MEMS Linear and Nonlinear Statics and Dynamics

MICROSYSTEMS

Series EditorsRoger T. Howe

Stanford University

Antonio J. RiccoNASA Ames Research Center

For further volumes:http://www.springer.com/series/6289

Mohammad I. Younis

MEMS Linear and NonlinearStatics and Dynamics

2123

Ph.D. Mohammad I. YounisDepartment of Mechanical EngineeringState University of New YorkBinghamton, NYUSAmyounis@binghamton.edu

ISSN 1389-2134ISBN 978-1-4419-6019-1 e-ISBN 978-1-4419-6020-7DOI 10.1007/978-1-4419-6020-7Springer New York Dordrecht Heidelberg London

Library of Congress Control Number: 2011930834

© Springer Science+Business Media, LLC 2011All rights reserved. This work may not be translated or copied in whole or in part without the writtenpermission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connectionwith any form of information storage and retrieval, electronic adaptation, computer software, or by similaror dissimilar methodology now known or hereafter developed is forbidden.The use in this publication of trade names, trademarks, service marks, and similar terms, even if they arenot identified as such, is not to be taken as an expression of opinion as to whether or not they are subjectto proprietary rights.

Printed on acid-free paper

Springer is part of Springer Science+Business Media (www.springer.com)

To my parents Ibrahim and HalemahMy wife Olaand our sonsIbrahim, Muhmoud, and Mutaz

Preface

Several decades have passed by since the discovery and development of micro-electro-mechanical systems (MEMS). This technology has reached a level of maturitythat, today, several MEMS devices are being used in our every-day life, ranging fromaccelerometers and pressure sensors in cars, micro-mirrors in Plasma TVs, radio-frequency (RF) switches and microphones in cell phones, and inertia sensors in videogames. Fabrication methods of MEMS, such as bulk and surface micromachining, arenow well-known and almost standardized. Nowadays, hundreds of foundries aroundthe world offer numerous fabrication services that can translate the imagination of aMEMS designer of a device into reality.

Even with the maturity of fabrication and commercialization, MEMS is stillone of the hottest evolving areas in science and engineering, where scientists fromacross various disciplines investigate, brainstorm, and collaborate to invent smarterdevices, develop new technologies, and innovate unique solutions. With the increas-ing pressure for sensors and actuators of sophisticated functionalities, which areself-powered, self-calibrated, and self-tested, MEMS are expected to remain thesought-after technology of scientists for many years to come.

However, with this growing demand on the MEMS technology come great chal-lenges. Designers are now aiming to achieve complicated objectives while meetinga long list of specifications related to sensitivity, fabrication, system integration,packaging, and reliability. These challenges have created a motivation to seek newsolutions and ideas, beyond changing the geometry of devices and making morecomplex configurations. Researchers are starting to realize the need to look into newmethods of improvement and innovation in MEMS beyond the static laws of designand the limitations of linear theories. It is realized now that linear theories are tooshallow to allow for bolder ideas and more aggressive design goals. More attention isbeing directed to investigate deeply the dynamics and motion aspects of MEMS andto explore the hidden opportunities of operating MEMS in the nonlinear regimes.

Most MEMS devices employ a structure or more that undergoes some sort ofmotion. Accelerometers, gyroscopes, micromirrors, microphones, resonators andoscillators, RF switches and filters, and thermal actuators are few examples of such.Hence, it comes no surprise that the motion characteristic of microstructures affectdirectly the specifications, quality, and limitations of MEMS devices. Unfortunately,

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viii Preface

however, understanding the motion aspects of these devices is not a trivial task, whichis one of the reasons that have delayed the research attention in this area.

Many microstructures are highly compliant. When actuated, they undergo largedeflection or deformation compared to their dimensions. This amplifies the geometricnonlinearity of the structures. Microstructures are commonly actuated by parallel-plate electrostatic forces, which are inherently nonlinear. When microstructures aredriven to motion, they may experience nonlinear dissipation mechanisms, such assqueeze-film damping. These various nonlinear sources in MEMS play significantrole in their response and performance. As a result, models and designs based onlinear theories can be inadequate, inaccurate, and incorrect. Further, the interactionamong inherent coupled-physical domains, such as mechanical, electrostatic, ther-mal, and fluidic, marks one of the key features of MEMS. This coupling can furthercomplicate the design process. In addition, new phenomena that are common in themicroscale range, such as squeeze-film damping and pull-in instability, add to thesedifficulties. Tackling multiphysics, nonlinear, and dynamic problems can be verychallenging especially in the presence of instabilities, such as pull-in, which cancause serious convergence problems in commercial simulation software.

