Tài liệu Mechanics of Microelectromechanical Systems pdf

Mechanics of

Microelectro￾mechanical

Systems

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Nicolae Lobontiu

Ephrahim Garcia

Mechanics of

Microelectromechanical

Systems

KLUWER ACADEMIC PUBLISHERS

NEW YORK, BOSTON, DORDRECHT, LONDON, MOSCOW

eBook ISBN: 0-387-23037-8

Print ISBN: 1-4020-8013-1

Print ©2005 Kluwer Academic Publishers

All rights reserved

No part of this eBook may be reproduced or transmitted in any form or by any means, electronic,

mechanical, recording, or otherwise, without written consent from the Publisher

Created in the United States of America

Boston

©2005 Springer Science + Business Media, Inc.

Visit Springer's eBookstore at: http://ebooks.kluweronline.com

and the Springer Global Website Online at: http://www.springeronline.com

To our families

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TABLE OF CONTENTS

Preface ix

STIFFNESS BASICS 1

1

1

6

14

21

43

58

60

1 INTRODUCTION

2

3

STIFFNESS DEFINITION

DEFORMATIONS, STRAINS AND STRESSES

4

5

6

7

MEMBERS, LOADS AND BOUNDARY CONDITIONS

LOAD-DISPLACEMENT CALCULATION METHODS:

CASTIGLIANO’S THEOREMS

COMPOSITE MEMBERS

PLATES AND SHELLS

Problems

MICROCANTILEVERS, MICROHINGES,

MICROBRIDGES

2

65

65

66

97

103

114

126

131

131

131

170

179

1

2

3

4

5

INTRODUCTION

MICROCANTILEVERS

MICROHINGES

COMPOUND MICROCANTILEVERS

MICROBRIDGES

Problems

3 MICROSUSPENSIONS

1

2

3

INTRODUCTION

MICROSUSPENSIONS FOR LINEAR MOTION

MICROSUSPENSIONS FOR ROTARY MOTION

Problems

4 MICROTRANSDUCTION: ACTUATION AND SENSING

1

2

183

183

184

195

212

223

230

232

238

249

256

257

INTRODUCTION

THERMAL TRANSDUCTION

3

4

5

6

7

8

9

10

ELECTROSTATIC TRANSDUCTION

ELECTROMAGNETIC/MAGNETIC TRANSDUCTION

PIEZOELECTRIC (PZT) TRANSDUCTION

PIEZOMAGNETIC TRANSDUCTION

SHAPE MEMORY ALLOY (SMA) TRANSDUCTION

BIMORPH TRANSDUCTION

MULTIMORPH TRANSDUCTION

OTHER FORMS OF TRANSDUCTION

Problems

1

viii

5 STATIC RESPONSE OF MEMS 263

1

2

3

4

5

6

7

8

INTRODUCTION 263

SINGLE-SPRING MEMS 263

TWO-SPRING MEMS 271

MULTI-SPRING MEMS 285

DISPLACEMENT-AMPLIFICATION MICRODEVICES 286

LARGE DEFORMATIONS 307

BUCKLING 315

COMPOUND STRESSES AND YIELDING 330

Problems 335

6 MICROFABRICATION, MATERIALS, PRECISION AND

SCALING 343

1 INTRODUCTION

2

3

4

5

MICROFABRICATION

MATERIALS

PRECISION ISSUES IN MEMS

SCALING

Problems

Index

343

343

363

365

381

390

395

PREFACE

This book offers a comprehensive coverage to the mechanics of

microelectromechanical systems (MEMS), which are analyzed from a

mechanical engineer’s viewpoint as devices that transform an input form of

energy, such as thermal, electrostatic, electromagnetic or optical, into output

mechanical motion (in the case of actuation) or that can operate with the

reversed functionality (as in sensors) and convert an external stimulus, such as

mechanical motion, into (generally) electric energy. The impetus of this

proposal stems from the perception that such an approach might contribute to

a more solid understanding of the principles governing the mechanics of

MEMS, and would hopefully enhance the efficiency of modeling and

designing reliable and desirably-optimized microsystems. The work

represents an attempt at both extending and deepening the mechanical-based

approach to MEMS in the static domain by providing simple, yet reliable

tools that are applicable to micromechanism design through current

fabrication technologies.

