1

Fig. 3.47 Impulse response of a cantilever beam with and without a mass at its free end and without an in-span spring when the impulse force is applied at η1 = 0.5 (a) Ki = 0, mR = 0, τ1 = 1.79 (b) Ki = 0, mR = 0.4, τ1 = 2.9 (c) Ki = 100, ηs = 0.75, mR = 0, τ1 = 0.477 (d) Ki = 100, ηs = 0.75, mR = 0.4, τ1 = 0.895. The quantity τ 1 is the non dimensional period of the lowest natural frequency

dimensional time τ that is displayed is chosen as the period τ 1 of the lowest natural frequency; that is, τ1 = 2π 21 . This time varies with the presence or absence of Ki and mR and is given for each case.

3.10.4 Time-Dependent Boundary Excitation We shall now examine the case of a cantilever beam of constant cross section whose clamped end it subjected to a time-dependent non dimensional displacement yo (τ ). It is assumed that the beam is carrying a concentrated mass mi at η = ηm , and the system is initially at rest; that is, the initial displacement and initial velocity are zero. It is further assumed that no external force is applied; that is, fˆ = 0. In this case, the governing equation is obtained from Eq. (3.358) with Ki = So = Kf = 0 and i (η) = a (η) = 1; thus, ∂ 2y ∂ 4y [1 + + m δ − η = 0. (η )] i m ∂η4 ∂τ 2

(3.385)

3.10

Forced Excitation

211 (b)

2

1

1

0.5 y(η,τ)

y(η,τ)

(a)

0 −1 −2 1 0.8 0.6 η 0.4 0.2 0

0

0.5

1.5

1

−1 1 0.8 0.6 η 0.4 0.2

2

0

τ

(c)

0

0.5

1

1.5

2

2.5

3

τ

(d) 0.4 0.2 0 −0.2

0.4 0.2 y(η,τ)

y(η,τ)

0 −0.5

0 −0.2

−0.4 −0.6 1 0.8 0.6 η 0.4 0.2 0

0

0.1

0.2 τ

0.3

0.4

0.5

−0.4 1 0.8 0.6 η 0.4 0.2 0

0

0.2

0.4

0.6

0.8

1

τ

Fig. 3.48 Impulse response of a cantilever beam with and without a mass at its free end and without an in-span spring when the impulse force is applied at η1 = 1.0 (a) Ki = 0, mR = 0, τ1 = 1.79 (b) Ki = 0, mR = 0.4, τ1 = 2.9 (c) Ki = 100, ηs = 0.75, mR = 0, τ1 = 0.477 (d) Ki = 100, ηs = 0.75, mR = 0.4, τ1 = 0.895. The quantity τ 1 is the non dimensional period of the lowest natural frequency

The boundary conditions at η = 0 are y (0, τ ) = yo (τ ) y (0, τ ) = 0

(3.386)

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

(3.387)

The technique that is used to solve Eqs. (3.385) to (3.387) is as follows. We first take the Laplace transform of Eq. (3.385) with respect to τ to obtain d4 y¯ + [1 + mi δ (η − ηm )] s2 y¯ = 0 dη4

(3.388)

where y¯ = y¯ (η, s), s is the Laplace transform parameter, and the over bar indicates the Laplace transform of the quantity. Taking the Laplace transform of the boundary

212

3

Thin Beams: Part I

conditions, Eqs. (3.386) and (3.387), respectively, we have at η = 0 y¯ (0, s) = y¯ o (s) y¯ (0, s) = 0

(3.389)

and at η = 1 y¯ (1, s) = 0 y¯ (1, s) = 0.

(3.390)

We assume a solution of the form y¯ (η, s) = y¯ s (η, s) + y¯ d (η, s)

(3.391)

and substitute Eq. (3.391) into Eq. (3.388) to obtain d4 y¯ s d 4 y¯ d + + [1 + mi δ (η − ηm )] s2 y¯ d = − [1 + mi δ (η − ηm )] s2 y¯ s . 4 dη dη4

(3.392)

Substituting Eq. (3.391 into Eq. (3.389), the boundary conditions at η = 0 become y¯ s (0, s) + y¯ d (0, s) = y¯ o (s) y¯ s (0, s) + y¯ d (0, s) = 0

(3.393)

and substituting Eq. (3.391) into Eq. (3.390) the boundary conditions at η = 1 become y¯ s (1, s) + y¯ d (1, s) = 0 y¯ ¯ s (1, s) + y d (1, s) = 0.

(3.394)

We break the resulting equations and boundary conditions into two parts and solve each part separately and in a specific order. In the first part, we solve the equation ∂ 4 y¯ s =0 ∂η4

(3.395)

subject to the boundary conditions at η = 0 y¯ s (0, s) = y¯ o (s) y¯ s (0, s) = 0

(3.396)

and at η = 1 y¯ s (1, s) = 0 y¯ s (1, s) = 0.

(3.397)

3.10

Forced Excitation

213

For the second part, we solve d4 y¯ d + [1 + mi δ (η − ηm )] s2 y¯ d = − [1 + mi δ (η − ηm )] s2 y¯ s dη4

(3.398)

subject to the boundary conditions at η = 0 y¯ d (0, s) = 0 y¯ d (0, s) = 0

(3.399)

and at η = 1 y¯ d (1, s) = 0 y¯ d (1, s) = 0.

(3.400)

The first part must be solved first. The solution to Eq. (3.395) is y¯ s (η, s) = c1 + c2 η + c3 η2 + c4 η3 .

(3.401)

Substituting Eq. (3.401) into Eqs. (3.396) and (3.397), it is found that y¯ s (η, s) = y¯ o (s) .

(3.402)

Next, Eq. (3.402) is substituted into Eq. (3.398), which results in d4 y¯ d + [1 + mi δ (η − ηm )] s2 y¯ d = − [1 + mi δ (η − ηm )] s2 y¯ o (s) . dη4

(3.403)

To solve Eq. (3.403) subject to the boundary conditions given by Eqs. (3.399) and (3.400), the procedure of Section 3.10.2 is used. We assume a solution of the form y¯ d (η, s) =

∞ 

An (s) Yn (η)

(3.404)

n=1

where Yn is a solution to d4 Yn − [1 + mi δ (η − ηm )] 4n Yn = 0 dη4

(3.405)

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

(3.406)

214

3

Thin Beams: Part I

For these boundary conditions, Yn (η) is given by Case 3 of Table 3.3. Thus, Yn (η) = Yn(3) (η) =

(3)

Hn (η, ηm ) − T (n [η − ηm ]) u (η − ηm ) D(3) (n )

(3.407)

where n are the solutions to Eq. (3.114) for p = 3; that is, D(3) (n ) + mi n Hn(3) (ηm , ηm ) = 0

(3.408)

and, from Case 3 of Table 3.3, Hn(3) (η, ηm ) = T (n η) [T (n ) R (n [1 − ηm ]) − Q (n ) Q (n [1 − ηm ])] + S (n η) [R (n ) Q (n [1 − ηm ]) − Q (n ) R (n [1 − ηm ])] D(3) (n ) = R (n ) T (n ) − Q2 (n ) . (3.409) Substituting Eq. (3.404) into Eq. (3.403) and using Eq. (3.405), we arrive at ∞ 

  An (s) s2 + 4n [1 + mi δ (η − ηm )] Yn (η)

n=1

(3.410)

= − [1 + mi δ (η − ηm )] s y¯ o (s) . 2

Multiplying Eq. (3.410) by Yl (η), integrating over the region 0 ≤ η ≤ 1, and using the fact that Yl (η) is an orthogonal function, we obtain An (s) = −

s2 y¯ o (s) Bn s2 + 4n Nn

(3.411)

where

1 Nn =

Yn2 (η) dη + mi Yn2 (ηm ) 0

(3.412)

1 Yn (η) dη + mi Yn (ηm ) .

Bn = 0

Therefore, Eq. (3.404) becomes y¯ d (η, s) = −

∞ 2  s y¯ o (s) Bn Yn (η). s2 + 4n Nn n=1

(3.413)

3.10

Forced Excitation

215

Substituting Eqs. (3.413) and (3.402) into Eq. (3.391), we obtain y¯ (η, s) = y¯ o (s) −

∞ 2  s y¯ o (s) Bn Yn (η). s2 + 4n Nn

(3.414)

n=1

The inverse Laplace transform of Eq. (3.414) is y (η, τ ) = yo (τ ) −

∞  Bn gn (τ ) Yn (η) Nn

(3.415)

n=1

where gn (τ ) = L−1



s2 y¯ o (s) s2 + 4n

(3.416)

and L−1 [. . .] indicates the inverse Laplace transform of its argument. If the displacement at the clamped boundary is the unit step function, then yo (τ ) = u (τ )

(3.417)

and, from pair 6 of Table C.1 of Appendix C, y¯ o (s) =

1 . s

(3.418)

Then, from Eq. (3.416), gn (τ ) = L−1



 s 2 = cos  τ n s2 + 4n

(3.419)

where the Laplace transform pair 11 from Table C.1 of Appendix C has been used. Therefore, Eq. (3.415) becomes y (η, τ ) = u (τ ) −

∞ 

 Bn Yn (η) cos 2n τ . Nn

(3.420)

n=1

The undamped response at the free end of a cantilever beam when the fixed end is subjected to a sudden change in displacement is shown in Fig. 3.49 for mi = 0 and for mi = 0.4 and ηm = 1. These figures were obtained by summing twelve mode shapes associated with the twelve lowest natural frequencies. The duration of the non dimensional time τ is chosen as the period τ1 of the lowest natural frequency; that is, τ1 = 2π 21 . Therefore, for mi = 0, τ1 = 1.787 and for mi = 0.4, τ1 = 2.898. Also shown is the response using only the first three modes for mi = 0 and the response using only the first two modes for mi = 0.4. It is seen that only a few modes are required to capture the main features of the response.

216

3 (b) 5

5

4

4

3

3

2

2

y(1,τ)

y(1,τ)

(a)

1

1

0

0

−1

−1

−2

0

0.2 0.4 0.6 0.8

1

−2

1.2 1.4 1.6

0

0.2 0.4 0.6 0.8

(d)

3

2.5

2

2

1.5

1.5

y(1,τ)

y(1,τ)

1.2 1.4 1.6

3

2.5

1

1

0.5

0.5

0

0

−0.5

−0.5

−1

1 τ

τ

(c)

Thin Beams: Part I

0

0.5

1

1.5 τ

2

−1

2.5

0

0.5

1

1.5 τ

2

2.5

Fig. 3.49 Response of the free end of a cantilever beam with and without a mass at its free end when the fixed end is subjected to a sudden change in displacement of non dimensional magnitude 1 (a) mi = 0, τ1 = 1.787 and 12 mode shapes are used (b) mi = 0, τ1 = 1.787 and 3 mode shapes are used (c) mi = 0.4 and τ1 = 2.898 and 12 mode shapes are used and (d) mi = 0.4 and τ1 = 2.898 and 2 mode shapes are used

3.10.5 Forced Harmonic Oscillations The forced harmonic response of beams can be obtained directly from Eq. (3.377) 2 by assuming that fˆ (η, τ ) = g (η) ej τ. Then, y (η, τ ) =

τ 1 ∞  Yn (η) n=1 j2 τ

=e

2n Nn

2τ

0

∞  Yn (η) n=1

= ej

0

 2 sin 2n ξ g (η) ej (τ −ξ ) Yn (η) dηdξ

∞  n=1

2n Nn

τ

1

g (η) Yn (η) dη

0

Yn (η)   Nn 4n − 4

 2 sin 2n ξ e−j ξ dξ

0

1 g (η) Yn (η) dη 0

(3.421)

3.10

Forced Excitation

217

where only the steady-state portion of the solution has been retained; that is, that portion that oscillates with frequency 2 . If the harmonically oscillating force is applied at η = η1 , then g (η) = δ (η − η1 ) and Eq. (3.421) becomes y (η, τ ) = ej

2τ

∞  Yn (η) Yn (η1 )  . 4 − 4 N  n n n=1

(3.422)

3.10.6 Harmonic Boundary Excitation The response of a beam to harmonic excitation of the ‘fixed’ end of a cantilever 2 beam of the form y (0, τ ) = Yo ej τ will be determined. This model describes the response of a typical base-excited cantilever probe used in atomic force microscopy. We assume a solution of the form y (η, τ ) = Y (η) ej τ . 2

(3.423)

With these assumptions, Eq. (3.385) becomes ∂ 4Y − [1 + mi δ (η − ηm )] 4 Y = 0. ∂η4

(3.424)

The boundary conditions at η = 0, which are given by Eq. (3.386), become Y (0) = Yo Y (0) = 0

(3.425)

and the boundary condition at η = 1, which are given by Eq. (3.387), become Y (1) = 0 Y (1) = 0.

(3.426)

Comparing Eqs. (3.424), (3.425), and (3.426), respectively, with Eqs. (3.388), (3.389), and (3.390), it is seen that one can replace y¯ o (s) with Yo and s2 with −4 and in Eq. (3.391) on can replace y¯ (η, s) with Y (η) , y¯ s (η, s) with Ys (η) , y¯ d (η, s) with Yd (η), Thus, Eq. (3.391) becomes Y (η) = Ys (η) + Yd (η)

(3.427)

Ys (η) = Yo .

(3.428)

and, from Eq. (3.402),

218

3

Thin Beams: Part I

To obtain Yd (η), we start with Eqs. (3.398) and (3.404), respectively, and rewrite them as d 4 Yd − [1 + mi δ (η − ηm )] 4 Yd = [1 + mi δ (η − ηm )] 4 Yo dη4

(3.429)

and Yd (η) =

∞ 

An Yn (η)

(3.430)

n=1

where Yn (η) is given by Eq. (3.407). Then, substituting Eq. (3.430) into Eq. (3.429), we obtain ∞ 

  An 4n − 4 [1 + mi δ (η − ηm )] Yn (η) = [1 + mi δ (η − ηm )] 4 Yo (3.431)

n=1

where we have used Eq. (3.405). Multiplying Eq. (3.431) by Yl (η), integrating over the region 0 ≤ η ≤ 1, and using the fact that Yl (η) is an orthogonal function, we obtain An =

4 Yo Bn 4n − 4 Nn

(3.432)

where Nn and Bn are given by Eq. (3.412). Using Eqs. (3.432), (3.430), and (3.428) in Eq. (3.427), we obtain the following expression for the magnitude of the displacement under harmonic oscillation Y (η) = Yo + Yo

∞  4 Bn Yn (η)  . N 4n − 4 n=1 n

(3.433)

References Abramovitch DY, Andersson SB, Pao LY, Schitter G (2007) A tutorial on the mechanisms, dynamics, and control of atomic force microscopes. Proceedings of the 2007 American Control Conference, pp 3488–3502 Balachandran B, Magrab EB (2009) Vibrations, 2nd edn. Cengage, Toronto, ON Banks DO, Kurowski GJ (1977) The transverse vibration of a doubly tapered beam. ASME J Appl Mech 44:123–126 Batra RC, Porfiri M, Spinello D (2006) Electromechanical model of electrically actuated narrow microbeams. J Microelectromech Syst 15(5):1175–1189 Blevins RD (1979) Formulas for natural frequency and mode shape. Van Nostrand Reinhold, New York, NY

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Chen GY, Warmack RJ, Thundat T, Allison DP, Huang A (1994) Resonance response of scanning force microscopy cantilevers. Rev Sci Instrum 65(8):2532–2537 Chonan S, Jiangy ZW, Koseki M (1996) Soft-handling gripper driven by piezoceramic bimorph strips. Smart Mater Struct 5:407–414 Cranch ET, Adler AA (1956) Bending vibrations of variable section beams. Trans Am Soc Mech Eng, J Appl Mech 23:103–108 Ekinci KL, Roukes ML (2005) Nanoelectromechanical systems. Rev Sci Instrum 76:061101 Hodges DH (1984) Proper definition of curvature in nonlinear beam kinematics. AIAA J 22(12):1825–1827 Ilanko S (2003) Comments on “On the eigenfrequencies of a two-part beam-mass system.” J Sound Vib 265:909–910 Karnovsky IG, Lebed OI (2000) Formulas for structural dynamics. McGraw Hill, New York, NY Kelly SG, Srinivas S (2009) Free vibrations of elastically connected stretched beams. J Sound Vib 326:883–893 Kirk CL, Wiedemann SM (2002) Natural frequencies and mode shapes of a free-free beam with large end masses. J Sound Vib 254(5):939–949 Kopmaz O, Telli S (2002) On the eigenfrequencies of a two-part beam-mass system. J Sound Vib 252(2):370–384 Krylov S, Maimon R (2004) Pull-in dynamics of an elastic beam actuated by continuously distributed electrostatic force. ASME J Vib Acoust 126:332–342 Lee SY, Kuo YH (1992) Exact solutions for the analysis of general elastically restained nonuniform beams. ASME J Appl Mech 59:S205–S212 Meirovitch L (2001) Fundamentals of vibrations. McGraw Hill, New York, NY Melcher J, Hu S, Raman A (2007) Equivalent point-mass models of continuous atomic force microscope probes. Appl Phys Lett 91:053101 Oniszczuk Z (2000) Free transverse vibrations of elastically connected simply supported doublebeam complex system. J Sound Vib 232(2):387–403 Popov EP (1990) Engineering mechanics of solids. Prentice Hall, Upper Saddle River, NJ, Chapter 6 Rabe U, Jasner K, Arnold W (1996) Vibrations of free and surface-coupled atomic force microscope cantilevers: theory and experiment. Rev Sci Instrum 67(9):3281–3293 Salehi-Khojin A, Bashash S, Jalili N (2008) Modeling and experimental vibration analysis of nanomechanical cantilever active probes. J Micromech Microeng 18:085008–085018 Seidel H, Csepregi L (1984) Design optimization for cantilever-type accelerometers. Sens Actuators 6:81–92 Shaker FJ (1975) Effect of axial load on mode shapes and natural frequencies of beams. NASA Technical Note TN D-8109 Singh A, Mukherjee R, Turner K, Shaw S (2005) MEMS implementation of axial and follower end forces. J Sound Vib 286:637–644 Tuma JJ (1979) Engineering mathematics handbook, 2nd edn. McGraw Hill, New York, NY Urey H, Kan C, Davis WO (2005) Vibration mode frequency formulae for micromechanical scanners. J Micromech Microeng 15:1713–1721 Wang H-C (1967) Generalized hypergeometric function solutions on a class of nonuniform beams. ASME J Appl Mech 34:702–708 Weisstein EW (2003) CRC concise encyclopedia of mathematics, 2nd edn. Chapman & Hall, Boca Raton, FL Yang Z, Yang J (2009) Connected vibrating piezoelectric bimorph beams as a wide-band piezoelectric power harvester. J Intell Mater Syst Struct 20:569–574 Yang YT, Heh D, Wei PK, Fann WS, Gray MH, Hsu JWP (1997) Vibration dynamics of tapered optical fiber probes. J Appl Phys 81(4):1623–1627 Yoon J, Ru CQ, Mioduchowski A (2003) Vibration of an embedded multiwall carbon nanotube. Composites Sci Technol 63:1533–1542

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Yuya PA, Wen Y, Turner JA, Dzenis YA (2007) Determination of Young’s modulus of individual electrospun nanofibers by microcantilever vibration method. Appl Phys Lett 90:111909 Zhang YQ, Liu X, Liu GR (2007) Thermal effect on transverse vibrations of double-walled carbon nanotubes. Nanotechnology 18:445701 Zhou D (1997) The vibrations of a cantilever beam carrying a heavy tip mass with elastic supports. J Sound Vib 206(2):275–279 Ziegler C (2004) Cantilever-based biosensors. Anal Bioanal Chem 379(7–8):946–959 Zou Q, Tan W, Kim ES, Loeb GE (2008) Single- and triaxis piezoelectric-bimorph accelerometers. J Microelectromech Syst 17(1):45–57

Chapter 4

Thin Beams: Part II

The amplitude response function for beams with structural damping, constant viscous damping, viscous air damping, viscous fluid damping, and squeeze film damping are determined at the MEMS scale and at a scale 1000 times larger. The natural frequencies are obtained for beams subject to a constant in-plane force and for beams subject to an electrostatic force. The maximum average power of a twolayer piezoelectric beam energy harvester is determined as a function of the load resistance and frequency.

4.1 Introduction In this chapter, we continue the analysis of vibrating thin beams, but under different special conditions. First, the effects that various types of damping have on the amplitude frequency response are determined at the MEMS scale and the macro scale; that is, at a scale in which the spatial dimensions are 1000 times larger than those at the MEMS scale. This analysis shows the damping effects at the different scales. Next, the effects of externally applied constant in-plane forces on the natural frequencies of a beam clamped at both ends are examined. This beam configuration is then analyzed when it is also subject to an electrostatic field. The analysis includes the effects of the beam’s transverse displacement-induced in-plane forces and illustrates the importance of including a fringe correction for the beam’s geometry. These electrostatic beam configurations can be found in such dynamic applications as resonators, filters, and chemical and mass sensors (Batra et al. 2007). In the last section of this chapter, the average power produced by a vibrating two layer piezoelectric beam is determined.

