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 ebmagrab@umd.edu 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 vii viii Preface 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 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 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 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 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 Preface ix 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 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 and u3 = w is governed by Case 1, the boundary conditions for these quantities at x = xl , l = 1, and are as follows ux (xl , , t) = either ă [Al11 ux (xl , t) + al11 ux (xl , t)] + (−1)l or y Gux,x x=xl =0 (B.126a) and uθ (xl , θ , t) = either or − [Al12 uθ (xl , t) + al12 uθ (xl , t)] + (1) ă l y Gu,x x=xl =0 (B.126b) and either ă [Al13 w (xl , t) + al13 w (xl , t)] + (−1)l or y Gw,x w (xl , θ , t) = ∂Gw,xθ ∂Gw,xx − =0 − ∂x ∂θ x=xj (B.126c) and w,x (xl , , t) = either or ă Al23 w,x (xl , t) + al23 w,x (xl , t) + (−1) l y Gw,xx x=xl =0 (B.126d) N = 2: Timoshenko Beams For the case of Timoshenko beams, it is found in Chapter that N = 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 that u1 = w and u2 = ψ are described by Case 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 − − = ∂x ∂t Gw − (B.127) The boundary conditions can be obtained from Case 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, w (xl , t) = either or − [Al11 w (xl , t) + al11 w (xl , t)] + (1) Gw,x ă l x=xl =0 (B.128a) and (xl , t) = either or ă Al12 ψ (xl , t) + al12 ψ (xl , t) + (−1) Gψ,x l x=xl =0 (B.128b) 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 ∞ e−st g (t) dt G (s) = (C.1) 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 > and that this function is such that this integral exists; that is, ∞ |g (t)| e−at dt < ∞ (C.2) 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 + ωn 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 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)] + ωn X (s) = ˙ 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 + ωn (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 + mωd e−ζ ωn η sin (ωd η) f (t − η) dη = x (0) e −ζ ωn t x (0) + ζ ωn x (0) −ζ ωn t ˙ cos (ωd t) + e sin (ωd t) ωd t + mωd e−ζ ωn (t−η) sin (ωd (t − η)) f (η) dη where ωd = ωn − ζ and we have used the relation sin (ωd t − ϕ) = sin (ωd t) cos (ϕ) − cos (ωd t) sin (ϕ) = ζ sin (ωd t) − − ζ cos (ωd t) (C.6) Appendix C: Laplace Transforms and the Solutions to Ordinary Differential Equations 479 since, from Table C.1, ϕ = cos−1 ζ = sin−1 − ζ ζ < 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 t −ζ ωn t x (t) = Ao e sin (ωd t + ϕd ) + mωd e−ζ ωn η sin (ωd η) f (t − η) dη (C.8) 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) ˙ 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 − 2β + Kδ (x − x1 ) y + k − dx4 dx y = f (x) (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 and 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 − which, upon rearrangement, yields Y (s) = F (s) (C.11) 480 Appendix C: Laplace Transforms and the Solutions to Ordinary Differential Equations Y (s) = D (s) s3 − 2βs y (0) + s2 − 2β y (0) + sy (0) −x1 s + y (0) − Ky (x1 ) e (C.12) + F (s) where the prime denotes the derivative with respect to x, D (s) = s4 − 2βs2 + k − ε2 = − β + δ2 = β + β2 + β2 + 4 = s2 − δ s2 + ε −k (C.13) −k and it is assumed that β + − k > It is noted that ε2 δ = − k and ε − δ = −2β