Lumped-parameter stiffness and compliance properties of flexible

components are derived both analytically (as closed-form solutions) and as

simplified (engineering) formulas. Also studied are the principal means of

actuation/sensing and their integration into the overall microsystem. Various

examples of MEMS are studied in order to better illustrate the presentation of

the different modeling principles and algorithms.

Through its objective, approach and scope, this book offers a novel

and systematic insight into the MEMS domain and complements existing

work in the literature addressing part of the material developed herein.

Essentially, this book provides a database of stiffness/compliance models for

various spring-type flexible connectors that transmit the mechanical motion in

MEMS, as well as of the various forces/moments that are involved in

microtransduction. In order to predict their final state, the microsystems are

characterized by formulating, solving and analyzing the static equilibrium

equations, which incorporate spring, actuation and sensing effects.

Chapter 1 gives a succinct, yet comprehensive review of the main

tools enabling stiffness/compliance characterization of MEMS as it lays the

foundation of further developments in this book. Included are basic topics

from mechanics of materials and statics such as load-deformation, stress￾strain or structural members. Presented are the Castigliano’s theorems as basic

tools in stiffness/compliance calculation. Straight and curved line elements

are studied by explicitly formulating their compliance/stiffness characteristics.

Composite micromembers, such as sandwiched, serial, parallel, and hybrid

(serial-parallel) are also treated in detail, as well as thin plates and shells. All

the theoretical apparatus presented in this chapter is illustrated with examples

of MEMS designs.

Chapter 2 is dedicated to characterizing the main flexible components

that are encountered in MEMS and which enable mechanical mobility through

x

their elastic deformation. Studied are flexible members such as microhinges

(several configurations are presented including constant cross-section, circular,

corner-filleted and elliptic configurations), microcantilevers (which can be

either solid or hollow) and microbridges (fixed-fixed mechanical components).

Each compliant member presented in this chapter is defined by either exact or

simplified (engineering) stiffness or compliance equations that are derived by

means of lumped-parameter models. Solved examples and proposed problems

accompany again the basic text.

Chapter 3 derives the stiffnesses of various microsuspensions

(microsprings) that are largely utilized in the MEMS design. Included are

beam-type structures (straight, bent or curved), U-springs, serpentine springs,

sagittal springs, folded beams, and spiral springs (with either small or large

number of turns). All these flexible components are treated in a systematic

manner by offering equations for both the main (active) stiffnesses and the

secondary (parasitic) ones.

Chapter 4 analyzes the micro actuation and sensing techniques

(collectively known as transduction methods) that are currently implemented

in MEMS. Details are presented for microtransduction procedures such as

electrostatic, thermal, magnetic, electromagnetic, piezoelectric, with shape

memory alloys (SMA), bimorph- and multimorph-based. Examples are

provided for each type of actuation as they relate to particular types of MEMS.

Chapter 5 is a blend of all the material comprised in the book thus far,

as it attempts to combine elements of transduction (actuation/sensing) with

flexible connectors in examples of real-life microdevices that are studied in

the static domain. Concrete MEMS examples are analyzed from the

standpoint of their structure and motion traits. Single-spring and multiple￾spring micromechanisms are addressed, together with displacement￾amplification microdevices and large-displacement MEMS components. The

important aspects of buckling, postbuckling (evaluation of large

displacements following buckling), compound stresses and yield criteria are

also discussed in detail. Fully-solved examples and problems add to this

chapter’s material.