4.2 Damping 4.2.1 Generation of Governing Equation In this section, we shall consider the effects that five types of damping have on the frequency response of a beam: constant viscous damping, viscous damping from 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_4, 

221

222

4 Thin Beams: Part II

motion of a beam in air, damping due to the submergence of a beam in a viscous fluid, structural damping, and squeeze film damping. For squeeze film damping and viscous fluid damping, respectively, we shall use the results of Sections 2.3 and 2.4. The subsequent results are restricted to the case of harmonic excitation. We shall start by examining the effects of damping on a beam of variable cross section that has no in-span attachments, elastic foundation, electrostatic field, or inplane force. With these simplifications, the governing equation of motion given by Eq. (3.47) becomes ∂2 ∂x2

 EI (x)

∂ 2w ∂x2

 + ρA (x)

∂ 2w = f (x, t) . ∂t2

(4.1)

The effects of damping on the beam are taken into account by assuming that f (x, t) consists of two parts, one part is due to the applied external force per unit length fapp (x, t) and the other is due to the damping force per unit length fdamp (t). Thus, Eq. (4.1) can be written as ∂2 ∂x2

  ∂ 2w ∂ 2w EI (x) 2 + ρA (x) 2 = fapp (x, t) − fdamp (t) ∂x ∂t

(4.2)

where the minus sign indicates that the damping force acts in a direction opposite to that of the applied force, which is positive when acting in the positive (downward) z-direction. In our case, the damping force per unit length is composed of five individual contributions of damping: constant viscous damping, denoted fc (t); viscous air damping, denoted fair (t); squeeze film damping, denoted fsq (t); and viscous fluid damping, denoted ffluid (t). Then, Eq. (4.2) can be written as ∂2 ∂x2



∂ 2w EI (x) 2 ∂x

 + ρA (x)

∂ 2w = fapp (x, t) − fc (t) − fair (t) − fsq (t) − ffluid (t) . ∂t2 (4.3)

Structural damping is introduced in a different manner. We now consider each of the five types of damping individually. Structural Damping To represent the structural damping of beams oscillating harmonically at a frequency ω (rad/s), the structural damping model of Section 2.2.4 is used. This model assumes that the stress is related to the strain as   γ ∂εx σx = E εx + (4.4) ω ∂t where γ is an empirically determined value: for steel, γ is on the order of 0.0004. Equation (4.4) is a representation of a very simple viscoelastic material. Recalling Eq. (3.2), it is seen that E in Eq. (4.3) can be replaced as follows

4.2

Damping

223

  γ ∂ . E →E 1+ ω ∂t

(4.5)

This relation can be inserted directly into Eq. (4.3) and is valid only for harmonic excitation. Constant Viscous Damping Constant viscous damping is represented by that introduced in Eq. (2.3); that is, fc (t) = c

∂w ∂t

(4.6)

  where c Ns/m2 is the damping coefficient per unit length. For structural elements, this type of damping is usually employed when the damping mechanism is unknown and the qualitative effects of damping are desired. This relation can be inserted directly into Eq. (4.3) and is valid for both harmonic excitation and for an arbitrary time-varying excitation. Viscous Air Damping Based on a model of closely spaced spheres oscillating harmonically at frequency ω (rad/s), the following expression has been obtained for the viscous damping force per unit length on a beam of width b in air (Hosaka et al. 1995) fair (t) = cair (ω)

∂w ∂t

(4.7)

where  cair (ω) = 3πμair + 0.75π b 2ρair μair ω Ns/m2

(4.8)

  is the frequency-dependent damping coefficient, μair Ns/m2 is the viscosity of the   air, and ρair kg/m3 is the density of the air. Equation (4.7) is valid only for a beam undergoing harmonic excitation. Viscous Fluid Damping The viscous fluid damping force per unit length on a beam oscillating at a frequency ω (rad/s) is given by Eq. (2.79), which in the present notation is ffluid (t) = ma (ω)

∂ 2w ∂w + cv (ω) 2 ∂t ∂t

(4.9)

where, from Eq. (2.80), ma (ω) = ρf AReal (rect (ω)) kg/m cv (ω) = − ρf AωImag (rect (ω)) Ns/m2

(4.10)

224

4 Thin Beams: Part II

and A is the cross sectional area of a rectangular beam. The hydrodynamic function rect (ω) is given by rect (ω) = corr (ω) cir (ω) where cir (ω) is given by Eq. (2.81). The function corr (ω) is a complex-valued correction function that corrects the results for a beam of circular cross section so that it can be applied to a beam of rectangular cross section. The correction function is given in Appendix 4.1. Equation (4.9) is valid only for a beam undergoing harmonic excitation. In addition, the hydrodynamic function was obtained for a very long beam surrounded by a fluid of infinite extent. Squeeze Film Damping The squeeze film damping force per unit length on a beam oscillating at a frequency ω (rad/s) is given by Eq. (2.48), which in the present notation is fsq (t) = kr,s (ω) w + cr,d (ω)

∂w ∂t

(4.11)

where, from Eqs. (2.49) and (2.50), Pa b Sr,k (L/b, σ (ω)) N/m2 ho Pa b cr,d (ω) = Sr,d (L/b, σ (ω)) Ns/m2 . ωho kr,s (ω) =

(4.12)

The quantities Sr,k and Sr,d are given by Eq. (2.51), b is the width of the beam, and ho is the film gap. Equation (4.11) is valid only for a beam undergoing harmonic excitation. It is pointed out that Eq. (4.12) does not take into account the fact that the gap may vary over the length of the beam. Governing Equation Upon substituting Eqs. (4.5), (4.6), (4.7), (4.9), and (4.11) into Eq. (4.3), we arrive at     ∂w  γ ∂ ∂ 2w EI (x) 1 + + c + cair (ω) + cv (ω) + cr,d (ω) 2 ω ∂t ∂x ∂t (4.13) ∂ 2w + kr,s (ω) w + (ρA (x) + ma (ω)) 2 = fo (x) e jωt ∂t

∂2 ∂x2



where, by the form of the externally applied force, we have explicitly limited Eq. (4.13) to harmonic excitation. It is noted that in practical terms, air and fluid damping can’t be considered simultaneously. However, we shall obtain the general solution Eq. (4.13) as it is and then retain only the relevant parameters that are

4.2

Damping

225

appropriate to the specific system being examined. Both harmonic excitation and time-varying excitation can be considered when cair = cv = cr,d = ma = kr,s = γ = 0; that is, when the effects of squeeze film damping, viscous fluid loading, air viscous air damping, and structural damping are ignored. In this case, Eq. (4.13) reduces to ∂2 ∂x2

  ∂ 2w ∂w ∂ 2w EI (x) 2 + c + ρA (x) 2 = fapp (x, t) ∂x ∂t ∂t

(4.14)

where the more general form for the applied force can now be used. Since it has been assumed that the system described by Eq. (4.13) is undergoing harmonic excitation, w is assumed to be w (x, t) = W (x) e jωt

(4.15)

Substituting Eq. (4.15) into Eq. (4.13), we obtain     d2 d2 W E (1 + jγ ) 2 I (x) + jω c + cair (ω) + cv (ω) + cr,d (ω) W 2 dx dx + kr,s (ω) W − ω2 (ρA (x) + ma (ω)) W = fo (x) .

(4.16)

Using the parameters defined in Eq. (3.53), Eq. (4.16) can be written in terms of non dimensional quantities as   d2 d2 W i + j2 (2ζc + 2ζair () + 2ζv () + 2ζsd ()) W (η) dη2 dη2 + Ksk () W − 4 (a (η) + Ma ()) W = fˆo (η) (4.17)

(1 + jγ )

where I (x) → I (η) = Io i (η), A (x) → A (η) = Ao a (η), cL4 ρair b2 3π μair L4 , rair = , ra = to EIo 8to μair to EIo  cair (ω) L4 √  2ζair () = = ra 1 +  rair to EIo ρf cv (ω) L4 2ζv () = = − 2 Imag (rect ()) to EIo ρ 4 ro cr,d (ω) L = 2 Sr,d (L/b, σ ()) 2ζsd () = to EIo  2ζc =

(4.18)

226

4 Thin Beams: Part II

and

2 = ωto ,

fo (η) L4 fˆo (η) = , EIo

 1 to = 2 L

ρAo s EIo

Pa bL4 kr,s (ω) L4 = ro Sr,k (L/b, σ ()) , ro = EIo ho EIo 2 2 ρf b 12μeff L σ () = σo 2 , σo = , rf = 4to μf Pa h2o to ρf Ma () = Real (rect ()) , Re = rf 2 [ Used in rect ()] . ρ

Ksk () =

(4.19)

In Eqs. (4.18) and (4.19), Sr,k and Sr,d are given by Eq. (2.51) and we have used Eqs. (2.59), (2.80), and (2.82). Representative values for the various non dimensional quantities appearing in Eqs. (4.18) and (4.19) for a typical MEMS beam are given in Table 4.1. Also included in this table are the values for a beam whose dimensions have been increased one thousand-fold. We denote a beam with these increased dimensions as a macro beam.

Table 4.1 Representative values of different damping parameters appearing in Eqs. (4.18) and (4.19) for a polycrystalline silicon beam of length L and rectangular cross section of height h and depth b. The fluid is water at standard conditions and the gas is air at standard conditions

Quantity (units)   E N/m2 a ρ kg/m3  3 ρf kg/m   ρair kg/m3 2 μf Ns/m   μair Ns/m2 h (m) b (m) L (m)   Pa N/m2 ho (m) γ ζc a

Value at MEMS scale 160 × 109 2330 1000 1.2 1.0 × 10−3 1.83×10−5 2 × 10−6 15 × 10−6 150 × 10−6 105 2.5 × 10−6 0.0004 0.05

(Sharpe et al. 1997)

Computed parameters at MEMS scale

Computed parameters at macro scale: [h = 1000h, b = 1000b, L = 1000L, ho = 1000ho ]

to = 4.7 × 10−6 s σo = 1.57 ro = 189.8 rf = 11.96 rair = 0.392 ra = 1.16 × 10−2 ρf/ρ = 0.429 L4/(EI) = 316 × 10−6 m2 /N –

to = 4.7 × 10−3 s σo = 1.57 × 10−3 ro = 189.8 rf = 11.96 × 103 rair = 392 ra = 1.16 × 10−5 ρf/ρ = 0.429 L4/(EI) = 316 × 10−6 m2 /N ζc = 0.00005

4.2

Damping

227

4.2.2 General Solution The techniques of Section 3.10.2 are employed to obtain the solution to Eq. (4.17), which is rewritten as   d2 d2 W + j2ζ () 2 W + Ksd () W (1 + jγ ) 2 i (η) (4.20) dη dη2 − 4 (a (η) + Ma ()) W = fˆo (η) where 2ζ () = 2ζc + 2ζair () + 2ζv () + 2ζsd () .

(4.21)

It is noted that a (η) and i (η) can have either of two forms. The first form is for beams with continuously variable cross sections such as those given by Eqs. (3.248) or (3.270). The second form is that for a uniform beam with a concentrated mass Mi located at x = Lm for which a (η) = 1 + mi δ (η − ηm ) and i (η) = 1. The boundary conditions are assumed to be those given by Eq. (3.359) and (3.360) with So = 0. For the case of harmonic excitation, these boundary conditions at η = 0 can be written as   i (0) W (0) = − KL W (0) 0 ≤ KL ≤ ∞ i (0) W (0) = KtL W (0) 0 ≤ KtL ≤ ∞

(4.22)

and those at η = 1 can be written as    i (1) W (1) = KR − mR 4 W (1) 

i (1) W (1) = jR 4 − KtR W (1)

0 ≤ KR ≤ ∞ 0 ≤ KtR ≤ ∞

(4.23)

where the prime denotes the derivative with respect to at η. We assume a solution to Eq. (4.20) of the form W (η) =

∞ 

An Wn (η)

(4.24)

n=1

where Wn (η), n = 1, 2, . . . , are orthogonal functions that are solutions to d2 dη2

  d 2 Wn i (η) − a (η) 4n Wn = 0 dη2

(4.25)

and the boundary conditions at η = 0   i (0) Wn (0) = − KL Wn (0) i (0) Wn (0) = KtL Wn (0)

(4.26)

228

4 Thin Beams: Part II

and those at η = 1    i (1) Wn (1) = KR − mR 4n Wn (1) 

i (1) Wn (1) = jR 4n − KtR Wn (1) .

(4.27)

For the general boundary conditions, the orthogonal function Wn (η) = Yn (η), where Yn (η) is given by Eq. (3.123). The corresponding natural frequency coefficients n are determined from Eq. (3.106) with G1 = 0. When special cases of the (p) boundary conditions are considered, Yn (η) → Yn (η), p = 1, 2, . . . , 8, where (p) Yn (η) is given by Eq. (3.124), the corresponding natural frequency coefficients are determined from Eq. (3.114), and Table 3.3 is used. Substituting Eq. (4.24) into Eq. (4.20) and using Eq. (4.25), we obtain ∞ 

An

   4n − 4 a (η) Wn (η) + Zn () Wn (η) = fˆo (η)

(4.28)

n=1

where 

Zn () = Ksd () − Ma () 4 + j 2ζ () 2 + γ a (η) 4n .

(4.29)

Substituting Eq. (4.24) into Eqs. (4.22) and (4.23), respectively, we obtain from the boundary conditions at η = 0 ∞ 

An

n=1 ∞ 

   i (0) Wn (0) + KL Wn (0) = 0 

An i (0) Wn (0) − KtL Wn (0)



(4.30) =0

n=1

and from the boundary conditions at η = 1 ∞ 

An

    i (1) Wn (1) − KR − mR 4 Wn (1) = 0

n=1 ∞ 

 

 An i (1) Wn (1) − jR 4 − KtR Wn (1) = 0.

(4.31)

n=1

From Eq. (4.26), it is seen that Eq. (4.30) is satisfied. Using Eq. (4.27) in Eq. (4.31), it is found that

4.2

Damping

229

  An mR 4n − 4 Wn (1) = 0

∞ 7 n=1 ∞ 7 n=1

  An jR 4n − 4 Wn (1) = 0

(4.32)

Following the procedure used in Section 3.10.2, Eq. (4.28) is multiplied by Wl (η) and integrated over the region 0 ≤ η ≤ 1 to obtain "   81 An 4n − 4 a (η) Wn (η) Wl (η) dη n=1 0 # 81 81 + Zn () Wl (η) Wn (η) dη = fˆo (η) Wl (η) dη. ∞ 7

0

(4.33)

0

Next, the first equation of Eq. (4.32) is multiplied by Wl (1) and the second equation of Eq. (3.32) is multiplied by Wl (1), which results in   An mR 4n − 4 Wn (1)Wl (1) = 0

∞ 7 n=1 ∞ 7 n=1

  An jR 4n − 4 Wn (1)Wl (1) = 0.

(4.34)

Upon adding Eqs. (4.33) and (4.34), collecting terms, and using Eqs. (3.356) with mi = mL = jL = 0, we arrive at ∞ 

An

   4n − 4 Nn δnl + Zn () wnl = gl

l = 1, 2, . . .

(4.35)

n=1

where δ nl is the kronecker delta, from Eq. (3.357)

1 Nn =

a (η) Wn2 (η) dη + mR Wn2 (1) + jR Wn 2 (1)

(4.36)

0

and wnl =

81

Wl (η) Wn (η) dη

0

gl =

81

(4.37) fˆo (η) Wl (η) dη.

0

Writing Eq. (4.35) in matrix form yields

230

4 Thin Beams: Part II

⎤⎧ ⎫ ⎧ ⎫ a11 a12 · · · ⎪ ⎨ A1 ⎪ ⎬ ⎪ ⎨ g1 ⎪ ⎬ ⎥ A2 ⎢ a21 a22 = g2 ⎦ ⎣ ⎪ .. ⎩ .. ⎪ ⎭ ⎪ ⎭ ⎩ .. ⎪ . . . ⎡

(4.38)

where 

aln = 4n − 4 Nn δnl + Zn () wnl .

(4.39)

Thus, it is seen from Eq. (4.35) that, in general, damping couples the modes; that is, the coefficient matrix is not diagonal. Special Case: a(η) = 1 and mi = mR = jR = 0 It is noted from Eqs. (4.36) that when mi = mR = jR = 0 and the beam has a constant cross section; that is, a (η) = 1, the orthogonality condition simplifies to

1 Wl (η) Wn (η) dη = Nn δnl

(4.40)

0

where

1 Nn =

Wn2 (η) dη

(4.41)

Wl (η) Wn (η) dη = Nn δnl .

(4.42)

0

and, therefore,

1 wnl = 0

For this special case, the matrix [a] in Eq. (4.38) becomes a diagonal matrix and it is found that An =

gn  . Nn 4n − 4 + Zn ()

(4.43)

To show the effects of the different types of damping, we shall consider the harmonic excitation of constant cross section beams without in-span and boundary attachments; that is, the special case where a (η) = 1 and mi = mR = jR = 0. For this case, we use Eqs. (4.43), (4.24), and (4.15) to obtain w (η, τ ) = e j

2τ

∞  gn Hn () Wn (η) Nn n=1

where

(4.44)

4.2

Damping

231

Hn () =

1 . 4n − 4 + Zn ()

(4.45)

Noting that Hn () can be written as Hn () =

e−jθn () |hn ()|

(4.46)

where |hn ()| =

 a2n () + dn2 ()

θn () = tan−1

dn () an ()

(4.47)

and an () = 4n − 4 (1 + Ma ()) + Ksk () dn () = 2ζ () 2 + γ 4n

(4.48)

we can write Eq. (4.44) as w (η, τ ) =

∞  n=1

  2 gn Wn (η)  e j  τ −θn () Nn a2n () + dn2 ()

(4.49)

where Nn is given by Eq. (4.41). The displacement amplitude response function G (, η) is obtained from Eq. (4.49) as ! !∞ ! g W (η) e−jθn () ! ! n n !  G (, η) = |w (η, τ )| = ! ! m. 2 2 ! Nn an () + dn () !

(4.50)

n=1

4.2.3 Illustration of the Effects of Various Types of Damping: Cantilever Beam Several special cases of damping are illustrated by considering a cantilever beam that is excited harmonically at η = η1 with a force of magnitude Fo . Hence, the applied force is fˆo (η) = Fˆ o δ (η − η1 ) where

(4.51)

232

4 Thin Beams: Part II

Fˆ o =

Fo L4 m. EI

From Eq. (4.37), it is found that

1 gn =

Fˆ o δ (η − η1 ) Wn (η) dη = Fˆ o Wn (η1 )

(4.52)

0

where, from Case 3 of Table 3.3, Wn (η) = Yn(3) (η) = −

T (n ) T (n η) + S (n η) Q (n )

(4.53)

and n is determined from D(3) (n ) = R (n ) T (n ) − Q2 (n ) = 0.

(4.54)

The values of n that satisfy Eq. (4.54) are given in Case 3 of Table 3.5. Substituting Eq. (4.52) into Eq. (4.50), we obtain ! !∞ G (, η) !! Wn (η1 ) Wn (η) e−jθn () !!  =! GF (, η) = ! ! Fˆ o Nn a2n () + dn2 () !