These new challenges facing MEMS designers and researches combined with thegrowing interest in MEMS and their dynamical behavior have been the motivationbehind this book. This book has two main goals: First is to provide the necessaryanalytical and computational tools that enable students and professionals to modelthe static and dynamic behavior of MEMS accurately in multiphysics fields andaccounting properly for their nonlinearities. The second goal is to present in-depthanalysis and treatment for the most common static and dynamic phenomena in MEMSencountered by MEMS engineers and researchers, especially those associated withelectrostatic MEMS.

The organization of the book material is as follows: Chapter 1 introduces MEMS,their features, and some of their modeling and simulation challenges and needs.Chapter 2 discusses the basic principles of the vibrations of single- and multiple-degrees-of-freedom systems. Free vibrations and forced vibrations in response toharmonic and arbitrary forcing are discussed. Chapter 3 introduces the common sens-ing and actuation methods in MEMS. These include electrothermal, piezoelectric,electromagnetic, piezoresistive, and electrostatic methods. The rest of the chap-ter is dedicated to illustrate the theory of electrostatic transduction in parallel-platecapacitors, torsional actuators, and comb-drive devices. Chapter 4 discusses thebasic elements of lumped-parameter modeling, which are the stiffness elements,effective mass, and damping mechanisms including squeeze-film, slide-film, andthermoelastic damping.

Chapter 5 builds on the background of Chaps. 1–4 to introduce the reader tobasic principles of nonlinear dynamics and stability analysis as applied to MEMSapplications. In doing so, several common phenomena at the microscale are intro-duced and illustrated, such as pull-in, side instability of comb fingers, collapse dueto capillary forces, dynamic pull-in, and hysteresis. Analytical methods, such as lin-earization and phase diagrams, are illustrated. Then the chapter discusses nonlinear

Preface ix

oscillations with emphasis on the qualitative features and main differences comparedto the linear vibrations of Chap. 2.

Moving from lumped-parameter to distributed-parameter modeling, Chap. 6 isdedicated to the most common and essential structures in MEMS: microbeams.Using a Newtonian approach, the chapter starts with a discussion on the derivationof the linear equation of motion and various kinds of boundary conditions. The staticproblem is discussed followed by illustration of solving the eigenvalue problem toextract the natural frequencies and modeshapes of common beams. Then, forcedvibrations and the modal analysis procedure are presented. The second half of thechapter deals with nonlinear models of beams with emphasis on midplane stretchingand electrostatic nonlinearities. The Galerkin procedure and reduced-order modelingare then discussed. As an application, universal pull-in curves of electrostaticallyactuated microbeams are presented. Following the static simulations, methods tosolve the eigenvalue problem of beams under electrostatic actuation and the forcedvibration response due to AC and DC actuation are discussed. Modeling of AtomicForce Microscopes is then presented. The chapter ends with discussions on themodeling of damping in beams.

Chapters 7 and 8 present special case studies of importance in MEMS, which aretreated in some depth both theoretically and experimentally. Chapter 7 discusses thenonlinear dynamics of electrically actuated resonators. Simulation methods, suchas the shooting technique and the basin-of-attraction analysis, are introduced anddemonstrated. Dynamic pull-in, its utilization, and control are discussed. Chapter 8deals with a reliability topic, which is the response of MEMS to mechanical shock.Modeling shock in MEMS, its interaction with electrostatic forces and printed circuitboards, and details on experimental testing are presented.

This book can be used by professionals of all levels who aim to model and simulatethe behavior of MEMS devices and structures or to improve their design for staticand dynamic considerations. In addition, the book serves as an excellent reference toenable full understanding of common MEMS phenomena that face MEMS engineersand researchers, such as squeeze-film damping, buckling, and pull-in instability. Thedepth of treatment of many of the topics covered in this book should appeal to MEMSresearchers and those who consider doing research in related fields.