and, therefore, when β = 0, ε = δ To obtain the inverse Laplace transform, we use partial fractions on the following quantities to find that ˆ ¯ Q (s) = s3 − 2βs = 2 − δ s2 + ε ε + δ2 s sε sδ + s2 − δ s + ε2 ˆ ¯ R (s) = s2 − 2β = ε + δ2 s2 − δ s2 + ε δ2 ε2 + s2 − δ s + ε2 ˆ ¯ S (s) = ˆ ¯ T (s) = s s2 − δ s2 + ε s2 − δ s2 + ε = ε + δ2 ε2 + δ 1 − s2 − δ s + ε2 (C.14) s s − 2 − δ2 s s + ε2 = Using pairs 11 to 14 in Table C.1, it is found that the inverse Laplace transform of Eq (C.14) is δ + ε2 ˆ R (x) = δ + ε2 ˆ S (x) = δ + ε2 ˆ T (x) = δ + ε2 ˆ Q (x) = δ cos (εx) + ε cosh (δx) δ2 ε2 sin (εx) + sinh (δx) ε δ [− cos (εx) + cosh (δx)] 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 ˆ Q (x) = δ ε2 T (x) ˆ ˆ R (x) = Q (x) ˆ ˆ ˆ S (x) = R (x) + δ − ε2 T (x) ˆ ˆ T (x) = S (x) ˆ ˆ Q (x) = δ ε2 S (x) ˆ ˆ R (x) = δ ε2 T (x) ˆ ˆ ˆ S (x) = Q (x) + δ − ε S (x) (C.16) ˆ ˆ ˆ T (x) = R (x) + δ − ε2 T (x) ˆ ˆ ˆ Q (x) = δ ε2 R (x) + δ − ε T (x) ˆ ˆ R (x) = δ ε2 S (x) ˆ ˆ S (x) = δ − ε R (x) + δ ε2 + δ − ε 2 ˆ T (x) ˆ ˆ (x) = Q (x) + δ − ε S (x) ˆ T where the prime denotes the derivative with respect to x Using pairs and of Table C.1 and Eqs (C.14) and (C.15), the inverse Laplace transform of Eq (C.12) is ˆ ˆ ˆ ˆ y (x) = y (0) Q (x) + y (0) R (x) + y (0) S (x) + y (0) T (x) x ˆ − Ky (x1 ) T (x − x1 ) u (x − x1 ) + (C.17) ˆ f (η) T (x − η) dη 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) T ( x)/ + (C.18) f (η) T ( [x − η]) dη − Ky (x1 ) T ( [x − x1 ]) u (x − x1 ) / x + y (0) S ( x)/ 482 Appendix C: Laplace Transforms and the Solutions to Ordinary Differential Equations where [cos ( x) + cosh ( x)] R ( x) = [sin ( x) + sinh ( x)] S ( x) = [− cos ( x) + cosh ( x)] T ( x) = [− sin ( x) + sinh ( x)] Q ( x) = (C.19) The derivatives of Eq (C.19) are Q ( x) = T ( x) Q ( x) = 2S ( x) Q ( x) = 3R ( x) R ( x) = Q ( x) R ( x) = 2T ( x) R ( x) = 3S ( x) S ( x) = R ( x) S ( x) = 2Q ( x) S ( x) = 3T T ( x) = S ( x) T ( x) = 2R ( x) T ( x) = ( x) 3Q ( (C.20) x) where the prime denotes the derivative with respect to x Equations (C.17) to (C.20) are used extensively in Chapter The following set of transformed quantities appears in Chapter [see Eq (5.66)]: Qαβ (s, α, β) = Rαβ (s, α, β) = Sαβ (s, α, β) = Tαβ (s, α, β) = s3 s2 − α2 s2 + β2 s2 s2 − α s2 + β s s2 − α2 s2 s2 − α2 + β2 s2 + β2 = = = = α2 + β2 + β2 α s2 α2 s β 2s + 2 −α s + β2 β2 α2 + s2 − α s + β2 + β2 s2 s s − 2 −α s + β2 α2 + β s2 1 − 2 −α s + β2 α2 (C.21) Using pairs 11 to 14 in Table C.1, it is found that the inverse Laplace transform of Eq (C.21) is α2 + β Rαβ (x, α, β) = α + β2 Sαβ (x, α, β) = α + β2 Tαβ (x, α, β) = α + β2 Qαβ (x, α, β) = β cos (βx) + α cosh (αx) [β sin (βx) + α sinh (αx)] (C.22) [− cos (βx) + cosh (αx)] − 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 Qαβ (x, α, β) = Vαβ (x, α, β) = α − β Rαβ (x, α, β) + α β Tαβ (x, α, β) Rαβ (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 G (s/a) ag (at) Scaling of variable sn G (s) dn g gn (t) = n dt nth-order derivative, n = 1, 2, g (t − to ) u (t − to ) Shifting n sn−k gk−1 (0) − k=1 e−to s G (s) G (s) H (s) t Convolution g (η) h (t − η) dη or t g (t − η) h (η) dη g (to ) e−sto e−sto g (t) δ (t − to ) Delta function s s−a u (t − to ) Unit step function eat Exponential s2 + 2ζ ωn s + ωn −ζ ωn t e sin (ωd t) ωd ωn s s2 + 2ζ ωn s + ωn 1− 10 11 s s2 + 2ζ ωn s + ωn 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 − ζ When t is time, g (t) is step response of single degree-of-freedom system: φ = cos−1 ζ ζ < φ = cos−1 ζ ζ