(4.55)

n=1

When all damping effects are ignored, Eq. (4.55) reduces to the absolute value of Eq. (3.422). We shall now examine Eq. (4.55) for each of the five types of damping independently. For the numerical evaluation of each set of analytical results, we shall consider the excitation point η1 = 0.45 and the observation point η = 1.0. For each type of damping, one set of geometric values corresponding to the MEMS scale and one set of values corresponding to the macro scale are examined. The values chosen for each of these sets are given in Table 4.1. The magnitude of the displacement response over a range of frequencies that includes the lowest three natural frequencies of the beam is determined. The values of the lowest three natural frequency coefficients 1 , 2 , and 3 are also included with the numerical results. In addition, the quality factor Q is also computed for each resonance using   Eq. (2.21). To convert Eq. (2.21) to the present notation, we let GF n,max , η = Gn,max indicate the maximum value of GF (, η) that is associated with n and we cutoff frelet n,max indicate the frequency at which the maximum occurs. The √ quencies are determined from GF (n.U , η) = GF (n.L , η) = Gn. max 2, where n,U and n,L , respectively, are the upper and lower cutoff frequencies. Thus, in the current notation,

4.2

Damping

233

Q=

2n,max 2n,U − 2n,L

where we have used the fact that the square of the frequency coefficient is proportional to the frequency in Hertz. Recall Eq. (3.65). It will be seen from the subsequent graphs that in the vicinity of each resonant peak for the five types of damping considered, each damped system behaves very much like a single degree-of-freedom system having similar damping properties. Structural Damping For the case of structural damping only, Eq. (4.48) simplifies to an () = 4n − 4

(4.56)

dn () = γ 4n and Eq. (4.55) becomes ! ! ! !∞ ! Wn (η1 ) Wn (η) e−jθn () ! ! !  GF (, η) = !    ! ! n=1 Nn 4 − 4 2 + γ 4 2 ! n

(4.57)

n

where θn () = tan−1

γ 4n . 4n − 4

(4.58)

When notational differences are taken into account, it is found that for each value of n, 1/|hn ()| and θn (), respectively, given by Eqs. (4.47) and (4.56) are identical to Hs () and θs () of a single degree-of-freedom system with structural damping given by Eq. (2.47). Equation (4.57) is plotted in Fig. 4.1 using the appropriate values in Table 4.1. Only the MEMS scale is shown; the macro scale yields identical non dimensional results since γ is independent of geometry. Also presented in the graph are the maximum values Gn,max of the response at each resonance. Constant Viscous Damping For the case of constant viscous damping only, Eq. (4.48) simplifies to an () = 4n − 4 dn () = 2ζc 2 and Eq. (4.55) becomes

(4.59)

234

4 Thin Beams: Part II

G1,max = 228.4

102

Q = 2492 G2,max = 14.75 Q = 2499.9

GF(Ω,1)

100

G3,max = 0.7649 Q = 2499.9

10−2

10−4

10−6

Ω2 = 4.694

Ω1 = 1.875 0

2

4

6

Ω3 = 7.855 8

10

Ω

Fig. 4.1 Amplitude response function of a cantilever beam with structural damping at the MEMS scale for the appropriate parameters appearing in Table 4.1 for η1 = 0.45 and η = 1.0

! ! ! ! −jθn () ! !∞ W W e (η ) (η) n 1 n ! !  GF (, η) = ! 2  2 !! !n=1 Nn 4 − 4 + 2ζc 2

(4.60)

n

where θn () = tan−1

2ζc 2 . 4n − 4

(4.61)

When notational differences are taken into account, it is found that for each value of n, 1/|hn ()| and θn (), respectively, given by Eqs. (4.47) and (4.59) are identical to H() and θ () of a single degree-of-freedom system with constant viscous damping given by Eq. (2.20). It is seen from the values in Table 4.1 and Eq. (4.18) that for a given value of c, the value of ζc for the macro scale is 200 to 1000 times less than that for the MEMS scale. Equation (4.60) is plotted in Fig. 4.2 for only the MEMS scale using the appropriate values from Table 4.1. The macro scale results look very similar except that the quality factor and corresponding maximum amplitude are on the order of 1000 times larger. Also presented in the graph are the maximum values Gn,max of the response at each resonance. It is noted that ζ given by Eq. (2.11) is related to ζc given by Eq. (4.18) as ζ = ζc /2n ; therefore, Q ≈ 2n / (2ζc ).

4.2

Damping

235

101 G1,max = 3.218 100

QMEMS = 35.15

G2,max = 1.3

QMacro = 34677

QMEMS = 220.3 QMacro = 112678

10−1

G3,max = 0.1888 QMEMS = 616.9

GF(Ω,1)

QMacro = 146434

10−2 10−3 10−4 10−5 Ω1 = 1.875 10−6

0

2

Ω2 = 4.694 4

6

Ω3 = 7.855 8

10

Ω

Fig. 4.2 Amplitude response function of a cantilever beam with constant viscous damping at the MEMS scale for the appropriate parameters appearing in Table 4.1 for η1 = 0.45 and η = 1.0. The maximum amplitude values are for the MEMS scale

Viscous Air Damping For the case of viscous air damping only, Eq. (4.48) simplifies to an () = 4n − 4  √  dn () = ra 1 +  rair 2 .

(4.62)

Then Eq. (4.55) becomes ! ! !∞ ! −jθ () n ! ! Wn (η1 ) Wn (η) e !  GF (, η) = !! ! !n=1 Nn 4 − 4 2 + ra 1 + √rair  2 2 !

(4.63)

n

where θn () = tan

−1 ra

 √  1 +  rair 2 . 4n − 4

(4.64)

Equation (4.63) has been plotted for the MEMS scale in Fig. 4.3 for the appropriate parameter values given in Table 4.1. The response for the macro scale is very similar to that for the MEMS scale except that the quality factor and corresponding

236

4 Thin Beams: Part II 102 G1,max = 12.75

101 100

QMEMS = 139.4

G2,max = 2.843

QMacro = 7831.3

QMEMS = 482 QMacro = 19980

QMEMS = 898.2 QMacro = 31675

10−1 GF(Ω,1)

G2,max = 0.2749

10−2 10−3 10−4 10−5 10−6

Ω1 = 1.875 0

2

Ω2 = 4.694 4

6

Ω3 = 7.855 8

10

Ω

Fig. 4.3 Amplitude response function of a cantilever beam with viscous air damping at the MEMS scale for the appropriate parameters appearing in Table 4.1 for η1 = 0.45 and η = 1.0. The maximum amplitude values are for the MEMS scale

maximum amplitudes are larger. To indicate the differences between the scales, the quality factor for the macro scale has also been computed and their values are also displayed in the figure. It is seen upon comparing the values of the quality factor that viscous air damping can be neglected at the macro scale. Viscous Fluid Damping For the case of viscous fluid damping only, Eq. (4.48) simplifies to an () = 4n − me () 4 dn () = − ce () 2

(4.65)

where   me () = 1 + ρf ρ Real (rect ())   ce () = ρf ρ 2 Imag (rect ()) . Then Eq. (4.55) becomes

(4.66)

4.2

Damping

237

! ! ! !∞ −jθ () n ! ! W e Wn (η1 ) n (η) ! GF (, η) = !!  !    ! n=1 Nn 4 − me () 4 2 + ce () 2 2 !

(4.67)

n

where θn () = tan−1

−ce () 2 . − me () 4

(4.68)

4n

When notational differences are taken into account, it is found that for each value of n, 1/|hn ()| and θn (), respectively, given by Eqs. (4.47) and (4.65) are identical to Hf () and ψ () of a single degree-of-freedom system with viscous fluid damping given by Eq. (2.89). It is seen that in the vicinity of one of the beam’s natural frequencies  , G will be close to a maximum value when  = fn , where   n  F 2 2 fn = n me fn . Thus, the added mass of the fluid decreases the value of the frequency at which the maximum values occur without the fluid present by a factor    of me fn . A plot of Eq. (4.67) is given in Fig. 4.4 for the appropriate parameter values given in Table 4.1. From Eqs. (4.18) and (4.19), it is seen that Imag (rect ()) is a function

102

MEMS scale Macro scale

101 QMacro = 333 100

QMEMS = 4.585

GF(Ω,1)

10−1

QMacro = 333 QMEMS = 16.07

QMacro = 332.9 QMEMS = 28.75

10−2 10−3 10−4 10−5 10−6

Ω1 = 1.875 0

2

Ω2 = 4.694 4

6

Ω3 = 7.855 8

10

Ω

Fig. 4.4 Amplitude response function of a cantilever beam with viscous fluid damping at the MEMS scale and at the macro scale for the appropriate parameters appearing in Table 4.1 for η1 = 0.45 and η = 1.0

238

4 Thin Beams: Part II

the Reynolds number, which is a function of rf . From the values given in Table 4.1, it is seen that rf at the macro scale is 1000 times larger than its value at the MEMS scale. Referring to Fig. 2.10, it is seen that a thousand-fold increase in the Reynolds number results in the magnitude of Imag (rect ()) approaching −0.01. Hence, the magnitude of ce changes in a similar manner. Consequently, upon comparing the results shown in Fig. 4.4, it is seen that viscous fluid damping significantly affects the quality factor of beams at the MEMS scale compared to the quality factor at the macro scale. However, the decrease in the resonance frequencies due to the added mass of the fluid is approximately the same for both beams. It has been found that the frequencies at which the resonances occur at the MEMS scale are 1 = 1.6878, 2 = 4.2248, and 3 = 7.099 whereas those at the macro scales are 1 = 1.7151, 2 = 4.2932, and 3 = 7.1837. This viscous fluid damping model has been used to fit experimentally determined responses of carbon nanotubes (Sawano et al. 2010). Squeeze Film Damping For the case of squeeze film damping only, Eq. (4.48) simplifies to an () = 4n − 4 + ro Sr,k (L/b, σ ()) dn () = ro Sr,d (L/b, σ ())

(4.69)

and Eq. (4.55) becomes ! ! ! !∞ −jθn () ! ! W e W (η ) (η) n 1 n ! !  GF (, η) = ! !    ! n=1 Nn 4 − 4 + ro Sr,k (L/b, σ ()) 2 + ro Sr,d (L/b, σ ()) 2 ! n (4.70) where θn () = tan−1

4n

ro Sr,k (L/b, σ ()) . − 4 + ro Sr,d (L/b, σ ())

(4.71)

In Eqs. (4.69) to (4.71), Sr,k and Sr,d are given by Eq. (2.51). When notational differences are taken into account, it is found that for each value of n, 1/|hn ()| and θn (), respectively, given by Eqs. (4.47) and (4.69) are identical to Hr () and θr () of a single degree-of-freedom system with squeeze film damping given by Eq. (2.70) with ζ = 0. From the values given in Table 4.1, it is seen that σo for the macro scale is 1000 times less than that for the MEMS scale. A plot of Eq. (4.70) is given in Fig. 4.5 for the appropriate parameter values given in Table 4.1 for the MEMS scale and for the macro scale. It is seen from this figure that squeeze film damping can have very significant effects at the MEMS scale. In addition to eliminating through its damping mechanism the peak response at the lowest natural frequency, it also has a stiffening effect that increases the values of the natural frequencies of the higher resonances at the MEMS scale. The frequencies at which the resonances occur at the MEMS scale are 2 = 5.007, 3 = 7.936. It is seen that

4.3

In-Plane Forces and Electrostatic Attraction

239

102

MEMS scale Macro scale

QMacro = 151

101 100

QMacro = 946.4

GF(Ω,1)

10−1

QMEMS = 13.3

QMacro = 2650 QMEMS = 138.1

10−2 10−3 10−4 10−5 Ω1 = 1.875 10−6

0

2

Ω2 = 4.694 4

Ω

6

Ω3 = 7.855 8

10

Fig. 4.5 Amplitude response function of a cantilever beam with squeeze film damping at the MEMS scale and at the macro scale for the appropriate parameters appearing in Table 4.1 for η1 = 0.45 and η = 1.0

at the macro scale the squeeze film damping has a moderate effect on the amplitude response and that there is a negligible increase in stiffness due to the squeeze film as indicated by the negligible effect on the value of the resonance frequency.

4.3 In-Plane Forces and Electrostatic Attraction 4.3.1 Introduction Beams of constant cross section that are subjected to in-plane forces and electrostatic attraction are now examined. It is assumed that the beams have no in-span attachments and there is no elastic foundation. In addition, is it assume that the axial tensile force is a constant; that is, p (η, τ ) = po . The governing equation is obtained from Eq. (3.54) with mi = Ki = Kf = 0 and a (η) = i (η) = 1; that is, e21 Vo2 Fˆ r (y) ∂ 2y ∂ 2y ∂ 4y − + P − + = fˆ (η, τ ) (S (y (τ ))) o ∂η4 ∂η2 ∂τ 2 (d − y)2 where the parameters are defined in Eqs. (3.53) and (3.55).

(4.72)

240

4 Thin Beams: Part II

To get a sense of the magnitude of So , it is known that the displacement δ o of a bar due to a tensile axial force po is δo = po L/Ao E. If this expression is used in the definition of So given by Eq. (3.61) and it is assumed that we have a beam with a rectangular cross section, then So = 12 (δo /L) (L/h)2 . Assuming that δo /L = 0.001 (0.1% strain) and L/h = 100, it is found that So = 120. Recall that P (y (τ )) and So represent tensile forces applied to the beam’s neutral axis and that, in general, Vo = Vo (τ ). When cantilever beams are examined, P (y (τ )) = 0. When P (y (τ )) = 0, and/or Vo = 0, then Eq. (4.72) is a nonlinear equation. The boundary conditions that are of the most interest are beams clamped at both ends and cantilever beams; therefore, we shall limit our discussion to these two cases. Before proceeding, it is more convenient when considering the electrostatic force to make the displacement non dimensional with respect to the gap do . Therefore, we define yˆ = w/do and Eq. (4.72) becomes     ∂ 2 yˆ E12 Vo2 Fˆ r yˆ ∂ 2 yˆ ∂ 4 yˆ  − So + P yˆ (τ ) −  2 + 2 = Fˆ (η, τ ) 4 2 ∂η ∂η ∂τ 1 − yˆ

(4.73)

where 



1 P yˆ (τ ) = 2



do r

  Fˆ r yˆ = 1 + c2

2 1  

0

do b

dˆy dη

0.76

2 dη,

Fˆ (η, τ ) =

 0.76 , 1 − yˆ

f (η, τ ) L4 do EIo

(4.74)

εo bL4 E12 = . 2EIdo3

and r2 = Io /Ao . We shall now consider several special cases of Eqs. (4.72) and (4.73).

4.3.2 Beam Subjected to a Constant Axial Force Consider a beam subject to a constant axial force only; that is, P (y (τ )) = Vo = 0. We limit our discussion to determining the natural frequencies and mode shapes of a beam clamped at both ends. To determine the natural frequencies, we set fˆ (η, τ ) = 0 and assume that yˆ (η, τ ) = Y (η) e j τ .

(4.75)

d2 Y d4 Y − S − 4 Y = 0. o dη4 dη2

(4.76)

2

. Then Eq. (4.72) becomes

4.3

In-Plane Forces and Electrostatic Attraction

241

The boundary conditions are given by case 1 of Table 1. Thus, after using Eq. (4.75), we have at η = 0 that Y (0) = 0

(4.77)

Y (0) = 0 and at η = 1 that Y (1) = 0 Y (1) = 0

(4.78)

where the prime denotes the derivative with respect to η. To obtain a solution to Eq. (4.76), we start with Eq. (3.75) with Gi = 0; that is, ˆ (η) + Y (0) Rˆ (η) + Y (0) Sˆ (η) + Y (0) Tˆ (η) Y (η) = Y (0) Q

(4.79)

ˆ (η), etc., are given by Eq. (C.15) of Appendix C and the definitions of ε where Q and δ appearing in Eq. (C.15) are given by [recall Eq. (3.74) with Kf = 0]    1 2 4 So + So + 4 δ = 2    1 2 2 4 −So + So + 4 . ε = 2 2

(4.80)

Using the boundary condition at η = 0 given by Eq. (4.77), Eq. (4.79) simplifies to Y (η) = Y (0) Sˆ (η) + Y (0) Tˆ (η) .

(4.81)

The remaining two constants, Y (0) and Y (0), are determined by substituting Eq. (4.81) into the boundary conditions at η = 1, which are given by Eq. (4.78). Performing the substitution, we obtain the following system of equations 

Sˆ (1) Tˆ (1) ˆR (1) + So Tˆ (1) Sˆ (1)



Y (0) Y (0)

 = 0.

(4.82)

In arriving at Eq. (4.82), we have used Eq. (C.16) of Appendix C. The natural frequency coefficients n are determined by setting the determinant of the coefficients of Y (0) and Y (0) in Eq. (4.82) to zero, which leads to the following frequency equation Sˆ n2 (1) − Rˆ n (1) Tˆ n (1) − So Tˆ n2 (1) = 0.

(4.83)

In Eq. (4.83), Sˆ n (1), etc., indicates that in the definition Sˆ (1), etc.,  is replaced by n in the expressions for ε and δ given by Eq. (4.80). The corresponding mode shape is

242

4 Thin Beams: Part II

Yn (η) = −

Sˆ n (1) Tˆ n (η) + Sˆ n (η) . Tˆ n (1)

(4.84) (2)

When So = 0, Eqs. (4.83) and (4.84), respectively, reduce to D(2) (n ) and Yn (η) given by Case 2 of Table 3.3. The mode shape given by Eq. (4.84) is an orthogonal function. The values of 1 /π as a function of So are shown in Fig. 4.6. As indicated in Eq. (3.199) for a beam hinged at both ends, it is seen that an in-plane tensile force increases the natural frequency whereas an in-plane compressive force decreases it. Also shown in Fig. 4.6 is the value of So = Sbuckling , the magnitude of the compressive force that causes the beam to buckle.1

2

Ω1/π

1.5

1

0.5

Sbuckling = −39.478 0 −50

0

50

100

150

So

Fig. 4.6 Lowest natural frequency coefficient of a beam clamped at both ends as a function of a non dimensional in-plane tensile force So > 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 <

1 γbs R4o

(5.69)

and that for all other cases F2 > 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 << 1. Therefore, when So = Kf = 0 and γbs → 0, it is found that α →  and β → . Using these limiting values in the expressions in the appropriate characteristic equations in Table 5.2, it will be found that they reduce to the characteristic equations of those corresponding to the same boundary conditions given in Table 3.3 for the Euler-Bernoulli beam. The mode shapes corresponding to n are obtained from Eq. (5.74) with P1 = P2 = 0 and from Eq. (5.72) with H1 = H2 = 0. Thus, d11n d21n f2n (η) = f1n (η) − f2n (η) d12n d22n d11n d21n g2n (η) = g1n (η) − g2n (η) n (η) = g1n (η) − d12n d22n Yn (η) = f1n (η) −

(5.86)

Clamped at both ends

Hinged at both ends

Cantilever

Cantilever with attachments at free end: jR = KtR = 0

Clamped-hinged

1

2

3

4

5

α β − β α sinh α sin β +Ccf cosh α cos β = 0

Ccf 1 cosh α sin β − Ccf 2 sinh α cos β = 0

   γbs R2o 4 mR − KR Ccf 1 cosh α sin β − Ccf 2 sinh α cos β + Dcf = 0

Dcf = (So + A co ) 2 +



 2   1  γbs R2o co α 2 + β 2 + α 2 − β 2 αβ

β 2 + α2

    γbs R2o R2o 4 − β 2 − 1  2     β + α 2 γbs R2o R2o 4 + α 2 − 1



Obtained from case 4 from limit as KR → ∞.

Ccf 2

co β co = α Ccf 1 =

 2 Ccf = γbs R2o c2o α 2 + β 2 − 2 (So + A co )

– 

sin β = 0 

Ccc =

Parametersb

2 − 2 cosh α cos β + Ccc sinh α sin β = 0

Characteristic equationa [Eq. (5.85)]

a

When Kf = 0 and So = 0, the algebraic form of the characteristic equations given in the third column differ from those given by (Huang 1961); however, the numerical results are equal for cases 1, 2, 3, and 5. Case 4 was not considered by (Huang 1961). b A = β 2 − α 2 − R2 4 ; c = 1 + γ R2 S .  o bs o o o

Boundary Conditions

Case

Table 5.2 Characteristic equations for a Timoshenko beam without in-span attachments on an elastic foundation and subject to a tensile axial force for several sets of boundary conditions

5.3

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

297

where we have arbitrarily set Y (0) = 1. In all the functions and quantities containing a subscript n,  is replaced by n . Several special cases of Eq. (5.86) are given in Table 5.3. One In-span Attachment with Translation Only: Ap = 0 and Bq = 0 For this case, Eq. (5.84) simplifies to   Ap C1,p ηp , ηp − DT = 0

p = 1, 2, 3.

(5.87)

The values of  that satisfy Eq. (5.87) are the natural frequency coefficients n . It is seen from Eq. (5.58) that: when p = 1, a mass is attached at ηm ; when p = 2, a single degree-of-freedom systems is attached at ηo ; and when p = 3, a translation spring is attached at ηs . For each of these attachments, Eq. (5.87) applies to the general boundary conditions given by Eqs. (5.60) and (5.61) and any of its special cases. The mode shapes corresponding to n are obtained from Eqs. (5.79) and (5.82). Thus,   Yn (η) = C1,pn η, ηp   n (η) = C3,pn η, ηp

p = 1, 2, 3

(5.88)

where, in all the functions and parameters containing ,  is replaced by n . Rigid in-span support: When A3 = −Ki and Ki → ∞, we have the case of a rigid in-span support for which the displacement is zero at η = ηs . This case is equivalent to an in-span hinged constraint and the natural frequency coefficients are determined from C1,3 (η3 , η3 ) = 0.

(5.89)

When the beam is free at one end, then n determined from Eq. (5.89) is the frequency coefficient for a beam with an overhang. One In-span Attachment with Rotation Only: Ap = 0, and Bq = 0 For this case, Eq. (5.84) simplifies to   Bq C4,q ηˆ q , ηˆ q − DT = 0

q = 1, 2.

(5.90)

The values of  that satisfy Eq. (5.90) are the natural frequency coefficients n . It is seen from Eq. (5.58) that: when q = 1, a mass with rotational inertia Ji is attached at ηm (but the translational effects of the mass are ignored); and when q = 2, a torsion spring is attached at ηt . For each of these attachments, Eq. (5.90) applies to the general boundary conditions given by Eqs. (5.60) and (5.61) and any of its special cases.