The book can be used as a text for two courses related to MEMS modeling anddesign or more specifically for courses in the statics and dynamics of MEMS. Chap-ters 1–4 and some of the material of Chap. 5 can be used for a first-year graduate orsenior undergraduate course. For students who are familiar with mechanical vibra-tions, many of the materials of Chap. 2 can be assigned for self-reading except fortopics specific to MEMS applications, such as MEMS gyroscopes, accelerometers,and band-pass filters. Chapters 5–8 suit a second-year graduate course. In addition,instructors are recommended to add research-oriented projects to encourage studentsto explore what is new in this highly dynamic field.

The author has relied on introducing and illustrating many new concepts andanalytical and numerical approaches through examples instead of introducing themas abstract theories. While this approach does not provide much mathematical rigor,from the author’s experience, it is easier for the students to digest. This is especially

x Preface

true for those outside the mechanical engineering and nonlinear dynamics disciplines.The examples of the book range in their complexity from simple to more difficult andresearch-oriented ones. These are not intended for beginners in the field but ratherfor advanced researchers and graduate students. The author aims of such examplesto stimulate deep thinking and motivate further research in the field.

A note worth to be mentioned here is regarding the cited references in the book.While the author has attempted to present numerous references for researchers andinterested scientists on the various discussed topics, these are not complete lists anddo not represent the full spectrum of the state of the art. These references should beconsidered only as a good starting point for those who want to follow research inrelated topics.

I would like to express my deep thanks for the people who supported me whilewriting this book. Many thanks go to my students whose curiosity, thirst, and interestto learn more about this exciting field have inspired me to pursue with this project. Iwould like to thank my colleagues from the Department of Mechanical Engineeringat SUNY Binghamton, Ronald Miles who supported me greatly in my researchin MEMS and vibrations, and James Pitarresi, the department chair, who offeredgreat help and supports throughout the period of this project. I would like to thankProfessor Stephen Senturia of the MIT for his feedback and fruitful comments onthe book draft. I am thankful to Professor Ali Nayfeh of Virginia Tech, whom I amin debited to him for everything nonlinear I know. Thanks also go to Mr. AndrewWillner of Sensata Technologies for his support. Many of the research that I hadthe opportunity to conduct in the field of dynamics of MEMS have been supportedthrough the Dynamical Systems Program of the National Science Foundation. Thissupport is acknowledged and highly appreciated. I would like to thank my parents,Ibrahim and Halemah, for their continuous encouragement and sacrifices. Last butnot least, my deep thanks and appreciation go to my wife Ola, who has supportedme continuously and endlessly, especially handling our three boys (Ibrahim, nine;Muhmoud, eight; and Mutaz, one), whom despite being little, like MEMS, are highlysophisticated and nonlinear.

December 2010 Mohammad I. YounisMechanical EngineeringState University of New York at Binghamton

Contents

1 MEMS, Their Features, and Modeling Challenges . . . . . . . . . . . . . . . . . 11.1 What Are MEMS and Why They Are Attractive? . . . . . . . . . . . . . . . . 11.2 Why We Need Modeling and Simulation Tools? . . . . . . . . . . . . . . . . . 41.3 Challenges of MEMS Modeling and Simulations . . . . . . . . . . . . . . . . 41.4 Coupled-Field MEMS Phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.4.1 Squeeze-Film Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.4.2 Thermoelastic Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.4.3 Pull-in Instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.4.4 Stiction Due to Capillary Forces . . . . . . . . . . . . . . . . . . . . . . . . 9

1.5 The State-of-the-Art of MEMS Modeling and Simulations . . . . . . . . 10Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2 Vibrations of Lumped-Parameter Systems . . . . . . . . . . . . . . . . . . . . . . . . 132.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.2 Free Vibration of Single-Degree-of-Freedom Systems . . . . . . . . . . . . 14

2.2.1 Undamped Vibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.2.2 Damped Vibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.3 Forced Harmonic Excitation of Single-Degree-of-FreedomSystems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.4 Vibrating MEMS Gyroscopes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.5 Base Excitations of SDOF Systems and