Cantilever

Cantilever with attachments at free end: jR = 0, KtR = 0

3

4

  Yn (η) = γbs R4o 4n − 1 Tαβ (η, αn , βn ) + γbs R2o Rαβ (η, αn , βn ) − en Sαβ (η, αn , βn ) n (η) = − Sαβ (η, αn , βn )  − en γbs R2o kn Tαβ (η, αn , βn ) + co Rαβ (η, αn , βn )

  Yn (η) = γbs R4o 4n − 1 Tαβ (η, αn , βn ) + γbs R2o Rαβ (η, αn , βn ) + en Sαβ (η, αn , βn ) n (η) = − Sαβ (η, αn , βn )  + en γbs R2o kn Tαβ (η, αn , βn ) + co Rαβ (η, αn , βn )

Yn (η) = sin βn η n (η) = cos βn η

  Yn (η) = γbs R4o 4n − 1 Tαβ (η, αn , βn ) + γbs R2o Rαβ (η, αn , βn ) − en Sαβ (η, αn , βn ) n (η) = − Sαβ (η, αn , βn )  − en γbs R2o kn Tαβ (η, αn , βn ) + co Rαβ (η, αn , βn )

Mode shapesa [Eq. (5.86)]

1

  2 4 Ro n Sαβ (1, αn , βn ) + Qαβ (1, αn , βn )

an , An = mR 4n − KR bn    2 an = An γbs R4o 4n −  1 Tαβ (1,  αn , βn ) + γbs Ro Rαβ (1, αn , βn ) 2 2 − co Ro bn + 1 γbs Ro Sαβ (1, αn , βn ) − co Qαβ (1, αn , βn ) bn = kn Tαβ (1, αn , βn ) + An Sαβ (1, αn , βn ) en =

γbs R2o kn Tαβ

(1, αn , βn ) Rαβ (1, αn , βn )  =  γbs R2o kn Sαβ (1, αn , βn ) + co Qαβ (1, αn , βn )

en =

–

   1 γbs R4o 4n − 1 Tαβ (1, αn , βn ) + γbs R2o Rαβ (1, αn , βn ) Sαβ (1, αn , βn ) Sαβ (1, αn , βn ) =− γbs R2o kn Tαβ (1, αn , βn ) + co Rαβ (1, αn , βn )

en =

Parameters

The quantities α n and β n are given by Eq. (5.68) with  replaced by n , co = 1 + γbs R2o So , and kn and bn are given by Eq. (5.59) with  replaced by n .

Hinged at both ends

2

a

Clamped at both ends

Boundary Conditions

1

Case

Table 5.3 Mode shapes for a Timoshenko beam without in-span attachments on an elastic foundation with a tensile in-plane force for several sets of boundary conditions. The values of n are obtained from the characteristic equation of the corresponding case in Table 5.2

5.3

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

299

The mode shapes corresponding to n are obtained from Eqs. (5.79) and (5.82). Thus,   Yn (η) = C2,qn η, ηˆ q   n (η) = C4,qn η, ηˆ q

q = 1, 2

(5.91)

where, in all the functions and quantities containing ,  is replaced by n . Rigid in-span support: When B2 = −Kti and Kti → ∞, we have the case of a rigid in-span support for which the rotation is zero at η = ηt . This case is equivalent to an intermediate clamped slider constraint and the natural frequency coefficients are determined from   C4,2 ηˆ 2 , ηˆ 2 = 0.

(5.92)

Two In-span Attachments with Translation Only: Ap = 0 and Bq = 0 For this case, Eq. (5.84) simplifies to      Ap C1,p ηp , ηp − DT Ar C1,r (ηr , ηr ) − DT     p, r = 1, 2, 3. −Ar Ap C1,p ηr , ηp C1,r ηp , ηr = 0

(5.93)

The values of  that satisfy Eq. (5.93) are the natural frequency coefficients n . The quantities Ap and Ar can each represent a mass, translation spring, or a single degreeof-freedom system. It is noted that in the notation used in Eq. (5.93), when Ap and Ar represent different types of attachments, the locations ηp and ηr can be equal. For example, a translation spring and a mass can be attached to the beam at the same location. Conversely, two equal masses or two identical single degree-of-freedom systems, for example, can each be attached at a different location. The mode shapes corresponding to n are obtained from Eqs. (5.79) and (5.82). Thus,   Yn (η) = C1,pn η, ηp − e1n C1,rn (η, ηr )   n (η) = C3,pn η, ηp − e1n C3,rn (η, ηr )

p, r = 1, 2, or 3

(5.94)

where e1n

  Apn C1,pn ηp , ηp − DT,n   = Apn C1,rn ηp , ηr

(5.95)

  and the scale factor Apn Y ηp /DT,n has been set to 1. In all the functions and quantities containing ,  is replaced by n . The quantity DT,n is given by Eq. (5.78) with  replaced by n .

300

5 Timoshenko Beams

Two In-span Attachments, One with Translation and the Other with Rotation: Ap = 0 and Bq = 0 For this system, Eq. (5.84) becomes        Ap C1,p ηp , ηp − DT Bq C4,q ηˆ q , ηˆ q − DT     −Bq Ap C3,p ηˆ q , ηp C2,q ηp , ηˆ q = 0

(5.96)

p = 1, 2, or 3 q = 1 or 2. This case can be used to examine, for example, the effects of a mass and its rotational inertia attached at ηp = ηˆ q and the case of a translation spring and a torsion spring attached at ηp = ηˆ q . The mode shapes corresponding to n are obtained from Eqs. (5.79) and (5.82). Thus,     Yn (η) = C1,pn η, ηp − e2n C2,qn η, ηˆ q (5.97)     n (η) = C3,pn η, ηp − e2n C4,qn η, ηˆ q where e2n =

  Apn C1,pn ηp , ηp − DT,n   Apn C2,qn ηp , ηˆ q

(5.98)

  and the scale factor Apn Y ηp /DT,n has been arbitrarily set to 1. In all the functions and parameters containing ,  is replaced by n . The quantity DT,n is given by Eq. (5.78) with  replaced by n . Rigid in-span support: If we let the stiffness of the translation spring approach infinity; that is, A3 → ∞, then Eq. (5.96) becomes        Bq C1,3 (η3 , η3 ) C4,q ηˆ q , ηˆ q − C3,3 ηˆ q , η3 C2,q η3 , ηˆ q −DC1,3 (η3 , η3 ) = 0

(5.99)

where q = 1 and ηˆ 1 = η3 . One special case of Eq. (5.99), an Euler-Bernoulli beam hinged at both ends, has been investigated (Ginsberg and Pham 1995).

5.3.4 Numerical Results To obtain the various special cases of the general boundary conditions derived in the previous section, one uses the limiting procedure discussed in Section 3.2.3. Unfortunately, for the Timoshenko beam, some of the equations that result from this limiting process are too lengthy to state explicitly. However, several of the more manageable cases have been given in Tables 5.2 and 5.3.

5.3

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

301

Before proceeding to the numerical examination of the different cases of the results of the previous section, we first examine the case of a Timoshenko beam that is hinged at both ends. It is seen from Case 2 of Table 5.2 that the natural frequency coefficients are determined as β = nπ , n = 1, 2, . . . . From Eq. (5.68), it is found that the lower natural frequency coefficient is  n =



1 2γbs R4o

Dn −

  D2n − 4γbs R4o En

1/4

(5.100)

where Dn = 1 + (nπ Ro )2 (1 + γbs ) + γbs R4o Kf + γbs So R2o (nπ Ro )2     En = (π n)4 1 + γbs So R2o + Kf 1 + γbs (nπ Ro )2 + So (nπ )2 .

(5.101)

It is noted that an elastic foundation and an in-plane tensile force raise the natural frequency. When Kf = So = 0, Eqs. (5.100) and (5.101), respectively, become  n =

1 2γbs R4o

  4 nR D n − D 2 − 4γ (π ) bs o n

1/4

(5.102)

where D n = 1 + (nπ Ro )2 (1 + γbs ) .

(5.103)

If Ro << 1, then Eq. (5.102) can be approximated by 

nπ n ≈  1/4 ≈ nπ 1 − 0.25 (1 + γbs ) (nπ Ro )2 . D n

(5.104)

The solution to the Euler-Bernoulli beam hinged at both ends is obtained from (E) Case 1 of Table 3.5; that is, n = nπ . Then, Eq. (5.104) can be written as n (E)

n

≈ 1 − 0.25 (1 + γbs ) (nπ Ro )2 .

(5.105)

Thus, from Eq. (5.105), it is seen that the natural frequency coefficient for a Timoshenko beam is always less than that of an Euler-Bernoulli beam and the magnitude of this difference increases as n increases and Ro increases. This difference between the theories will also be found to be true when other boundary conditions are considered. Using Eq. (5.102), the percentage difference between the results from the Euler-Bernoulli and the Timoshenko beam theories is defined as (T) = 100



n(E)

2

− 2n

9 2 n(E) .

(5.106)

302

5 Timoshenko Beams

When γbs = 3.12 and n = 1, (T) is greater than 5% for Ro > 0.052 and is less than 0.01% for Ro < 0.0022. For a beam with a rectangular cross section of height h and length L, Ro = 0.2887 (h/L). Then a value of Ro = 0.0022 corresponds to L/h = 131 and a value of Ro = 0.052 corresponds to L/h = 5.6. The latter case indicates a short, stubby beam. When n = 2, (T) is greater than 5% for 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 << a and h << b. The plate has a density ρ (kg/m3 ), a Young’s modulus E (N/m2 ) and a Poisson’s ratio ν. As with beams, we shall primarily be concerned with the transverse deflections of the plate; that is, displacements in the z-direction. In addition, the stresses normal to the surface of the plate, denoted σz , are assumed to be small compared to the in-plane stresses, denoted σx and σy , and to the in-plane shear stress, denoted τxy . Therefore, we shall assume that σz = τxz . = τyz . = 0, which is known as the plane stress assumption in linear elasticity theory. The surface of a uniformly thick plate that lies equidistant from the top and bottom surfaces of the plate is called the middle plane; it is analogous to the neutral axis of a beam. Additional assumptions governing the bending of thin plates are similar to those for the Euler-Bernoulli beam: the middle plane remains unaltered after bending and plane sections normal to the middle plane remain normal and their lengths unchanged. Based on the plane stress assumption, we start with Eqs. (A.6) and (A.7) of Appendix A; that is,  E  εx + νεy 2 1−ν  E  εy + νεx σy = 2 1−υ E γxy τxy = Gγxy = 2 (1 + ν) σx =

(6.1)

a b σy

τxy

z z

Ny y

τyx

τxy

x

h

σx z z σx

Nx

τyx

σy z

Middle surface

Fig. 6.1 Geometry of a thin rectangular plate and the stresses and in-plane forces acting on it

6.2

Derivation of Governing Equation and Boundary Conditions: Rectangular Plates

343

where, from Eq. (A.3), the normal strains ε x and εy and the shear strain γxy are given by εx =

∂u , ∂x

εy =

∂v , ∂y

γxy =

∂u ∂v + ∂y ∂x

(6.2)

and u is the in-plane displacement in the x-direction and v is the in-plane displacement in the y-direction. If the plate is aligned with the coordinate axes as shown in Fig. 6.1, then these axes can be considered principal axes. Extending the development of bending of a beam given in Section 3.2.1 to that for a plate, the curvature κ x of the xz-plane is [recall Eq. (3.7)] κx =

1 ∂ 2w = 2 Rx ∂x

(6.3)

where Rx is the radius of curvature of the xz-plane. Similarly, the curvature κ y of the yz-plane is κy =

1 ∂ 2w = 2 Ry ∂y

(6.4)

where Ry is the radius of curvature of the yz-plane. Extending the reasoning that was used to obtain Eq. (3.1) to two dimensions, the strains ε x and ε y are ∂ 2w ∂x2 ∂ 2w εy = − z 2 ∂y εx = − z

(6.5)

where w = w (x, y, t) and for our cross section a = 0 in Eq. (3.1). Comparing Eq. (6.2) with Eq. (6.5), it is seen that ∂w ∂x ∂w v=−z ∂y

u=−z

(6.6)

and, therefore, from Eqs. (6.6) and (6.2), γxy = −2z

∂ 2w . ∂y∂x

(6.7)

344

6 Thin Plates

Substituting Eqs. (6.5) and (6.7) into Eq. (6.1), it is found that Ez σx = − 1 − ν2

 

∂ 2w ∂ 2w + ν ∂x2 ∂y2 ∂ 2w ∂ 2w +ν 2 2 ∂y ∂x

σy = −

Ez 1 − ν2

τxy = −

Ez ∂ 2 w . (1 + ν) ∂y∂x

  (6.8)

The normal stresses and shear stresses distributed on the xz and yx surfaces cause moments per unit length Mx , My , and Mxy , which are defined as

h/2 Mx = −h/2

h/2 My = −h/2

E σz zdz = − 1 − ν2 E σy zdz = − 1 − ν2

h/2 Mxy = −h/2

τxy zdz = −





∂ 2w ∂ 2w + ν ∂x2 ∂y2 ∂ 2w ∂ 2w + ν ∂y2 ∂x2

∂ 2w E (1 + ν) ∂y∂x

 h/2









∂ 2w ∂ 2w z dz = −D + ν ∂x2 ∂y2 2

−h/2

 h/2

∂ 2w ∂ 2w z dz = −D + ν ∂y2 ∂x2 2

−h/2

h/2 z2 dz = −D (1 − ν) −h/2

∂ 2w = −Myx ∂y∂x (6.9)

where D=

Eh3   Nm 12 1 − ν 2

(6.10)

is called the flexural rigidity of the plate. It is analogous to the product EIo of a beam. We shall now use these results to determine the various contributions to the total energy of a thin plate.

6.2.2 Contributions to the Total Energy The total energy of the plate is composed of the kinetic energy, the strain energy, the work performed by an externally applied transverse force and in-plane forces, an elastic foundation, and attachments to the interior of the plate and to its boundaries. We shall examine each of these contributions separately.

6.2

Derivation of Governing Equation and Boundary Conditions: Rectangular Plates

345

Strain Energy The strain energy is determined by substituting Eqs. (6.1), (6.5) and (6.7) into Eq. (A.11) of Appendix A, which results in 1 Up = 2



  σx εx + σy εy + τxy γxy dV

V



 E 1−ν 2 2 2  ε dV + ε + 2νε ε + γ x y x y 2 xy 2 1 − ν2 V  2 2 , 2 2  2 2 ∂ w ∂ 2w ∂ 2w ∂ w D ∂ w + + 2ν 2 2 + 2 (1 − ν) = dA. 2 2 2 ∂y∂x ∂x ∂y ∂x ∂y

=



A

(6.11) Kinetic Energy: Plate Element and Concentrated Mass The kinetic energy of a plate element is given by 1 Tρ = 2





ρ V

∂w ∂t

2

h dV = 2





ρ A

∂w ∂t

2 dA.

(6.12)

The kinetic energy of a concentrated mass Mi (kg) located at x = am and y = bm is TMi

1 = Mi 2



∂w (am , bm , t) ∂t

2

which can be written as TMi =

1 2





Mi A

∂w ∂t

2 δ (x − am ) δ (y − bm ) dA.

(6.13)

External Forces: Transverse and In-plane

  The work done by an external transverse force Fa (x, y, t) N/m2 normal to the surface of the plate is

WF =

Fa (x, y, t) w (x, y, t) dA.

(6.14)

A

It is assumed that there are two constant in-plane tensile forces per unit length acting on the plate: Nx (N/m), which acts in a direction parallel to the x-axis, and Ny (N/m), which acts in a direction parallel to the y-axis. To determine the work

346

6 Thin Plates

done by these two forces, we use Eq. (3.16) to arrive at the following changes in length of the middle surface x ≈

1 2

1 y ≈ 2

 

∂w ∂x ∂w ∂y

2 dx 2 dy.

Therefore, the work of the in-plane tensile forces is 1 WN = − 2

,

 Nx

A

∂w ∂x



2 + Ny

∂w ∂y

2 dA.

(6.15)

Elastic Foundation and In-span Translation Spring When an elastic translation spring of constant ki (N/m) is placed at x = as and y = bs , the energy stored by the spring is Uki =

1 2 ki w (as , bs , t) 2

which can be written as 1 Uki = − 2



ki w2 (x, y, t) δ (x − as ) δ (y − bs ) dA.

(6.16)

A

  When the plate is placed on an elastic foundation of constant kf N/m3 , the energy stored in the foundation can be represented by a relation similar to that given by Eq. (3.20); that is, Ukf = −

1 2



kf w2 (x, y, t) dA.

(6.17)

A

Attachments on the Boundaries It is assumed that along the edges of the plate the following elements are attached: along the edge x = 0, there is a mass MxL (kg), a translation spring kxL (N/m), and a torsion spring kxtL (Nm/rad); along the edge x = a, there is a mass MxR (kg), a translation spring kxR (N/m), and a torsion spring kxtR (Nm/rad); along the edge y = 0, there is a mass MyL (kg), a translation spring kyL (N/m), and a torsion spring kytL (Nm/rad); and along the edge y = b, there is a mass MyR (kg), a translation spring kyR (N/m), and a torsion spring kytR (Nm/rad). Then, using the notation of Eq. (B.36) of Appendix B, the difference between the kinetic energy and potential energy of the attachments on the edges perpendicular to the x-axis is

6.2

Derivation of Governing Equation and Boundary Conditions: Rectangular Plates

F

(C1 )

,     ∂w (0, y, t) 2 ∂w (a, y, t) 2 1 MxL = + MxR 2 ∂t ∂t   1 kxL w2 (0, y, t) + kxR w2 (a, y, t) − 2,     ∂w (0, y, t) 2 ∂w (a, y, t) 2 1 + kxtR kxtL − 2 ∂x ∂x

347

(6.18)

and the difference between the kinetic energy and potential energy of the attachments on the edges perpendicular to the y-axis is F

(C2 )

,     ∂w (x, 0, t) 2 ∂w (x, b, t) 2 1 = + MyR MyL 2 ∂t ∂t   1 − kyL w2 (x, 0, t) + kyR w2 (x, b, t) 2,     1 ∂w (x, 0, t) 2 ∂w (x, b, t) 2 − . + kytR kytL 2 ∂y ∂y

(6.19)

Minimization Function From Eqs. (6.12) and (6.13), the total kinetic energy of the system is T = Tρ + TMi =

1 2





(ρh + Mi δ (x − am ) δ (y − bm )) A

∂w ∂t

2 dA.

(6.20)

The total external work is obtained from the sum of Eqs. (6.14) and (6.15). Thus,

W = WF + WN =

Fa (x, y, t) w (x, y, t) dA A

1 − 2

,

 Nx

A

∂w ∂x



2 + Ny

∂w ∂y

2 -

(6.21) dA.

The total stored elastic energy is the sum of Eqs. (6.11), (6.16), and (6.17). Thus, U = Up + Uki + Ukf  2 2 , 2 2  2 2 ∂ w ∂ w D ∂ 2w ∂ 2w ∂ w dA = + + 2ν 2 2 + 2 (1 − ν) 2 2 2 ∂x ∂y ∂x ∂y ∂y∂x A

  1 + kf w2 (x, y, t) + ki w2 (x, y, t) δ (x − as ) δ (y − bs ) dA. 2 A

(6.22)

348

6 Thin Plates

Then, from Eqs. (6.18) to (6.22),

T − U + W + F (C1 ) + F (C2 ) =

FdA + F (C1 ) + F (C2 )

(6.23)

A

where F=

 2 1 ∂w (ρh + Mi δ (x − am ) δ (y − bm )) 2 ∂t 1 + Fa (x, y, t) w (w, y, t) − kf w2 (x, y, t) 2 , 2  2 2  2 2 - (6.24) D ∂ w ∂ 2w ∂ 2w ∂ w ∂ 2w − + + 2ν 2 2 + 2 (1 − ν) 2 2 2 ∂x ∂y ∂x ∂y ∂y∂x ,     1 ∂w 2 ∂w 2 2 Nx − + Ny + ki w (x, y, t) δ (x − as ) δ (y − bs ) . 2 ∂x ∂y

6.2.3 Governing Equations The governing equation is obtained by using Eq. (B.120) of Appendix B; that is, Fw −

∂Fw, y ∂ 2 Fw, yy ∂ 2 Fw, xy ∂ 2 Fw, xx ∂Fw, x ∂Fw˙ − + − =0 + + 2 2 ∂x ∂y ∂x∂y ∂t ∂x ∂y

(6.25)

where we have replaced u in Eqs. (B.120) with w. Thus, from Eq. (6.24), it is found that Fw = Fa (x, y, t) − ki w (x, y, t) δ (x − as ) δ (y − bs ) − kf w (x, y, t) ∂w Fw, x = − Nx ∂x ∂w Fw,y = − Ny ∂y   2 ∂ w ∂ 2w Fw, xx = − D +ν 2 ∂x2 ∂y   2 ∂ w ∂ 2w Fw,yy = − D + ν ∂y2 ∂x2 Fw, xy = − 2D (1 − ν)

∂ 2w ∂x∂y

Fw˙ = (ρh + Mi δ (x − am ) δ (y − bm ))

∂w . ∂t

(6.26)

6.2

Derivation of Governing Equation and Boundary Conditions: Rectangular Plates

349

Substituting Eq. (6.26) into Eq. (6.25), performing the indicated operations, and collecting terms results in the following governing equation of motion of a rectangular plate  ∂ 2w ∂ 2w  − N + kf + ki δ (x − as ) δ (y − bs ) w (x, y, t) y 2 2 ∂x ∂y ∂ 2w + (ρh + Mi δ (x − am ) δ (y − bm )) 2 = Fa (x, y, t) ∂t

4 w − Nx D∇xy

(6.27)

where ∂2 ∂2 + 2 2 ∂x ∂y 4 ∂ ∂4 ∂4 = 4 +2 2 2 + 4. ∂x ∂x ∂y ∂y

2 = ∇xy 4 ∇xy

(6.28)

It is noted that the in-plane forces Nx and Ny in Eq. (6.27) are tensile; when they are compressive, Nx is replaced with –Nx and Ny with –Ny .

6.2.4 Boundary Conditions Upon comparing Eqs. (6.18) and (6.19) with Eq. (B.36), it is found that a11 = MxL ,

a21 = MxR ,

a12 = a22 = 0

A11 = kxL ,

A21 = kxR ,

A12 = kxtL ,

b11 = MyL ,

b21 = MyR ,

b12 = b22 = 0

B11 = kyL ,

B21 = kyR ,

B12 = kytL ,

A22 = kxtR

(6.29)

B22 = kytR .

The boundary conditions are obtained from Eqs. (B.121a,b) and (B.122a,b). In Eqs. (B.121a) and (B.122a), respectively, it is noted upon using Eq. (6.26) that  ∂Fw, xy ∂Fw, xx − Fw, x − ∂x ∂y



x=xj

∂w ∂ = −Nx +D ∂x ∂x



∂ 2w ∂ 2w + ν ∂x2 ∂y2  2  ∂ ∂ w + 2D (1 − ν) ∂y ∂x∂y x=xj

 ∂w + Vx (x, y, t) = −Nx ∂x



j = 1, 2 x=xj

(6.30a)

350

6 Thin Plates

 ∂Fw,xy ∂Fw,yy − Fw,y − ∂y ∂x

y=yj

   ∂ ∂ 2w ∂w ∂ 2w +D = −Ny + ν ∂y ∂y ∂y2 ∂x2  2  ∂ w ∂ + 2D (1 − ν) (6.30b) ∂x ∂x∂y y=yj  ∂w = −Ny j = 1, 2 + Vy (x, y, t) ∂y y=yj

where x1 = 0, x2 = a, y1 = 0, y2 = b, and ∂ 2  ∇xy w − Vx (x, y, t) = D ∂x ∂ 2  ∇xy w − Vy (x, y, t) = D ∂y

 3  ∂Mxy ∂ 3w ∂ w + (2 − ν) =D ∂y ∂x3 ∂x∂y2   3 ∂Mxy ∂ 3w ∂ w + (2 − ν) =D ∂x ∂y3 ∂y∂x2

(6.31)

are the shear forces per unit length along the indicated boundary. These shear forces are sometimes referred to as the Kelvin-Kirchhoff edge reactions. In arriving at Eq. (6.31), Eq. (6.9) was used. From Eqs. (B.121b) and (B.122b), it is noted upon using Eqs. (6.26) and (6.9) that   2 !   ∂ w ∂ 2w Fw, xx x=x = − D +ν 2 = Mx (x, y, t)!x=x 2 j j ∂x ∂y x=xj  2  !   ∂ w ∂ 2w + ν = My (x, y, t)!y=y Fw,yy y=y = − D 2 2 j j ∂y ∂x y=yj

(6.32) j = 1, 2.

Using Eqs. (6.29) to (6.32) in Eqs. (B.121a),b) and (B.122a,b), we obtain the following boundary conditions along the edges of a rectangular plate. In obtaining these results, we have used the fact that in Eqs. (B.121a,b) and (B.122a,b), respectively, y = b and x = a. At x = 0 Either w (0, y, t) = 0 or  ∂w ∂ 2w + bVx (x, y, t) kxL w + MxL 2 − Nx b ∂x ∂t

=0

(6.33a)

x=0

and either ∂w (0, y, t)/∂x = 0 or  kxtL

∂w + bMx (x, y, t) ∂x

=0 x=0

(6.33b)

6.2

Derivation of Governing Equation and Boundary Conditions: Rectangular Plates

351

At x = a Either w (a, y, t) = 0 or  ∂w ∂ 2w − bVx (x, y, t) kxR w + MxR 2 + Nx b ∂x ∂t

=0

(6.34a)

x=a

and either ∂w (a, y, t)/∂x = 0 or  ∂w kxtR − bMx (x, y, t) ∂x

=0

(6.34b)

x=a

At y = 0 Either w (x, 0, t) = 0 or  kyL w + MyL

∂w ∂ 2w + Ny a − aVy (x, y, t) ∂y ∂t2

=0

(6.35a)

y=0

and either ∂w (x, 0, t)/∂y = 0 or  ∂w kytL + aMy (x, y, t) ∂y

=0

(6.35b)

y=0

At y = b Either w (x, b, t) = 0 or  ∂w ∂ 2w + aVy (x, y, t) kyR w + MyR 2 − Ny a ∂y ∂t

=0

(6.36a)

y=b

and either ∂w (x, b, t)/∂y = 0 or  ∂w kytR − aMy (x, y, t) ∂y

=0

(6.36b)

y=b

As was the case with the Euler-Bernoulli beam theory, the either/or formalism is not required since, for example, the case of w (0, y, t) = 0 can be obtained by taking the limit as kxL → ∞. Similar reasoning applies for each of the other spring constants along their respective boundary. It is pointed out that the form of boundary conditions given by Eqs. (6.33) and (6.34), respectively, is analogous to the boundary conditions for an Euler-Bernoulli beam given by Eqs. (3.51) and (3.52).

352

6 Thin Plates

6.2.5 Non Dimensional Form of the Governing Equation and Boundary Conditions To place the governing equations and boundary conditions into a non dimensional form, the following parameters are introduced x y t w ρha4 2 , η = , ξ = , τ = , tp2 = s a a a tp D as am bs bm b η s = , ηm = , ξ s = , ξm = , α= a a a a a kf a4 ki a2 Mi , Ki = , mi = Kf = , mp = ρabh kg D D mp wˆ =

Nˆ x =

Nx a2 , D

Nˆ y =

Ny a2 , D

Fˆ a =

(6.37)

Fa a3 . D

Using Eq. (6.37), the governing equation given by Eq. (6.27) can be written as   ∂ 2 wˆ ∂ 2 wˆ ˆy − N + K + K δ − η δ − ξ wˆ (η ) (ξ ) f i s s ∂η2 ∂ξ 2 ∂ 2 wˆ + (1 + αmi δ (η − ηm ) δ (ξ − ξm )) 2 = Fˆ a (η, ξ , τ ) ∂τ

4 ∇ηξ wˆ − Nˆ x

(6.38)

where we have used the relations δ(x) = δ (aη) = δ (η)/|a| , δ(y) = δ (aξ ) = δ (ξ )/|a|, and ∂2 ∂2 + ∂η2 ∂ξ 2 4 ∂ ∂4 ∂4 = 4 +2 2 2 + 4. ∂η ∂η ∂ξ ∂ξ

2 = ∇ηξ 4 ∇ηξ

(6.39)

The boundary conditions can be written in non dimensional form by introducing the additional parameters kxtβ a bD kytβ = D Myβ = mp

Kxtβ =

Kyβ

Kytβ

mxβ and the quantities

kxβ a3 , bD kyβ a2 = , D Mxβ = , mp

Kxβ =

myβ

(6.40) β = L, R

6.2

Derivation of Governing Equation and Boundary Conditions: Rectangular Plates

Vˆ η (η, ξ , τ ) =

Vx a2 = D

Vˆ ξ (η, ξ , τ ) =

Vy a2 = D

ˆ η (η, ξ , τ ) = M

Mx a D

ˆ ξ (η, ξ , τ ) = M

My a D

 

∂ 3 wˆ ∂ 3 wˆ + (2 − ν) ∂η3 ∂η∂ξ 2

∂ 3 wˆ ∂ 3 wˆ + (2 − ν) 3 ∂ξ ∂ξ ∂η2  2  ∂ wˆ ∂ 2 wˆ =− +ν 2 ∂η2 ∂ξ  2  ∂ w ˆ ∂ 2 wˆ =− . + ν ∂ξ 2 ∂η2

353

  (6.41)

Then, the boundary conditions can be written as follows. At η = 0  ∂ 2 wˆ ∂ wˆ ˆ + mxL 2 − Nˆ η + Vˆ η (η, ξ , τ ) KxL w ∂η ∂τ  ∂w ˆ ˆ η (η, ξ , τ ) KxtL +M ∂η

η=0

 ∂ 2 wˆ ∂ wˆ ˆ + mxR 2 + Nˆ η − Vˆ η (η, ξ , τ ) KxR w ∂η ∂τ  ∂ wˆ ˆ η (η, ξ , τ ) KxtR −M ∂η

η=1

η=0

=0 (6.42) =0

At η = 1

η=1

=0 (6.43) =0

At ξ = 0  ∂ 2 wˆ ∂ wˆ KyL wˆ + αmyL 2 + Nˆ ξ − Vˆ ξ (η, ξ , τ ) ∂τ ∂ξ  ∂w ˆ ˆ ξ (η, ξ , τ ) KytL +M ∂ξ

ξ =0

 ∂ 2 wˆ ∂ wˆ + Vˆ ξ (η, ξ , τ ) KyR wˆ + αmyR 2 − Nˆ ξ ∂ξ ∂τ  ∂w ˆ ˆ ξ (η, ξ , τ ) KytR −M ∂ξ

ξ =α

ξ =0

=0 (6.44) =0

At ξ = α

ξ =α

=0 (6.45) =0

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6.3 Governing Equations and Boundary Conditions: Circular Plates To convert the preceding results to those of an annular circular plate of inner radius b and outer radius a, the following transformations for Cartesian to polar cylindrical coordinates are used ∂2 ∂ ∂ ∂ ∂ ∂2 → → , → , ∂x ∂r ∂y r∂θ ∂x2 ∂r2 ∂2 ∂2 ∂ ∂2 ∂ ∂2 + 2 2, → − 2 . → 2 r∂r r ∂θ ∂x∂y r∂r∂θ ∂y r ∂θ

(6.46)

Then, if it is assumed that for a circular plate ki = Mi = 0, Eq. (6.27) can be written as 4 2 D∇rθ wr − Nr ∇rθ wr + kf wr + ρh

∂ 2 wr = Fa (r, θ , t) ∂t2

(6.47)

where wr = w (r, θ , t), we have assumed that Nx = Ny = Nr , and 2 = ∇rθ

∂2 ∂ ∂2 + . + ∂r2 r∂r r2 ∂θ 2

(6.48)

The applicable moment and shear force, respectively, become    2 ∂wr ∂ 2 wr ∂ wr +ν Mx → Mr (r, θ , t) = − D + 2 2 ∂r2 r∂r r ∂θ ∂ 2  ∂Mxy (6.49) ∇rθ wr − Vx → Vr (r, θ , t) = D ∂r r∂θ    3  ∂ ∂ 2 wr ∂ wr 2 =D . wr + (1 − ν) 2 − ∇rθ ∂r r ∂r∂θ 2 r3 ∂θ 2 The boundary conditions are obtained from Eqs. (6.33) and (6.34) with the following changes in notation: x = 0 → r = b; x = a → r = a; kxL → kb ; kxtL → ktb ; kxR → ka ; kxtR → kta ; MxL → Mb ; and MxR → Ma . In addition, in the notation of Eqs. (B.121) and (B.122) of Appendix B, along the inner boundary C1 , y = 2π b and along the outer boundary of C1 , y = 2π a. Then Eqs. (6.33a,b) and (6.34a,b), respectively, become At r = b > 0  kb wr + Mb

∂ 2 wr ∂wr − 2π bNr + 2π bVr (r, θ , t) 2 ∂t ∂r  ∂wr + 2π bMr (r, θ , t) ktb ∂r

=0 r=b

(6.50) =0

r=b

6.3

Governing Equations and Boundary Conditions: Circular Plates

355

When the plate is a solid plate b = 0 and Eq. (6.50) is not applicable; it is replaced by the requirement that the displacement at the center of the plate remains finite. At r = a  ka wr + Ma

∂ 2 wr ∂wr + 2π aNr − 2π aVr (r, θ , t) ∂t2 ∂r  ∂wr − 2π aMr (r, θ , t) kta ∂r

=0 r=a

(6.51) =0

r=a

Non Dimensional Form The governing equation can be put in terms of non dimensional quantities by introducing the parameters rˆ =

r , a

wr , a

wˆ r =

kf a4 Kf = , D

Nˆ r =

Nr a2 D

Fa a3 fˆa = D

(6.52)

into Eq. (6.47) to arrive at ∇rˆ4θ wˆ r − Nˆ r ∇rˆ2θ wˆ r + Kf wˆ r +

  ∂ 2 wˆ r = fa rˆ , θ , τ 2 ∂τ

(6.53)

where ∇rˆ2θ =

∂2 ∂ ∂2 + + ∂ rˆ 2 rˆ ∂ rˆ rˆ 2 ∂θ 2

(6.54)

and τ is given in Eq. (6.37). In the definition of tp in Eq. (6.37), a now denotes the outer radius of the plate. The boundary conditions can be placed in non dimensional form by introducing the additional parameters α=

b , a

mb =

kb a2 , Kb = 2π αD

Mb , mp α 2

ma =

ka a2 Ka = , 2π D

Ma , mp

mp = ρhπ a2

ktb Ktb = , 2π αD

kta Kta = 2π D

(6.55)

and the quantities    2   Ma ∂ wˆ r ∂ wˆ r ∂ 2w ˆr ˆ rˆ rˆ , θ , τ = r = − + ν M + D rˆ ∂ rˆ ∂ rˆ 2 rˆ 2 ∂θ 2     Vr a2 ∂ 3 wˆ r ∂ 2  ∂ 2 wˆ r ˆ Vrˆ rˆ , θ , τ = = ∇rˆ θ wˆ r + (1 − ν) 2 − 3 2 . D ∂ rˆ rˆ ∂ rˆ ∂θ 2 rˆ ∂θ Then Eqs. (6.50) and (6.51), respectively, become as follows.

(6.56)

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6 Thin Plates

At rˆ = α > 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.

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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 << 1. 12a2

(7.14)

406

7 Cylindrical Shells and Carbon Nanotube Approximations

Substituting Eq. (7.8) into Eq. (7.10) and using the approximation    2 ξ −1 ξ ξ 1+ ≈1− + a a a

(7.15)

we obtain E  Nθ =  1 − ν2

h/2 ,

 νεx0

+ ξ νχx + εθ0

−h/2

ξ 1− + a

 2  ξ a

 2 ξ ξ dξ + χθ ξ 1 − + a a

 = E˜ νεx0 + εθ0 + ha χθ a 

= E˜ N¯ θ Nθx

 2   2    

h/2   ξ ξ ξ ξ ξ 0 1− + εxθ 1 − + +ξ 1+ τs dξ =G a a 2a a a −h/2



= Gh

0 εxθ

ha a − τs 2



  ha a 0 ˜ = EKo εxθ − τs 2

˜ o N¯ θx . = EK (7.16) In Eq. (7.16), we have set 1 ± ha = 1. Upon substituting Eqs. (7.8) into Eqs. (7.11) and (7.12), respectively, we arrive at E  Mx =  1 − ν2

˜ a εx0 = Eah

h/2   εx0 ξ + ξ 2 /a + χx ξ 2 (1 + ξ/a) + νεθ0 ξ + νχθ ξ 2 dξ −h/2

+ aχx + νaχθ



˜ aM ¯x = Eah Mxθ

h/2  0 εxθ =G ξ + ξ 2 (1 + ξ/ (2a)) τs dξ −h/2

Gh3 ˜ a Ko (aτs ) τs = Eah 12 ¯ xθ ˜ a Ko M = Eah =

(7.17) and

7.2

Derivation of Governing Equations and Boundary Conditions: Flügge’s Theory

E  Mθ =  1 − ν2

h/2 ,

 νεx0 ξ

+ νχx ξ

2

+ εθ0 ξ

−h/2

ξ 1− + a

407

 2  ξ a



 2 ξ + χθ ξ 2 dξ a

 ˜ a νaχx − εθ0 + aχθ = Eah ξ 1− + a

˜ aM ¯θ = Eah

h/2 , 0 εxθ ξ

Mθx = G −h/2



ξ 1− + a

  2   2   ξ ξ ξ ξ 2 +ξ 1 + τs dξ 1− + a 2a a a



 Gh3 0 0 ˜ a Ko −εxθ + aτs −εxθ + aτs = Eah 12a ˜ ¯ θx = Eaha Ko M

=

(7.18) where, in Eq. (7.18), we have used the approximation given by Eq. (7.15). Substituting Eq. (7.6) in Eqs. (7.13), (7.16), (7.17), and (7.18), we obtain the following expressions for the non dimensional force resultants and moment resultants in terms of the shell displacements

N¯ x =

 ∂ 2 w¯ ∂ u¯ θ ∂ u¯ x − ha 2 + ν + w¯ , ∂η ∂η ∂θ

∂ 2 w¯ ∂ u¯ x ∂ u¯ θ + − ha , ∂θ ∂η ∂η∂θ  ∂ u¯ θ ∂ u¯ ∂ 2 w¯ ∂ 2 w¯ ¯x = x − , + ν − M ∂η ∂η2 ∂θ ∂θ 2  ∂ u¯ θ ∂ 2 w¯ ¯ − , Mxθ = 2 ∂η ∂η∂θ N¯ xθ =

N¯ θ = ν N¯ θx =

∂ 2 w¯ ∂ u¯ x ∂ u¯ θ + + w¯ + ha 2 ∂η ∂θ ∂θ

∂ 2 w¯ ∂ u¯ x ∂ u¯ θ + + ha ∂θ ∂η ∂η∂θ

¯ θ = −ν M

∂ 2 w¯ ∂ 2 w¯ − w ¯ − ∂η2 ∂θ 2

¯ xθ − ¯ θx = M M

∂ u¯ x ∂ u¯ θ − ∂θ ∂η

(7.19)

where, again, we have set 1 ± ha = 1.

7.2.2 Contributions to the Total Energy The total energy of the shell is composed of the kinetic energy, the strain energy, and the work performed by an externally applied radial pressure. We shall examine each of these contributions separately.

408

7 Cylindrical Shells and Carbon Nanotube Approximations

Strain Energy The strain energy in the shell is determined from Eq. (A.11) of Appendix A, which is the current notation is expressed as

a2 U= 2

h/2 (σx ex + σθ eθ + τxθ γxθ ) (1 + ξ/a) dξ dηdθ .

(7.20)

−h/2

A

In arriving at Eq. (7.20), we have determined from Fig. 7.1 that dV = dsx dAθ = (a + ξ ) dξ dxdθ = a2 (1 + ξ/a) dξ dηdθ . Upon substituting Eq. (7.7) into Eq. (7.20), we obtain

a2 E  U=  2 1 − ν2

h/2  2 e2x + e2θ + 2νex eθ + Ko γxθ (1 + ξ/a) dξ dηdθ . (7.21) A

−h/2

Using Eqs. (7.5) and (7.15) in Eq. (7.21), it is found that

a2 E  U=  2 1 − ν2

: h/2 "

A

εx0 + ξ χx

2

(1 + ξ/a)

−h/2

 2 

  ξ + + 2ν εx0 + ξ χx εθ0 + ξ χθ + ξ χθ a   2 #;

2 ξ ξ 0 + Ko εxθ 1− + dξ dηdθ + ξ (1 + ξ/ (2a)) τs a a

= a2 E˜ V (F) dηdθ

εθ0

2



ξ 1− + a

(7.22)

A

where

V

(F)

1 = 2h =

and



h/2  ξ2 Qo + Q2 2 dξ a

−h/2

1 (Qo + ha Q2 ) 2

(7.23)

7.2

Derivation of Governing Equations and Boundary Conditions: Flügge’s Theory

409

2

2 2 0 Qo = εx0 + εθ0 + 2νεx0 εθ0 + Ko εxθ

 2  Q2 = (aχx )2 + 2ν (aχx ) (aχθ ) + (aχθ )2 + 2 (aχx ) εx0 − (aχθ ) εθ0 + εθ0 

2 0 0 . + Ko (aτs )2 + εxθ − (aτs ) εxθ (7.24) In Eq. (7.23), we have intentionally omitted the term containing ξ since the integral of this term is zero due to the symmetry of the shell surfaces about its middle surface. To determine an expression for V(F) in terms of the shell displacements, Eq. (7.6) is substituted into Eq. (7.24) to obtain V

(F)

" ha ∂ 2 w¯ ∂ u¯ θ ∂ 2 w¯ ∂ u¯ x ∂ 2 w¯ 2 =V + + w ¯ + 2 w ¯ − 2ν −2 2 ∂η ∂η2 ∂θ 2 ∂θ ∂η2 :  ;#    ∂ u¯ θ 2 ¯ ∂ u¯ x 2 ∂ u¯ θ ∂ 2 w ∂ u¯ x ∂ 2 w¯ + Ko 3 + −6 +2 ∂η ∂θ ∂η ∂η∂θ ∂θ ∂η∂θ (Don)

(7.25)

and V

(Don)

1 = 2

,

∂ u¯ x ∂η "

+ ha

  2    ∂ u¯ x ∂ u¯ θ ∂ u¯ θ ∂ u¯ θ 2 ∂ u¯ x + 2ν + w¯ + + w¯ + Ko + ∂η ∂θ ∂θ ∂θ ∂η #2  2 2  2 2 ∂ 2 w¯ ¯ ∂ w¯ ∂ w ∂ 2 w¯ ∂ 2 w¯ . + 2ν + + 4K o 2 2 2 2 ∂η∂θ ∂η ∂η ∂θ ∂θ (7.26) 2

The significance of V(Don) will become apparent subsequently. Kinetic Energy The kinetic energy is given by T=

1 2



V

a2 = 2





2 2 2 ρ u˙¯ x + u˙¯ θ + w˙¯ (a + ξ ) dξ dxdθ

 2 2 2 ρha2 u¯˙ x + u˙¯ θ + w˙¯ dθ dη

A

=

a2 E˜ 2



(7.27)

 2 2 2 to2 u¯˙ x + u˙¯ θ + w˙¯ dθ dη

A

where to =

a , cp

c2p =

E  . ρ 1 − ν2

(7.28)

410

7 Cylindrical Shells and Carbon Nanotube Approximations

External Work

  Considering an external pressure qn N/m2 that is normal to the surface and acting outward on the middle surface of the cylindrical shell, the external work on the middle surface of the shell is

Wq =



qn (x, θ , t) wadθ dx = a3

A

˜ = Ea

qn (η, θ , t) wdθ ¯ dη A



(7.29)

¯ dη q¯ n (η, θ , t) wdθ

2 A

where q¯ n =

qn a . E˜

(7.30)

Minimization Function Combining Eqs. (7.22), (7.27), and (7.29), it is found that ˜ 2 T − U + Wq = Ea



¯ dη Vdθ

(7.31)

A

where V¯ = V¯ (1) − V (F) + V¯ (3)

(7.32)

and to2 ˙ 2 ˙ 2 ˙ 2  u¯ + u¯ θ + w¯ 2 x = q¯ n w. ¯

V¯ (1) = V¯ (3)

(7.33)

7.2.3 Governing Equations The governing equations of motion can be determined from Eq. (B.125) of ¯ Thus, Appendix B with G = V.