Accelerometers Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.6 Response of SDOF Systems to Arbitrary Excitation . . . . . . . . . . . . . . 362.7 Vibrations of Two-Degree-of-Freedom Systems . . . . . . . . . . . . . . . . . 40

2.7.1 Undamped Free Vibration and Eigenvalue Problem . . . . . . . . 402.7.2 Modal Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432.7.3 Resonances in 2-DOF Systems . . . . . . . . . . . . . . . . . . . . . . . . . 47

2.8 Numerical Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

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2.9 MEMS Band-Pass Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

3 Sensing and Actuation in MEMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573.1 Electrothermal Actuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.1.1 U-Shaped Actuator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 583.1.2 V-Beam Actuator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 593.1.3 Bimorph Actuator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

3.2 Piezoelectric Actuation and Detection . . . . . . . . . . . . . . . . . . . . . . . . . . 613.3 Electromagnetic and Magnetic Actuation . . . . . . . . . . . . . . . . . . . . . . . 653.4 Piezoresistive Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 683.5 Electrostatic Actuation and Detection . . . . . . . . . . . . . . . . . . . . . . . . . . 70

3.5.1 Simple Parallel-Plate Capacitors . . . . . . . . . . . . . . . . . . . . . . . . 723.5.2 Torsional Actuators and Micromirrors . . . . . . . . . . . . . . . . . . . 783.5.3 Comb-Drive Devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

3.6 Resonant Sensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

4 Elements of Lumped-Parameter Modeling in MEMS . . . . . . . . . . . . . . . 974.1 Stiffness of Microstructures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

4.1.1 Experimental Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 984.1.2 Computational Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1014.1.3 Analytical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

4.2 Spring–Mass Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1074.3 Damping in MEMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

4.3.1 Mechanisms of Energy Losses . . . . . . . . . . . . . . . . . . . . . . . . . 1124.3.2 Air Damping Fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1144.3.3 Damping Dependence on Pressure: Newell’s

Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1164.3.4 Drag Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1174.3.5 Squeeze-Film Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1184.3.6 Slide-Film Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1394.3.7 Intrinsic Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1424.3.8 Extracting Damping Coefficients Experimentally . . . . . . . . . . 143Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

5 Introduction to Nonlinear Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1555.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1555.2 Nondimensionalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1575.3 Fixed Points and Linearization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1615.4 Bifurcations of Fixed Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

5.4.1 Saddle-Node Bifurcation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1695.4.2 Transcritical Bifurcation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

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5.4.3 Pitchfork Bifurcation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1785.4.4 Hopf Bifurcation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

5.5 Phase Portraits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1845.5.1 Phase Diagram of a Parallel-Plate Capacitor

and the Dynamic Pull-in Concept . . . . . . . . . . . . . . . . . . . . . . . 1905.5.2 Phase Diagram of a Double-Sided Capacitor . . . . . . . . . . . . . . 194

5.6 Step-Input Actuation of Capacitive RF Switches . . . . . . . . . . . . . . . . . 1975.7 Dynamics of Torsional Actuators and Micromirrors . . . . . . . . . . . . . . 201

5.7.1 Single-Degree-of-Freedom Model . . . . . . . . . . . . . . . . . . . . . . 2025.7.2 Two-Degree-of-Freedom Model . . . . . . . . . . . . . . . . . . . . . . . . 206

5.8 Nonlinear Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2115.8.1 The Effect of a Constant Force . . . . . . . . . . . . . . . . . . . . . . . . . 2125.8.2 Free Vibration in the Presence of Nonlinearities . . . . . . . . . . . 2165.8.3 Forced Harmonic Vibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2175.8.4 Parametric Excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2305.8.5 Self-Excited Oscillators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237

5.9 Modal Interaction, Chaos, and Fractal Behavior . . . . . . . . . . . . . . . . . 239Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245

6 Microbeams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2516.1 The Linear Equation of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251

6.1.1 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2546.1.2 Beams Made of Different Material Layers . . . . . . . . . . . . . . . . 259

6.2 The Static Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2606.3 Residual Stresses and Nonideal Supports

of Cantilever Microbeams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2626.4 Natural Frequencies and Modeshapes . . . . . . . . . . . . . . . . . . . . . . . . . . 267