7.2

Derivation of Governing Equations and Boundary Conditions: Flügge’s Theory

V¯ w¯ +

∂ 2 V¯ w¯ ,ηη ∂η2

∂ V¯ u˙¯ x ∂ V¯ u¯ x,η ∂ V¯ u¯ x,θ + + =0 ∂η ∂θ ∂t ∂ V¯ u˙¯ θ ∂ V¯ u¯ θ,η ∂ V¯ u¯ θ,θ + + =0 ∂η ∂θ ∂t ∂ 2 V¯ w¯ ,ηθ ∂ 2 V¯ w¯ ,θθ ∂ V¯ ˙ + + − w¯ = 0 2 ∂θ ∂η∂θ ∂t

411

(7.34)

where we have used the fact that V¯ is independent of u¯ x , u¯ θ , w¯ ,η , and w¯ ,θ and the higher-order derivatives u¯ x,ηη , u¯ θ,ηη , u¯ x,ηθ , u¯ θ,ηθ , u¯ x,θθ , and u¯ θ,θθ . Using Eq. (7.32) in Eq. (7.34), we obtain the following three coupled non dimensional partial differential equations governing the motions in a cylindrical shell according to Flügge’s theory  ∂ 3 w¯ ∂ w¯ ∂ 2 u¯ x ∂ 2 u¯ x ∂ 2 u¯ θ ∂ 3w ¯ ∂ 2 u¯ x + K + K − =0 + ν + h K − o p a o 2 2 2 3 ∂η∂θ ∂η ∂η ∂θ ∂η∂θ ∂η ∂τ 2 ∂ 2 u¯ x ∂ 2 u¯ θ ∂ 2 u¯ θ ∂w ¯ ∂ 3 w¯ ∂ 2 u¯ θ + =0 + Ko 2 + − h + K − (1 ) a o ∂η∂θ ∂θ ∂η ∂θ 2 ∂η2 ∂θ ∂τ 2 ∂ u¯ θ ∂ 3 u¯ x ∂ 3 u¯ x ∂ u¯ x ∂ 3 u¯ θ − ha 3 − ha Ko − ν + h + K (1 ) a o 2 2 ∂η ∂η∂θ ∂η ∂η ∂θ ∂θ ∂ 2 w¯ ∂ 2 w¯ −w¯ − 2ha 2 − ha ∇ 4 w¯ − 2 = − q¯ n ∂θ ∂τ (7.35) where τ = t/to , Kp

∇2 =

∂2 ∂2 + , ∂η2 ∂θ 2

Kp =

1+ν 2

(7.36)

and we have set 1 + ha = 1.

7.2.4 Boundary Conditions We shall be considering complete cylindrical shells only; therefore, there are no boundaries along θ . For this case, the boundary conditions can be obtained from Eqs. (B.126a-d) of Appendix B. If it is assumed that the length of the shell is L, then we are interested in the boundary conditions at η = 0 and η = L/a. Assuming that there are no attachments to the shell at η = 0 and η = L/a, we set Alkn = Blkn = alkn = blkn = 0 in Eqs. (B126a-d). Furthermore, it is seen from Eqs. (7.25) and (7.26) that only w¯ is a function of spatial derivatives higher than the first derivative.

412

7 Cylindrical Shells and Carbon Nanotube Approximations

Consequently, Eqs. (B.126a-d) simplify to the following four conditions that must be satisfied at η = 0 and at η = L/a: (F)

either

u¯ x = 0

or

Vu¯ x,η = 0

(7.37a)

either

u¯ θ = 0

or

Vu¯(F) =0 θ,η

(7.37b)

and

and

either

w¯ = 0

or

∂Vw(F) ¯ ,ηη ∂η

(F)

+

∂Vw¯ ,θη ∂θ

=0

(7.37c)

and either

∂ w¯ =0 ∂η

or

Vw(F) ¯ ,ηη = 0

(7.37d)

where, based on the functional dependencies of V¯ (1) , V (F) , and V¯ (3) , we have recognized that only V (F) is a function of u¯ x,η , u¯ θ,η , w¯ ,ηη , and w ¯ ,ηθ . Using Eqs. (7.25) in Eqs. (7.37a-d), we obtain the following boundary conditions at η = 0 and η = L/a either

u¯ x = 0

or

N¯ x = 0

(7.38a)

either

u¯ θ = 0

or

T¯ xθ = 0

(7.38b)

w¯ = 0

or

V¯ x = 0

(7.38c)

∂ w¯ =0 ∂η

or

¯x =0 M

(7.38d)

and

and either and either

¯ x are given in Eq. (7.19) and where N¯ x and M

7.2

Derivation of Governing Equations and Boundary Conditions: Flügge’s Theory

T¯ xθ =

∂ u¯ θ ∂ u¯ x + + 3ha ∂θ ∂η



∂ 2w ¯ ∂ u¯ θ − ∂η ∂η∂θ

413



  ∂ 2 w¯ ∂ u¯ θ ∂ u¯ x ∂ 2 w¯ ∂ u¯ θ + (1 + ha ) − ha + 2ha − ∂θ ∂η ∂η∂θ ∂η ∂η∂θ ¯ ¯ = Nxθ + ha Mxθ     ∂ 2 w¯ ∂ 2 w¯ ∂ u¯ x ∂ 2 w¯ ∂ u¯ θ ∂ ∂ u¯ θ ∂ ∂ u¯ x ¯ − 2 +ν − 2 + +4 −3 Vx = − Ko ∂η ∂η ∂θ ∂θ ∂η ∂θ ∂η∂θ ∂η ∂θ    2 ¯x ∂M ∂ u¯ θ ∂ w¯ ∂ u¯ θ ∂ u¯ x ∂ = − Ko −4 − + + ∂η ∂θ ∂η ∂η∂θ ∂η ∂θ ¯x ¯x  ∂M  ∂M ∂  ∂  ¯ xθ + M ¯ xθ + M ¯ xθ − M ¯ θx = ¯ θx . − Ko + Ko −2M M = ∂η ∂θ ∂η ∂θ (7.39) =

The quantity V¯ x and T¯ xθ are the shear force resultants that act on the edge as shown in Fig. 7.1. From Eqs. (7.38a-d), it is seen that one can specify many different combinations of boundary conditions at η = 0 and η = L/a. We shall limit our examination to the following three boundary conditions. Clamped At a clamped end, the displacements and the slope of the radial displacement in the axial direction are zero; hence, the boundary conditions are u¯ x = u¯ θ = w¯ =

∂w ¯ = 0. ∂η

(7.40)

Hinged (Simply Supported, Pinned) and No Normal Constraint No normal constraint means that u¯ x does not have to equal zero at the boundary. Hence, the boundary conditions are ¯ x = 0. N¯ x = u¯ θ = w¯ = M

(7.41)

These conditions indicate that the shell remains circular at the boundary and there is no axial force on the shell. This combination of boundary conditions has been referred to as a shell with a shear diaphragm; that is, one that has a thin flat circular plate attached at the end. Such a plate would be very stiff in the plane of the plate so that w¯ ≈ 0 and u¯ θ ≈ 0 and the plate would have relatively little stiffness to bending so that it is reasonable to assume that Mx ≈ 0. Free In a free edge, the following stress resultants and moment resultant are zero ¯ x = 0. N¯ x = T¯ xθ = V¯ x = M

(7.42)

414

7 Cylindrical Shells and Carbon Nanotube Approximations

7.2.5 Boundary Conditions and the Generation of Orthogonal Functions From the discussion in Section B.2.2 of Appendix B, any combination of the boundary conditions given by Eq. (7.38) will allow one to generate orthogonal functions since V¯ is a symmetric quadratic when qn = 0. A solution to Eq. (7.35) from which the natural frequencies and modes shapes are obtained can be found by assuming that ¯ x,n (η) cos nθ eiωt u¯ x (η, θ , t) = U ¯ θ,n (η) sin nθ eiωt u¯ θ (η, θ , t) = U ¯ n (η) cos nθ e w¯ (η, θ , t) = W

iωt

(7.43)

.

If Eq. (7.43) is substituted into Eq. (7.35) and the boundary conditions are given by any combination of Eqs. (7.40) to (7.42), then, in principle, one can solve the resulting system of three coupled ordinary differential equations as a function of the spatial variable η for the natural frequencies ω = ωnm and the corresponding mode ¯ x,nm (η), U ¯ θ,nm (η), and W ¯ nm (η). In addition, it is noted upon comparing shapes U V¯ (1) of Eq. (7.33) with the first term of the first equation of Eq. (B.95) of Appendix B that in the current non dimensional notation pij (x) → pij (η) = δij to2

(7.44)

where δ ij is the kronecker delta. Consequently, from Eq. (B.117) of Appendix B, the orthogonality condition is given by

L/a   ¯ θ,nm U ¯ nm W ¯ x,nm U ¯ x,nl + U ¯ θ,nl + W ¯ nl dη = δml Nnm U

(7.45)

0

where, in Eq. (B.117), B j = 0 since it has been assumed that there are no attachments on the boundaries and from Eq. (B.118)

Nnm

L/a  2 2 2 ¯ x,nm ¯ θ,nm ¯ nm U dη. = +U +W 0

(7.46)

7.3

Derivation of Governing Equations and Boundary Conditions: Donnell’s Theory

415

7.3 Derivation of Governing Equations and Boundary Conditions: Donnell’s Theory 7.3.1 Introduction The Donnell theory assumes that 1 + ξ/a = 1 in Eq. (7.3) and that uθ can be ignored in the expression for β θ given in Eq. (7.2). With these assumptions, Eq. (7.5) becomes ex = εx0 + ξ χx eθ = εθ0 + ξ χθ γxθ =

0 εxθ

(7.47)

+ ξ τs

where now ∂ u¯ x ∂ u¯ θ , εθ0 = +w ¯ ∂η ∂θ 1 ∂ 2 w¯ 1 ∂ 2 w¯ χx = − , χθ = − 2 a ∂η a ∂θ 2 ∂ u¯ x 2 ∂ 2 w¯ ∂ u¯ θ 0 εxθ = + , τs = − . ∂θ ∂η a ∂η∂θ εx0 =

(7.48)

Substituting Eq. (7.47) into Eq. (7.7), yields  

E  εx0 + ξ χx + ν εθ0 + ξ χθ σx =  1 − ν2  

E  εθ0 + ξ χθ + ν εx0 + ξ χx σθ =  1 − ν2 

0 + ξ τs . τxθ = G εxθ

(7.49)

Equation (7.49) is now used to obtain the force resultants and moment resultants from Eqs. (7.9) to (7.12). For the Donnell theory, the definitions of the force and moment resultants given by Eqs. (7.9) to (7.12) are simplified since it has been assume that 1 + ξ/a = 1. Then the force resultant definitions become ⎫ ⎧ ⎧ ⎫ ⎪ ⎪

h/2 ⎨ σx ⎬ Nx(D) ⎬ ⎨ σθ dξ = (7.50) Nθ(D) ⎩ ⎭ ⎪ ⎭ ⎩ (D) (D) ⎪ τ xθ Nxθ = Nθx −h/2 and moment resultants become ⎫ ⎧ ⎧ ⎫ ⎪ ⎪

h/2 ⎨ σx ⎬ Mx(D) ⎬ ⎨ (D) σθ ξ dξ . = Mθ ⎩ ⎭ ⎪ ⎭ ⎩ (D) (D) ⎪ τ xθ Mxθ = Mθx −h/2

(7.51)

416

7 Cylindrical Shells and Carbon Nanotube Approximations

Substituting Eq. (7.49) into Eqs. (7.50) and (7.51), respectively, we obtain Nx(D)

E  = 1 − ν2

h/2

 εx0 + χx ξ + νεθ0 + νξ χθ dξ

−h/2



= E˜ εx0 + νεθ0 = E˜ N¯ x(D) (D) Nθ

E  = 1 − ν2

h/2

 νεx0 + ξ νχx + εθ0 + χθ ξ dξ

−h/2

 = E˜ νεx0 + εθ0

(7.52)

(D)

= E˜ N¯ θ (D) Nxθ

= N¯ xθ

h/2  0 εxθ =G + ξ τ dξ −h/2

0 = Ghεxθ

0 ˜ o εxθ = EK

˜ o N¯ (D) = EK ˜ o N¯ (D) = EK xθ θx and Mx(D)

E  = 1 − ν2

h/2  εx0 ξ + χx ξ 2 + νεθ0 ξ + νχθ ξ 2 dξ −h/2

˜ a (aχx + νaχθ ) = Eah ˜ aM ¯ x(D) = Eah (D) Mθ

E  = 1 − ν2

h/2  νεx0 ξ + νχx ξ 2 + εθ0 ξ + χθ ξ 2 dξ −h/2

˜ a (νaχx + aχθ ) = Eah ¯ (D) ˜ aM = Eah θ (D) Mxθ

=M

(D)

Mθx

h/2  0 εxθ =G ξ + ξ 2 τs dξ −h/2

Gh3

˜ a Ko (aτs ) τs = Eah 12 ˜ a Ko M ¯ (D) = Eah ¯ (D) . ˜ a Ko M = Eah xθ θx =

(7.53)

7.3

Derivation of Governing Equations and Boundary Conditions: Donnell’s Theory

417

Substituting Eq. (7.48) into Eqs. (7.52) and (7.53), we obtain the following expressions for the non dimensional force resultants and moment resultants N¯ x(D) =

 ∂ u¯ θ ∂ u¯ x ∂ 2 w¯ ∂ 2 w¯ ¯ x(D) = − +ν + w¯ , M −ν 2 2 ∂η ∂θ ∂η ∂θ

∂ u¯ θ ∂ u¯ x +w ¯ +ν , ∂θ ∂η ∂ u¯ x ∂ u¯ θ (D) = N¯ θx = + , ∂θ ∂η

∂ 2 w¯ ∂ 2 w¯ − ν ∂θ 2 ∂η2 ¯ ∂ 2w ¯ (D) = −2 =M . θx ∂η∂θ

(D) N¯ θ =

¯ θ(D) = − M

(D) N¯ xθ

(D) ¯ xθ M

(7.54)

7.3.2 Contribution to the Total Energy The strain energy in the shell is determined from Eq. (7.20) by setting 1 + ξ/a = 1. Thus, a2 U= 2

h/2 (σx ex + σθ eθ + τxθ γxθ ) dξ dηdθ .

(7.55)

−h/2

A

Upon substituting Eq. (7.7) into Eq. (7.55), we obtain

a2 E  U=  2 1 − ν2

h/2  2 e2x + e2θ + 2νex eθ + Ko γxθ dξ dηdθ . A

(7.56)

−h/2

Substituting Eq. (7.47) into Eq. (7.56) yields

a2 E  U=  2 1 − ν2 + 2ν 2˜





=a E A

where

εx0

: h/2

2 2 εx0 + ξ χx + εθ0 + ξ χθ A

+ ξ χx

−h/2



V (D) dηdθ

εθ0



+ ξ χθ +

0 Ko εxθ

+ ξ τs

; 2 

dξ dηdθ

(7.57)

418

7 Cylindrical Shells and Carbon Nanotube Approximations

V

(D)

1 = 2h

h/2  2 (D) ξ (D) Qo + Q2 2 dξ a

(7.58)

−h/2

 1 (D) Qo + ha Q2(D) = 2 and

2

2 2 0 Qo(D) = εx0 + εθ0 + 2νεx0 εθ0 + Ko εxθ (D) Q2

(7.59)

= (aχx ) + 2ν (aχx ) (aχθ ) + (aχθ ) + Ko (aτs ) . 2

2

2

In Eq. (7.58), we have intentionally omitted the term containing ξ since the integral of this term is zero. Upon substituting Eq. (7.48) into Eq. (7.59), it is found that V (D) = V (Don) where V (Don) is given by Eq. (7.26). Thus, the minimization function given by Eq. (7.31) becomes

2 ˜ V¯ (D) dθ dη T − U + Wq = Ea

(7.60)

(7.61)

A

where V¯ (D) = V¯ (1) − V (Don) + V¯ (3)

(7.62)

and V¯ (1) and V¯ (3) are given in Eq. (7.33).

7.3.3 Governing Equations To obtain the governing equations, we again use Eq. (B.125) of Appendix B with G = V¯ (D) . Therefore, using Eqs. (7.34) with V¯ replaced by V¯ (D) , we obtain the following three coupled non dimensional partial differential equations governing the motions in a cylindrical shell according to Donnell’s theory ∂ 2 u¯ x ∂ 2 u¯ θ ∂ 2 u¯ x ∂w ¯ ∂ 2 u¯ x + Ko 2 + Kp =0 +ν − 2 ∂η ∂θ ∂η∂θ ∂η ∂τ 2 ∂ 2 u¯ θ ∂ 2 u¯ x ∂ 2 u¯ θ ∂ 2 u¯ θ ∂ w¯ + Ko 2 + − Kp + =0 ∂η∂θ ∂θ ∂η ∂θ 2 ∂τ 2 ∂ u¯ θ ¯ ∂ u¯ x ∂ 2w + +w ¯ + ha ∇ 4 w¯ + 2 = q¯ n ν ∂η ∂θ ∂τ where ∇ 2 and Kp are given by Eq. (7.36) and Ko is given by Eq. (7.14).

(7.63)

7.3

Derivation of Governing Equations and Boundary Conditions: Donnell’s Theory

419

7.3.4 Boundary Conditions We shall be considering complete cylindrical shells only; therefore, there are no boundaries along θ . In addition, it is assumed that the length of the shell is L so that we are interested in the boundary conditions at η = 0 and η = L/a. The boundary conditions can be obtained from Eq. (B.126a-d) of Appendix B. Since it has been assumed that there are no attachments to the shell at η = 0 and η = L/a, we set Alkn = Blkn = alkn = blkn = 0. Furthermore, it is seen from Eqs. (7.60) and (7.26) that only w¯ is a function of spatial derivatives higher than the first derivative. Consequently, Eqs. (B.126a-d) simplify to the following four conditions that must be satisfied at η = 0 and at η = L/a: (Don)

=0

(7.64a)

(Don)

=0

(7.64b)

either

u¯ x = 0

or

Vu¯ x,η

either

u¯ θ = 0

or

Vu¯ θ,η

and

and (Don)

either

w ¯ =0

or

∂Vw¯ ,ηη ∂η

(Don)

+

∂Vw¯ ,θη ∂θ

=0

(7.64c)

and either

∂ w¯ =0 ∂η

or

(Don)

Vw¯ ,ηη = 0

(7.64d)

where, based on the functional dependencies of V¯ (1) , V (Don) , and V¯ (3) , we have recognized that only V (Don) is a function of u¯ x,η , ..., w¯ ,θθ but not w¯ ,η , which is reflected Eq. (7.64c). Using Eq. (7.26) in Eq. (7.64a-d), we obtain the following boundary conditions at η = 0 and η = L/a either

u¯ x = 0

or

N¯ x(D) = 0

(7.65a)

either

u¯ θ = 0

or

(D) N¯ xθ = 0

(7.65b)

either

w¯ = 0

or

V¯ x(D) = 0

(7.65c)

and

and

and

420

7 Cylindrical Shells and Carbon Nanotube Approximations

either

∂w ¯ =0 ∂η

or

¯ x(D) = 0 M

(7.65d)

where ¯ (D) ¯ x(D) ∂M ∂M xθ − 2Ko . (7.66) ∂η ∂θ From Eqs. (7.65a-d), it is seen that one could specify many different combinations of boundary conditions at η = 0 and η = L/a. We shall limit our examination to the following three boundary conditions. V¯ x(D) = −

Clamped At a clamped end, the displacements and the slope of the radial displacement in the axial direction are zero; hence, the boundary conditions are u¯ x = u¯ θ = w¯ =

∂w ¯ = 0. ∂η

(7.67)

Hinged (Simply Supported, Pinned) and No Normal Constraint No normal constraint means that u¯ x does not have to equal zero at the boundary. Hence, the boundary conditions are ¯ x(D) = 0. N¯ x(D) = u¯ θ = w¯ = M

(7.68)

This combination of boundary conditions has been referred to as a shell with a shear diaphragm; see the discussion following Eq. (7.41). Free In a free edge, the following force resultants and moment resultant are zero (D) ¯ x(D) = 0. N¯ x(D) = N¯ xθ = V¯ x(D) = M

(7.69)

7.4 Natural Frequencies of Clamped and Cantilever Shells: Single-Wall Carbon Nanotube Approximations 7.4.1 Rayleigh-Ritz Solution We shall determine the natural frequencies of a cylindrical shell that is clamped at both ends and one that is clamped at one end and free at the other. These boundary conditions are usually employed for the cylindrical shell model of singlewall carbon nanotubes. The hinged shear diaphragm boundary condition has little application to carbon nanotubes and is, therefore, not considered for single-wall nanotubes. However, when double-wall carbon nanotubes are considered we shall use these boundary conditions in order to reduce considerably the mathematical complexity while still providing some insight into such a system’s characteristics.