6.4.1 Nondimensionalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2726.4.2 Flexible (Nonideal) Supports . . . . . . . . . . . . . . . . . . . . . . . . . . . 2746.4.3 Cantilever Beam with a Lumped Mass at the Tip . . . . . . . . . . 277

6.5 The Effect of Axial Load on the Natural Frequencyand the Buckling Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280

6.6 The Orthogonality of Modeshapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2846.7 Forced Vibrations and Modal Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 286

6.7.1 Undamped Response with no Axial Load . . . . . . . . . . . . . . . . 2866.7.2 Adding Axial Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2906.7.3 Adding Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291

6.8 A Nonlinear Model of Beams with Midplane Stretching . . . . . . . . . . 2926.9 Other Nonlinear Models of Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2966.10 The Galerkin Discretization and Reduced-Order Modeling . . . . . . . . 297

6.10.1 The Galerkin Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2976.10.2 Beams with Midplane Stretching . . . . . . . . . . . . . . . . . . . . . . . 299

6.11 Reduced-Order Model of Beams Under Electrostatic Force . . . . . . . . 302

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6.12 The Static Behavior of Beams Under Electrostatic Force . . . . . . . . . . 3066.12.1 Cantilever Microbeams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3066.12.2 Clamped–Clamped Microbeams . . . . . . . . . . . . . . . . . . . . . . . . 3086.12.3 Microbeams with Partial Electrodes

and Initial Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3126.13 The Natural Frequencies Under Electrostatic Force . . . . . . . . . . . . . . . 3146.14 Pull-in Time of RF Switches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3196.15 Resonators Under AC + DC Excitation . . . . . . . . . . . . . . . . . . . . . . . . . 3216.16 Atomic Force Microscopes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323

6.16.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3236.16.2 Interaction Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3256.16.3 AFM Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3276.16.4 AFM Under Lennard–Jones Force . . . . . . . . . . . . . . . . . . . . . . 330

6.17 Beams Under Capillary Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3386.18 Coupled-Field Damping of Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343

6.18.1 Squeeze-Film Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3436.18.2 Thermoelastic Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352

7 Nonlinear Dynamics of an Electrically Actuated Resonator . . . . . . . . . 3597.1 The Device and Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . 3597.2 Initial Characterization and Parameters Extraction . . . . . . . . . . . . . . . 3607.3 Experimental Data for Large DC and AC Excitations . . . . . . . . . . . . . 363

7.3.1 Primary Resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3637.3.2 Subharmonic Resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365

7.4 Simulations Using Long-Time Integration . . . . . . . . . . . . . . . . . . . . . . 3687.5 Simulations Using the Shooting Technique . . . . . . . . . . . . . . . . . . . . . 369

7.5.1 The Shooting Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3697.5.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372

7.6 Basin of Attraction Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3767.7 Attractors Tracking and the Integrity Factor . . . . . . . . . . . . . . . . . . . . . 3797.8 Remarks on Resonant Dynamic Pull-in . . . . . . . . . . . . . . . . . . . . . . . . 3847.9 Mass Detection Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3867.10 Controlling Resonant Dynamic Pull-in . . . . . . . . . . . . . . . . . . . . . . . . . 391

7.10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3917.10.2 Simulation and Experimental Results . . . . . . . . . . . . . . . . . . . . 3927.10.3 Integrity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398

8 Mechanical Shock in MEMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4018.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4018.2 Mechanical Shock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4038.3 Modeling Shock in Lumped-Parameter Models . . . . . . . . . . . . . . . . . . 405

Contents xv

8.4 The Shock-Response Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4078.5 Modeling Shock on Microbeams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4098.6 Computationally Efficient Approach for Microstructures . . . . . . . . . . 4118.7 High-g Shock Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4138.8 The Combined Effect of Shock and Electrostatic Forces . . . . . . . . . . 414

8.8.1 Single-Degree-of-Freedom Model . . . . . . . . . . . . . . . . . . . . . . 4148.8.2 Beam Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4188.8.3 Switch Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4218.8.4 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421

8.9 Resonators Under Shock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4268.9.1 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4268.9.2 Experimental Results and Comparison with Simulations . . . . 429

8.10 The Effect of the PCB Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 440

Appendix A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451