7.4

Natural Frequencies of Clamped and Cantilever Shells: Single-Wall. . .

421

Because of the complexity of the governing equations and boundary conditions, we shall use the Rayleigh-Ritz procedure and consider two shell theories: that of Flügge and that of Donnell. For both theories, we set qn = 0, and therefore, Wq = 0. To employ the Rayleigh-Ritz method, it is assumed that the shell is vibrating at a frequency ω and with the displacement amplitudes u¯ x (η, θ ), u¯ θ (η, θ ), and w¯ (η, θ ). Then, for the Flügge theory, we form the quantity [recall Eq. (3.280)]

 = Tmax − Umax

ηL 2π   ˜ 2  2 2 Ea  u¯ x + u¯ 2θ + w = ¯ 2 − 2V (F) dθ dη 2 0

(7.70)

0

where  = ωto , to is given by Eq. (7.28), ηL = L/a, and V (F) is given by Eq. (7.25). We assume that ¯ x,n (η) cos nθ u¯ x (η, θ ) = U ¯ θ,n (η) sin nθ u¯ θ (η, θ ) = U

(7.71)

¯ n (η) cos nθ w¯ (η, θ ) = W ¯ x,n , U ¯ θ,n , and W ¯ n will be defined subsequently. Substituting Eq. (7.71) into where U Eqs. (7.70), (7.25), and (7.26), we obtain ηL   ˜ 2π  2 2 Ea 2 ¯ x,n + U ¯ θ,n ¯ n2 − 2Vn(F) dη  U +W = 2

(7.72)

0

where Vn(F)

,  ¯n ¯n ¯ x,n ∂ 2 W ∂ 2W h ∂U a 2 ¯2 ¯ θ,n W = Vn(Don) + + 1 − 2n − 2νn U −2 n 2 2 ∂η ∂η ∂η2 :  ;- (7.73)  ¯ θ,n 2 ¯n ¯ θ,n ∂ W ¯n ∂U ∂W ∂U 2 ¯2 2¯ + Ko 3 + n Ux,n + 6n + 2n Ux,n ∂η ∂η ∂η ∂η

and Vn(Don)

,

 ¯ x,n 2 ¯ x,n     ∂U ∂U ¯ θ,n + W ¯ θ,n + W ¯ n + nU ¯n 2 nU + 2ν ∂η ∂η "    ¯ θ,n 2 ¯n 2 ¯n ∂ 2W ∂U ∂ 2W ¯ x,n + ¯n + Ko −nU + ha − 2νn2 W 2 ∂η ∂η ∂η2 2 # ¯ ∂ W n ¯ n2 + 4Ko n2 + n4 W . ∂η

1 = 2

(7.74)

422

7 Cylindrical Shells and Carbon Nanotube Approximations

It is mentioned that all terms in Eq. (7.72) contain the definite integrals of either cos2 (nθ ) or sin2 (nθ ) over the interval 0 to 2π . When these integrals are evaluated, the results are equal to π . ¯ x,n , U ¯ θ,n and W ¯ n are of the form We assume that U

¯ x,n (η) = U ¯ θ,n (η) = U ¯ n (η) = W

M  m=1 M  m=1 M 

Anm ϕm (η) Bnm ϕm (η)

(7.75)

Cnm ϕm (η)

m=1

where Anm , Bnm , and Cnm are unknown constants, M is an appropriately chosen integer, the prime denotes the derivative with respect to η, and ϕm (η) is the mth mode shape of an Euler-Bernoulli beam of length L. This beam function will be chosen to satisfy approximately the boundary conditions of the shell. In other words, if the shell is clamped at both ends, then the beam mode shape will be that for a beam clamped at both ends. Equation (7.75) is substituted into Eqs. (7.72) to (7.74), and the Rayleigh-Ritz procedure discussed in Section 3.8.5 is applied for three sets of unknown constants; that is, ∂ = 0, ∂Ani

∂ = 0, ∂Bni

∂ =0 ∂Cni

i = 1, 2, ..., M.

(7.76)

After performing the indicated operations and a considerable amount of algebra, we obtain the following system of equations M  m=1 M  m=1 M 

a1km (n) Anm + a2km (n) Anm + a3km (n) Anm +

m=1

M  m=1 M  m=1 M  m=1

b1km (n) Bnm + b2km (n) Bnm + b3km (n) Bnm +

M  m=1 M  m=1 M  m=1

c1km (n) Cnm − 2 c2km (n) Cnm − 2 c3km (n) Cnm − 2

M  m=1 M  m=1 M 

I2km Anm = 0 I1km Bnm = 0 I1km Cnm = 0

m=1

k = 1, 2, ..., M (7.77) where

Natural Frequencies of Clamped and Cantilever Shells: Single-Wall. . .

7.4

423

a1km (n) = I3mk + Ko n2 I2mk b1km (n) = nνI4km − nKo I2mk   c1km (n) = νI4km + ha Ko n2 I2mk − I3mk a2km (n) = nνI4mk − nKo I2mk b2km (n) = Ko I2mk + n2 I1mk c2km (n) = nI1mk + ha [3Ko nI2mk − nνI4mk ]   a3km (n) = νI4mk + ha Ko n2 I2mk − I3mk

(7.78)

b3km (n) = nI1mk + ha [3Ko nI2mk − nνI4km ]

c3km (n) = I1mk + ha I3mk − νn2 (I4km + I4mk ) + n4 I1mk  

+ 4Ko n2 I2mk + 1 − 2n2 I1mk and

ηL I1mk =

ηL ϕm ϕk dη,

I2mk =

0

ηL I3mk = 0

ϕm ϕk dη

0

ϕm ϕk dη,

ηL I4mk =

(7.79) ϕm ϕk dη.

0

In arriving at Eq. (7.77), we have set 1 ± ha = 1 and 1 + 3ha = 1. It is noted that I1mk , I2mk , and I3mk are symmetric quantities, whereas I4mk is not. Thus, the order of the subscripts in the quantity I4mk is significant. When the Donnell shell theory is used, these coefficients reduce to a1km (n) = I3mk + Ko n2 I2mk b1km (n) = nνI4km − nKo I2mk c1km (n) = νI4km a2km (n) = nνI4mk − nKo I2mk b2km (n) = Ko I2mk + n2 I1mk c2km (n) = nI1mk a3km (n) = νI4mk b3km (n) = nI1mk

 c3km (n) = I1mk + ha I3mk − νn2 (I4km + I4mk ) + n4 I1mk + 4Ko n2 I2mk . Equation (7.77) can be written in matrix form as follows

(7.80)

424

7 Cylindrical Shells and Carbon Nanotube Approximations

⎫ ⎤⎧ [a1km (n)] [b1km (n)] [c1km (n)] ⎨ [An1 , ..., AnM ]T ⎬ ⎣ [a2km (n)] [b2km (n)] [c2km (n)] ⎦ [Bn1 , ..., BnM ]T ⎭ ⎩ [a3km (n)] [b3km (n)] [c3km (n)] [Cn1 , ..., CnM ]T ⎫ ⎡ ⎤⎧ [I2km ] [0] [0] ⎨ [An1 , ..., AnM ]T ⎬ −2 ⎣ [0] [I1km ] [0] ⎦ [Bn1 , ..., BnM ]T = 0 ⎩ ⎭ [0] [0] [I1km ] [Cn1 , ..., CnM ]T ⎡

(7.81)

where each sub matrix in Eq. (7.81) is an M×M matrix whose elements are given by Eq. (7.78) or by Eq. (7.80) as the case may be. The superscript T indicates the transpose of the matrix. It is seen that Eq. (7.81) is a standard eigenvalue formulation that can be solved by readily available procedures yielding, for each value of n, 3M natural frequency coefficients nm , m = 1, 2, . . . , 3M and the corresponding mode shapes. When n = 0, Eq. (7.81) reduces to 

 [a1km (0)] [c1km (0)] [I2km ] 0 − 2 [a3km (0)] [c3km (0)] 0 [I1km ]



[An1 , ..., AnM ]T [Cn1 , ..., CnM ]T

 = 0 (7.82)

where, for the Flügge theory, a1km (0) = I3mk c1km (0) = νI4km − ha I3mk a3km (0) = νI4mk − ha I3mk

(7.83)

c3km (0) = I1mk + ha I3mk and for the Donnell theory, a1km (0) = I3mk c1km (0) = νI4km a3km (0) = νI4mk

(7.84)

c3km (0) = I1mk + ha I3mk . This type of vibration is referred to as the breathing mode; that is, the radial displacement is independent of the angular location.

7.4.2 Numerical Results When a shell that is clamped at one end and free at the other end is considered, we use for ϕ m the mth mode shape of a cantilever beam. This beam function satisfies exactly the boundary conditions at the clamped end given by either Eq. (7.40) for the Flügge theory or Eq. (7.67) for the Donnell theory, but only approximately satisfies the boundary conditions given by Eq. (7.42) for the Flügge theory or Eq. (7.69) for the Donnell theory at the free end. From Case 3 of Table 3.3, we have for a cantilever beam that

7.4

Natural Frequencies of Clamped and Cantilever Shells: Single-Wall. . .

ϕm (η) = −Dm T (λm η) + S (λm η)

425

(7.85)

where Dm =

T (m ) , Q (m )

λm =

m ηL

(7.86)

and m are the roots of cos (m ) cosh (m ) + 1 = 0.

(7.87)

The values of m are given in Case 3 of Table 3.5. The functions T(x), etc. are given by Eq. (C.19) of Appendix C. Using Eq. (C.20) of Appendix C, it is found that ϕm (η) = λm [−Dm S (λm η) + R (λm η)] ϕm (η) = λ2m [−Dm R (λm η) + Q (λm η)] .

(7.88)

When a shell is clamped at both ends, we select for ϕ m the mth mode shape for a beam clamped at both ends. This function satisfies exactly the shell boundary conditions at both ends, which are given by Eq. (7.40) for the Flügge theory or (7.67) for the Donnell theory. From Case 2 of Table 3.3, we have for a beam clamped at both ends that ϕm (η) = −Dm T (λm η) + S (λm η)

(7.89)

where Dm =

S (m ) T (m )

(7.90)

and m are the roots of cos (m ) cosh (m ) − 1 = 0.

(7.91)

The values of m are given in Case 2 of Table 3.5. Using these relations, the lowest natural frequencies  = n1 , n = 1, 2, 3, are determined from Eq. (7.81) for a cantilever shell and for a shell clamped at both ends. The results are shown in Figs. 7.2 and 7.3, respectively, for the Flügge and Donnell theories. For each of these cases M = 4 and ν = 0.2. It is seen from these figures that the boundary conditions greatly influence the magnitude of the natural frequency coefficient corresponding to n = 1. However, the boundary conditions have a very small effect on the natural frequency coefficient n1 when n ≥ 2 and L/a ≥ 16. It is also noted that the lowest natural frequency coefficient isn’t always associated with n = 1; as L/a → 0, it is seen that when n = 2 and n = 3 lower natural frequencies can occur.

426

7 Cylindrical Shells and Carbon Nanotube Approximations

100

100

n=3 10−1

10−1 n=3

Δn1

Δn1

n=2

n=2

n=1 10−2

10−3 0 10

10−2

Donnell Flugge Euler beam Timoshenko beam

n=1

101

102

10−3 0 10

Donnell Flugge Euler beam

n=1

Timoshenko beam

n=1

101

L/a

L/a

(a)

(b)

102

Fig. 7.2 Natural frequency coefficient n1 for a cantilever shell according to the Flügge and Donnell theories as a function of L/a for v = 0.2 and n = 1, 2, 3 (a) a/h = 15 and (b) a/h = 50

100

100

n=3 −1

10

10−1

n=3

Δn1

Δn1

n=2

n=2

n=1 −2

10

Donnell Flugge Euler beam

−3

10

100

10

−2

10

−3

n=1

Timoshenko beam 101

102

Donnell Flugge Euler beam

n=1

Timoshenko beam

100

101

L/a

L/a

(a)

(b)

n=1 102

Fig. 7.3 Natural frequency coefficients n1 for a shell clamped at both ends according to the Flügge and Donnell theories as a function of L/a for v = 0.2 and n = 1, 2, 3 (a) a/h = 15 and (b) a/h = 50

Also appearing in these figures are the lowest natural frequency coefficients for an Euler-Bernoulli beam and for a Timoshenko beam. The lowest natural frequency coefficient of the Euler-Bernoulli beam 1 in terms of the shell natural frequency coefficient  is determined from  = 21 tplate /tbeam , where tplate is given by to in Eq. (7.28) and tbeam is given by to in Eq. (3.53). Thus,

a2  = 21 L



  Io 1 − ν 2 . Ao

(7.92)

7.4

Natural Frequencies of Clamped and Cantilever Shells: Single-Wall. . .

427

For a thin-walled beam with a circular cross section that has an outer radius equal to a + h/2 and inner radius equal to a − h/2, 1 Io = Ao 4

,      a2 h 2 h 2 h2 a2 + a− a+ 1+ 2 ≈ . = 2 2 2 2 4a

(7.93)

Then Eq. (7.92) becomes 2  =  1 2 L a

  1 − ν2 . 2

(7.94)

From Table 3.5, it is found that for a cantilever beam, 1 = 0.5969π and for a beam clamped at both ends 1 = 1.5056π . It is seen in Figs. 7.2 and 7.3 that there are certain ranges of L/a where the Euler-Bernoulli beam predicts the shell’s lowest natural frequency coefficient for n = 1 for the Flügge’s theory. However, the ranges in which the shell theory and the Euler-Bernoulli beam theory agree vary substantially with the value of L/a, the boundary conditions, and the value of a/h and, therefore, the thin beam approximation is difficult to apply. For the Timoshenko beam theory, Eq. (7.94) becomes

2    1(T) 1 − ν2  =  2 2 L a

(7.95)

(T)

where, for the beam clamped at both ends, 1 are the roots of the characteristic equations for Case 1 of Table 5.2 that are shown in Fig. 5.3. For the case of (T) a cantilever beam, 1 are the roots of the characteristic equations for Case 3 of (T) Table 5.2 that are shown in Fig. 5.4. In addition, 1 is a function of Ro = ro /L, which upon using Eqs. (5.38) and (7.93) can be written as 1 Ro = √   . 2 L a

(7.96)

It is seen in Figs. 7.2 and 7.3 that the Timoshenko beam theory predicts reasonably well the lowest natural frequency coefficient for n = 1 of Flügge’s theory over the range L/a > 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 << 1, η (x, y, t) is arbitrary to within a class of twice differentiable continuous functions and subject to the requirement that η (x, y, t) and its derivatives are zero at t1 and t2 , and we have again employed the shorthand notation η = ~   η (x, y, t)|Cj = η Cj to indicate functions defined on the boundaries C1 and C2 . In addition, both η and/or its normal derivative ∂η/∂n = η,n (an outward normal)

2

When F and F (Cj ) are independent of y, Eq. (B.4) is written as

x2 L= x1

Fdx + F (C1 ) .

Appendix B: Variational Calculus: Generation of Governing Equations, Boundary. . .

445

may be subject to constraints consistent with the restrictions on Cj imposed upon   the functions eligible for making I u, ~ u an extremum. We define the variation of u and ~ u as δu = εη = u¯ − u

(B.7)

δu u. u¯ − ~ ~ = εη = ~   Then, the necessary condition to make I u, ~ u an extreme is ! !!   d d !! = = 0. I u¯ , ~ I u + εη, ~ u + εη !! δ I u¯ , ~ u¯ ! u¯ = ~ ε=0 dε dε ε=0

(B.8)

Upon substituting Eq. (B.6) into Eq. (B.5), we obtain



t2 :

I u + εη, ~ u + εη = ~ t1

; ¯ Fdxdy + F¯ (C1 ) + F¯ (C2 ) dt

(B.9)

R

where  F¯ = F x, y, t, u + εη, u˙ + εη, ˙ u,x + εη,x , u,y + εη,y , u,xx + εη,xx , u,xy  + εη,xy , u,yy + εη,yy   + ε , u u + εη, ~ + εη , + ε F¯ (C1 ) = F (C1 ) x, y, t, ~ u˙ u˙ η˙ η˙ ~ ~,x ~,x ~ ~,x ~,x   F¯ (C2 ) = F (C2 ) x, y, t, ~ u,y + εη , ~ u + εη, ~ u˙ + ε η˙ , ~ u˙ ,y + ε η˙ . ~ ~,y ~ ~,y

(B.10)

Using Eq. (B.8) and noting that ∂α ∂ F¯ ∂ F¯ ∂α = = F¯ α ∂ε ∂α ∂ε ∂ε C C ) ) ( ( ∂ F¯ j ∂ F¯ j ∂β (C ) ∂β = = F¯ β j ∂ε ∂β ∂ε ∂ε

α = u¯ , u˙¯ , u¯ ,x , ..., u¯ ,yy (B.11) β=~ u¯ x , ..., ~ u¯ , ~ u˙¯ ,y u˙¯ , ~

j = 1, 2

where F¯ α =

∂ F¯ ∂α

and

(C ) F¯ β j =

∂ F¯ (Cj ) ∂β

(B.12)

446

Appendix B: Variational Calculus: Generation of Governing Equations, Boundary. . .

the variation of Eq. (B.9) can be written as   !! d δ I u¯ , ~ I u + εη, ~ u + εη !! u¯ = ~ ε=0 dε :

t2

 = ˙ u˙ + η,x Fu,x + η,y Fu,y + η,xx Fu,xx + η,xy Fu,xy ηFu + ηF t1

R

 (C ) (C ) (C ) + η,yy Fu,yy dxdy + η (C1 ) F u 1 + η˙ (C1 ) F u˙ 1 + η (C1 ) F u 1 ~ ~ ,x ~ ~ ~,x ~

! since F¯ !ε→0

(C ) (C ) (C ) (C ) + η˙ (C1 ) F u˙ 1 +η (C2 ) F u 2 + η˙ (C2 ) F u˙ 2 + η (C2 ) F u 2 ~ ~ ,y ~ ~,x ~ ~,y ~,x ~ < (C2 ) + η˙ (C2 ) F u˙ dt ~,y ~,y (B.13) ! ! = F, F¯ u¯ !ε→0 = Fu , etc., and u¯ |ε→0 = u, u¯ ,x !ε→0 = u,x , etc. The

quantities η,α and η,β are given by Eq. (B.3) with u replaced by η and η as the case ~     may be. The notations η Cj and η Cj indicate that η and η are given on the ~,α ~ ~,α ~ specified portion of the boundary. We shall now go through several steps to put Eq. (B.13) into a more useful form. We start by noting that

t2

t2 ηF ˙ u˙ dt = −

t1

t2

t1

  (Cj ) η˙ Cj Fu˙ dt = − ~

t1

t2 t1

t2 t1

η

t2 t1

(C1 ) η˙ (C1 ) F u˙ dt ~,x ,x

~

(C2 ) η˙ (C2 ) F u˙ dt ~,y ,y

~

t2 =− t1

t2 =− t1

∂Fu˙ dt, ∂t

C   ∂Fu(˙ j ) dt η C ~ j ∂t 1) ∂F (C u˙ ,x η (C ) ~ dt ~,x 1 ∂t

(B.14)

(C )

∂F u˙ 2 ~,y dt η (C ) ~,y 2 ∂t

since we have required that η and η and their spatial derivatives vanish at t1 and ~ t2 . We shall restrict our discussion to a rectangular region that is aligned such that its respective edges are parallel to the (x,y) coordinate axes. Substituting Eq. (B.14) into Eq. (B.13), we obtain

Appendix B: Variational Calculus: Generation of Governing Equations, Boundary. . .



t2 :  ∂Fu˙ + η,x Fu,x + η,y Fu,y + η,xx Fu,xx η Fu − δ [I (¯u)] = ∂t t1 R < + η,xy Fu,xy + η,yy Fu,yy dxdy + B12 (C1 , C2 ) dt = 0

447

(B.15)

where B12 (C1 , C2 ) =

2 2               η xj , y, t H3 u xj , t , 1 + η,x xj , y, t H3 u,x xj , t , 1 j=1

+

j=1

2 

2              η x, yj , t H3 u yj , t , 2 + η,y x, yj , t H3 u,y yj , t , 2

j=1

j=1

(B.16) and l) ∂Fα(C ˙ . ∂t

H3 (α, l) = Fα(Cl ) −

(B.17)

In arriving at Eq. (B.16), it has been assumed that C1 corresponds to x1 and x2 and  C2 corresponds to y1 and y2 and, therefore, on C1 , we have set η (C1 ) = η xj , y, t ,       η,x (C1 ) = η,x xj , y, t , ~ u = u xj , t , and ~ u,x = u,x xj , t and on C2 we have set         u = u yj , t and ~ u,y = u,y yj , t . η (C2 ) = η x, yj , t , η,y (C2 ) = η,y x, yj , t , ~ Next, Green’s theorem, which is given as

 R

∂P ∂Q + dxdy = ∂x ∂y

(Pdy − Qdx)

(B.18)

C

is used to further simplify Eq. (B.15) by considering the following four cases for P and Q. i) When P = ηFu,x and Q = ηFu,y , then



  η,x Fu,x + η,y Fu,y dxdy = −

R





η R

+

∂Fu,y ∂Fu,x + dxdy ∂x ∂y

  η Fu,x dy − Fu,y dx

(B.19a)

C



ii) When P = η,x Fu,xx − η∂Fu,xx ∂x and Q = 0, then





η,xx Fu,xx dxdy =

R

R

∂ 2 Fu,xx η dxdy + ∂x2



 ∂Fu,xx dy (B.19b) η,x Fu,xx − η ∂x

C

Appendix B: Variational Calculus: Generation of Governing Equations, Boundary. . .

448

iii) When P = 0 and Q = η,y Fu,yy − η∂Fu,yy ∂y, then





η,yy Fu,yy dxdy = R

η R



iv) When P = 0.5 η,y Fu,xy then

R

∂ 2 Fu,yy ∂y2



 ∂Fu,yy dxdy − dx (B.19c) η,y Fu,yy − η ∂y C

   − η∂Fu,xy ∂y and Q = 0.5 η,x Fu,xy − η∂Fu,xy ∂x ,



 ∂Fu,xy 1 η,xy Fu,xy dxdy = η dxdy + dy η,y Fu,xy − η ∂x∂y 2 ∂y R C 

 ∂Fu,xy 1 η,x Fu,xy − η − dx 2 ∂x



∂ 2 Fu,xy

C

(B.19d) Upon substituting Eqs. (B.19a-d) into Eq. (B.15), we obtain

t2 :

δ [I (¯u)] = t1

;   η L1 (u (x, y, t)) − L2 (˙u (x, y, t)) dxdy + BC (η, u) dt = 0

R

(B.20) where L1 (u (x, y, t)) = Fu − ∂Fu˙ L2 (˙u (x, y, t)) = ∂t

∂ 2 Fu,yy ∂ 2 Fu,xy ∂Fu,y ∂Fu,x ∂ 2 Fu,xx + + − + ∂x ∂y ∂x2 ∂y2 ∂x∂y

(B.21)

and  

 ∂Fu,yy 1 1 ∂Fu,xy − η,x Fu,xy − η,y Fu,yy + η + − Fu,y dx BC (η, u) = 2 2 ∂x ∂y C  

 ∂Fu,xx 1 1 ∂Fu,xy + η,x Fu,xx + η,y Fu,xy − η + − Fu,x dy 2 ∂x 2 ∂y C

+ B12 (C1 , C2 ) . (B.22) The boundary integrals in Eq. (B.22) can be simplified further as follows. Referring to Fig. B.1, if the outward normal to C at a point p is n and s is in a direction tangent to C at a point p , then

Appendix B: Variational Calculus: Generation of Governing Equations, Boundary. . .

449

y

Fig. B.1 Normal and tangent to boundary C at point p

s p′

θ x

C

dx = sin θ dn dx = cos θ ds

dy = − cos θ dn dy = sin θ ds

n

(B.23)

and ∂ ∂ ∂ = sin α − cos α ∂n ∂x ∂y ∂ ∂ ∂ = cos α + sin α ∂s ∂x ∂y

(B.24)

where θ is the angle between tangent s and the x-axis. Equation (B.24) has as its inverse ∂ ∂ ∂ = sin θ + cos θ ∂x ∂n ∂s ∂ ∂ ∂ = − cos θ + sin θ . ∂y ∂n ∂s

(B.25)

Upon using Eqs. (B.24) and (B.25) in Eq. (B.22), noting that dx = (dx/ds) ds = cos (θ ) ds and dy = (dy/ds) ds = sin (θ ) ds, and collecting terms, we arrive at 

 ∂Fu,xx 1 ∂Fu,xy BC (η, u) = η Fu,x − − sin θ 2 ∂y ∂x C   ∂Fu,yy 1 ∂Fu,xy − Fu,y − − cos θ ds 2 ∂x ∂y 

  1 ∂η  + Fu,xx − Fu,yy sin θ cos θ + Fu,xy sin2 θ − cos2 θ ds ∂s 2 C

  + η,n Fu,xx sin2 θ + Fuyy cos2 θ − Fu,xy sin θ cos θ ds

C

+ B12 (C1 , C2 ) . (B.26)

Appendix B: Variational Calculus: Generation of Governing Equations, Boundary. . .

450

Since C is a closed curve, we have that

C

∂η [...] ds = − ∂s

η C

∂ [...] ds. ∂s

Then Eq. (B.26) becomes 

 ∂Fu,xx 1 ∂Fu,xy − sin θ 2 ∂y ∂x C   ∂Fu,yy 1 ∂Fu,xy − Fu,y − − cos θ 2 ∂x ∂y 

   1 ∂  ds − Fu,xx − Fu,yy sin θ cos θ + Fu,xy sin2 θ − cos2 θ ∂s 2

  + η,n Fu,xx sin2 θ + Fu,yy cos2 θ − Fu,xy sin θ cos θ ds

BC (η, u) =

η

Fu,x −

C

+ B12 (C1 , C2 ) . (B.27) When R is a rectangular region and we set θ = 90◦ , it is seen from Fig. B.1 that n coincides with the x-axis and s with the y-axis. On the other hand, if we set θ = 0◦ , it is seen that n coincides with the y-axis and s with the x-axis. In addition, C1 is composed of two parallel lines, one located at x1 and the other at x2: x2 > 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 <1

Special case of 10: ζ = 0

484

Appendix C: Laplace Transforms and the Solutions to Ordinary Differential Equations Table C.1 (continued) G (s)

12 13 14 15 16 17 18

ω s2 + ω2 s s2 − ω2 ω s2 − ω2

g (t)

Description

sin (ωt)

Special case of 8: ζ = 0

cosh (ωt) sinh (ωt)

2a3 s4 − a4

sinh (at) − sin (at)

2a2 s − a4

cosh (at) − cos (at)

2as2 − a4

sinh (at) + sin (at)

2s3 s4 − a4

cosh (at) + cos (at)

s4 s4

Reference Roberts GE, Kaufman H (1966) Table of Laplace transforms. Saunders, Philadelphia, PA

Index

A Added mass, 27, 237 AFM, see Atomic force microscope Amplitude response, 11, 16, 22–23, 29, 40, 62–64, 72, 231–239 Atomic force microscope, 26, 31, 38, 182 beam, Euler-Bernoulli, 217 motion scanner, 73 Axial force, see Force Axial strain, 84, 87, 274 Axial stress, 85, 87, 274 B Bandwidth, 12 Base excitation single degree-of-freedom system, 10, 15, 22, 48, 54–55 two degree-of-freedom system, 56, 64 See also Boundary excitation Beam approximation of cylindrical shell, 424 Beams, Euler-Bernoulli, 84 axial force, 4, 89, 98, 149, 180, 240 boundary attachments, 93, 126, 130, 137, 144, 151, 167, 178, 183, 259–260 boundary conditions, 96, 102–103, 121, 144, 170, 187, 205, 211, 227, 260 cantilever, 126, 133, 137, 140, 167, 209, 215, 250 cantilever with extended mass, 150 cantilever with step change in properties, 183 cantilever, damped, 231 cantilever, tapered, 178, 182 characteristic equation, 111, 138, 149, 154, 160, 171, 174, 178, 189, 195, 200, 203, 241 clamped-clamped, 126, 137, 140, 240, 244 clamped-clamped with in-span mass, 130 clamped-clamped with in-span spring, 130

clamped-clamped with in-span spring and mass, 130 concentrated mass, 88, 134, 152, 163, 201, 210, 227 damped, 221, 224 elastic foundation, 4, 90, 149 elastically connected, 197, 199, 201 electrostatic attraction, 239, 243, 250 energy harvester, piezoelectric, 252, 260 forced excitation, 203, 216, 224, 231, 260 free-free, 131 governing equation, 95, 101, 103, 144, 165, 170, 187, 198, 201, 205, 210, 239, 243, 260 hinged-hinged, 137, 150 in-span attachments, 90, 92, 96, 101, 103, 109, 111, 143–144, 154, 163, 201 in-span rigid support, 115–116, 131, 193 mode shape, 103, 116–118, 139–140, 149, 163, 172, 174, 189, 193, 200, 241 natural frequency, 103, 119, 138, 140, 148, 160, 171, 174, 180, 189, 191, 232, 241, 247 single degree-of-freedom system, 92, 96, 101, 137 squeeze film damping, 224, 238 structural damping, 233 two degree-of-freedom system, 144 variable cross section, 164, 168, 173, 182, 183 viscous air damping, 235 viscous damping, 233 viscous fluid damping, 236 Beams, Timoshenko, 273 axial force, 277, 282, 286–287 boundary attachments, 281, 285, 287, 312–313, 326, 330 boundary conditions, 281, 284, 318, 328–329

485

486 Beams, Timoshenko (cont.) cantilever, 304, 313–314 characteristic equation, 294–300, 312, 322, 327 clamped-clamped, 304 comparison with Euler-Bernoulli theory, 284, 301 concentrated mass, 276, 297, 314 elastic foundation, 277, 287 elastically connected, 325 forced excitation, 328 governing equations, 284, 325, 329 hinged-hinged, 300–301 in-span attachments, 276–277, 287, 296, 305, 308 in-span rigid support, 297, 299 mode shape, 287, 295–300, 326 natural frequency, 287, 301–302, 305–310, 314, 322 single degree-of-freedom system, 281, 297 variable cross section, 310 Beams, variable cross section, 165, 310 exponential taper, 166, 173, 313 linear taper, 166, 168, 314 step change in properties, 314 triangular taper, 182 Bending moment, 84, 275, 344, 354, 402, 405, 413, 415 Bending stiffness, effective, 257 Bimorph beam energy harvester, 201 Boundary attachments, see Beams; Plates Boundary condition, electric, 261 Boundary conditions, see Beams; Plates; Shells, cylindrical Boundary excitation harmonic, 217, 261 time-dependent, 210 Breathing mode, 424 Buckling, 150, 182, 242, 363 C Capacitance, 30, 45, 48, 91 Capacitor, 91 Carbon nanotube properties, 428 Carbon nanotubes, see Shells, cylindrical; Double-wall carbon nanotube Center frequency, 12 Center of mass, 150, 156 Charge, 31, 43, 45, 254, 258 Circular plates, see Plates Coil impedance, 54, 56 Compliance, piezoelectric, 44 Concentrated mass, see Beams; Plates

Index Continuity conditions, 155, 157, 184, 188, 193, 317, 318 Coupling coefficient, 41 Critically damped, 10 Current, 259 Curvature, 86, 343 Cutoff frequency, 12 Cylindrical shells, see Shells, cylindrical D Damping Macro scale, 232 MEMS scale, 232 squeeze film, 17, 224, 238 structural, 15, 222, 233 viscous, 8, 223, 233, 260 viscous air, 223, 235 viscous fluid, 26, 223, 236 Damping coefficient squeeze film, 17, 20 viscous, 8 viscous fluid, 27 Damping controlled region, 11 Damping factor, 10 equivalent, 26 Dielectric constant, see Permittivity Double-wall carbon nanotube beam approximation, 199, 325 cylindrical shell approximation, 432 E Elastic foundation, see Beams; Plates Elastically connected beams, see Beams Electric charge, 43, 254 Electric enthalpy, 252–253 Electric field strength, 43, 254 Electric potential, 255 Electromagnetic coupling factor, 53 Electromechanical coupling coefficient, 44 alternative, 47 Electromechanical coupling constant, 43 Electrostatic attraction beam, 91 clamped-clamped beam, 243 effect on natural frequency, 37, 247 single degree-of-freedom system, 31 Energy harvester enhanced piezoelectric, 65 permanent magnet, 53 piezoelectric layered beam, 252, 260 piezoelectric single degree-of-freedom system, 42

Index Energy, contributions to beam, Euler-Bernoulli, 84, 252 beam, Timoshenko, 275 cylindrical shell, Donnell’s theory, 417 cylindrical shell, Flügge’s theory, 407 plate, 344, 391 Euler-Bernoulli law, 87 Euler-Lagrange equation, 95, 280, 392, 411, 451, 453 Extensional vibrations, see Plates External pressure, 410 F Filter high pass RC, 67 MEMS, 71 Flexural rigidity, 342 Force axial, 89, 103, 149, 205, 240, 277 axial due to transverse load, 90 in-plane, 239, 343, 346, 356, 359 shear, 87, 275, 350, 354 transverse, 89, 205, 222, 258, 276, 344 Force resultant, 404, 407, 415, 417 shear, Flügge’s theory, 413 Forced excitation beam, Euler-Bernoulli, 203, 216, 227, 231, 260 beam, Timoshenko, 329 circular plate, 382 Frequency response function, 67, 72, 264 G Geometric mean frequency, 12 Grippers, 201 H Hamaker constant, 38 Hydrodynamic function, 26, 80, 224 I Impulse force two degree-of-freedom system, 59 Impulse response beam, Euler-Bernoulli, 208 beam, Timoshenko, 333 plate, circular, 387 In-span attachments, see Beams Inviscid fluid, 28

487 K Kelvin-Kirchhoff edge reactions, 350 Kinetic energy beam, Euler-Bernoulli, 88 beam, Timoshenko, 275 beam, two layer, 253 concentrated mass, 88, 275 cylindrical shell, 409 maximum, 176, 310, 371 plate, in-plane, 390 plate, transverse, 345 rotational inertia, 276 single degree-of-freedom system, 92, 278 Kirchhoff’s hypothesis, 403 Knudsen number, 19 L Laplace transform, 57, 73, 105, 147, 201, 211, 289, 319, 477 Load impedance, 46 Load resistor, 48, 50, 55, 65, 68, 259, 266 M Mass controlled region, 12 Mass, equivalent, 136 Maxwell equations, 254 MEMS biosensor, 133 damping, 16 mechanical filter, 71 piezoelectric energy harvester, 252 resonator, 149 RF filter, 390 Middle plane, 342 See also Middle surface Middle surface, 346, 402 twist of, 404 Minimization function beam, Euler-Bernoulli, 93, 259 beam, Timoshenko, 280 cylindrical shell, 410 plate, in-plane motion, 392 plate, transverse motion, 347 Modal amplitude, 206, 382 Moment of inertia, 86 Moment, externally applied, 276 N Neutral axis, 84–85, 89, 94, 240, 243, 252, 257, 274, 342 Nodal circle, 363 Nodal diameter, 363

488 Node point, 119, 131, 197, 208, 309 strain, 119 Non dimensional form beam, Euler-Bernoulli, 101, 186, 198, 225, 240, 262 beam, Timoshenko, 284, 318, 325 cylindrical shells, 411, 418, 434 plate, 352, 355 single degree-of-freedom system, 10 two degree-of-freedom system, 57, 66 Nonlinear beam, 96, 240 electrostatic force, 32, 240 squeeze film damping, 25 O Open circuit, 49, 67, 264, 267 Orthogonal function, 100, 203, 206, 227, 242, 263, 328, 385, 414 Orthogonality condition, 463, 467, 472 beam, Euler-Bernoulli, 204, 229–230 beam, Timoshenko, 328 cylindrical shell, 414 plate, circular, 385 Overdamped, 10 P Pass band ripple, 72 Period beam, 210 undamped oscillations, 10 Permittivity, 32, 44, 91 Phase response, 11, 15, 22, 29, 40 Piezoelectric 33 and 31 modes, 42, 252 constitutive relations, 43, 254 coupling constant, 43, 254 Plane stress, 342 Plates, 341 annular carrying rigid disk, 369 annular clamped-clamped, 366 annular clamped-free, 366 annular clamped-free with mass, 367 annular free-clamped, 366 annular free-free, 366 annular hinged-hinged, 366 annular with rigid disk, 363 axisymmetric motion, 357, 363, 386–387, 389 boundary attachments, 346, 353–355, 357, 359, 362 boundary conditions, 349–351, 354, 357, 393

Index cantilever, 374 characteristic equation, 360, 373, 396 circular, 354 comparison with beams, 379 concentrated mass, 345, 367, 379, 389 elastic foundation, 346, 354, 356, 359 extensional motion, 390 forced excitation, 382 governing equation, 348, 392 in-plane force, 349, 356, 359, 363 interior attachments, 344–345 mode shape, 356, 359, 361, 363, 374, 381, 395, 397 natural frequency, 356, 360, 363, 373, 396, 397 rectangular, 342, 370 rectangular all edges clamped, 374 rectangular opposite edges free, 379 rectangular opposite edges hinged, 374 solid circular clamped, 363 solid circular free, 363, 394–395 solid circular hinged, 363 solid circular with translation and torsion springs, 367 Poling direction, 42 Potential energy maximum, 176, 310, 371 single degree-of-freedom system, 35, 92, 278 Power dissipated by viscous damper, 13 enhanced energy harvester, 68 permanent magnet harvester, 55 piezoelectric energy harvester, 48, 266 Proof mass, 150 Q Quality factor, 12, 16, 29, 232–238 R Radius of curvature, 85, 343 Rayleigh beam theory, 287, 302 Rayleigh-Ritz method, 175, 310 beam exponential double taper, 179, 313 beam linear double taper, 179, 313 beam triangular taper, 182 cylindrical shells, 420 rectangular plates, 370 Rectangular plates, see Plates Resistance, optimum, 49 Reynolds number, 27, 77 Rigid body mode, 119, 121

Index Rotation due to bending, 274 Rotational inertia, 287 S Scale factor, 117–128, 208, 299, 361 Shear coefficient, see Shear correction factor Shear correction factor, 274 Shear force, see Force Shear strain, 274, 287, 343 Shear stress, 274 Shells, cylindrical boundary condtions, Donnell’s theory, 419 boundary conditions, Flügge’s theory, 412 cantilever, 424 characteristic equation, 424, 436 clamped-clamped, 425 Donnell’s theory, 401, 415 double-wall carbon nanotube, 401, 432 Flügge’s theory, 402 governing equations, Donnell’s theory, 418 governing equations, Flügge’s theory, 411, 433 hinged-hinged, 434 mode shape, 427 natural frequency, 425, 436 single-wall carbon nanotube, 401 strain-displacement relations, 403 stress-strain relations, 404 Short circuit, 47, 49–50, 67, 264, 267 Shunt impedance, 46 Single degree-of-freedom system, 7, 96, 233, 280 constant force, 8 harmonically varying force, 10 natural frequency, 9 reaction force, 8 spring force, derivate of, 8 time-varying force, 8, 24 See also Beams Spring constant piezoelectric element, 45 squeeze film, 17, 20 torsion, 9, 277 translation, 277

489 Spring stiffness, equivalent, 35, 134 Square plates, see Plates Squeeze film damping, see Damping Squeeze number, 18 critical, 18 Static displacement, 7, 33 Static equilibrium position, 9, 33 Stiffness controlled region, 11 Strain, 43, 254 Strain energy cylindrical shell, Donnell’s theory, 417 cylindrical shell, Flügge’s theory, 408 Euler-Bernoulli beam, 85, 88, 253 plate, 345 plate, in-plane, 391 Timoshenko beam, 276 Stray impedance, 46 Stress, 43, 85, 254, 344 Stress resultants, 391 Symmetric quadratic, 95, 100, 204, 328, 385, 414, 452, 461 T Time constant, 48 Transfer function, 58, 72 Two degree-of-freedom system, 56, 137, 140 in-span, 144 natural frequency, 60 time-domain response, 73 U Undamped, 10 V van der Waals force, 31, 38, 432 Vibration absorber, 62 Viscosity dynamic, 18, 26, 75 effective, 19, 25 Viscous fluid, 26, 75 Viscous fluid damping, see Damping Voltage, 31, 42, 66 pull-in, 31, 33–34, 245