The final chapter, Chapter 6, includes a presentation of the main

microfabrication procedures that are currently being used to produce the

microdevices presented in this book. MEMS materials are also mentioned

together with their mechanical properties. Precision issues in MEMS design

and fabrication, which include material properties variability,

microfabrication limitations in producing ideal geometric shapes, as well as

simplifying assumptions in modeling, are addressed comprehensively. The

chapter concludes with aspects regarding scaling laws that apply to MEMS

and their impact on modeling and design.

This book is mainly intended to be a textbook for upper￾undergraduate/graduate level students. The numerous solved examples

together with the proposed problems are hoped to be useful for both the

student and the instructor. These applications supplement the material which

xi

is offered in this book, and which attempts to be self-contained such that

extended reference to other sources be not an absolute pre-requisite. It is also

hoped that the book will be of interest to a larger segment of readers involved

with MEMS development at different levels of background and

proficiency/skills. The researcher with a non-mechanical background should

find topics in this book that could enrich her/his customary modeling/design

arsenal, while the professional of mechanical formation would hopefully

encounter familiar principles that are applied to microsystem modeling and

design.

Although considerable effort has been spent to ensure that all the

mathematical models and corresponding numerical results are correct, this

book is probably not error-free. In this respect, any suggestion would

gratefully be acknowledged and considered.

The authors would like to thank Dr. Yoonsu Nam of Kangwon

National University, Korea, for his design help with the microdevices that are

illustrated in this book, as well as to Mr. Timothy Reissman of Cornell

University for proof-reading part of the manuscript and for taking the pictures

of the prototype microdevices that have been included in this book.

Ithaca, New York

June 2004

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Chapter 1

STIFFNESS BASICS

1. INTRODUCTION

Stiffness is a fundamental qualifier of elastically-deformable mechanical

microcomponents and micromechanisms whose static, modal or dynamic

response need to be evaluated. This chapter gives a brief introduction to the

stiffness of microeletromechanical structural components by outlining the

corresponding linear, small-deformation theory, as well as by studying

several concrete examples. The fundamental notions of elastic deformation,

strain, stress and strain energy, which are all related to stiffness, are briefly

outlined. Energy methods are further presented, specifically the Castigliano’s

theorems, which are utilized herein to derive stiffness or compliance

equations.

A six degree-of-freedom lumped-parameter stiffness model is proposed

for the constant cross-section fixed-free straight members that are sensitive to

bending, axial and torsion loading. A similar model is developed for curved

members, both thick and thin, by explicitly deriving the compliance

equations. Composite beams, either sandwiched or in serial/parallel

configurations, are also presented in terms of their stiffnesses. Later, the

stiffness of thin plates and membranes is approached and equations are

formulated for circular and rectangular members. Problems that are proposed

to be solved conclude this chapter.

2. STIFFNESS DEFINITION

MEMS mainly move by elastic deformation of their flexible components.

One way of characterizing the static response of elastic members is by

defining their relevant stiffnesses. The simple example of a linear spring is

shown in Fig. 1.1, where a force is applied by slowly increasing its

magnitude from zero to a final value over a period of time such that the

force is in static equilibrium with the spring force at any moment in time.

The force necessary to extend the spring by the quantity is calculated

as:

2 Chapter 1

where is the spring’s linear stiffness, which depends on the material and

geometrical properties of the spring. This simple linear-spring model can be

used to evaluate axial deformations and forced-produced beam deflections of

mechanical microcomponents. For materials with linear elastic behavior and

in the small-deformation range, the stiffness is constant. Chapter 5 will

introduce the large-deformation theory which involves non-linear

relationships between load and the corresponding deformation. Another way

of expressing the load-deformation relationship for the spring in Fig. 1.1 is

by reversing the causality of the problem, and relating the deformation to the

force as:

where is the spring’s linear compliance, and is the inverse of the stiffness,

as can be seen by comparing Eqs. (1.1) and (1.2).

Figure 1.1 Load and deformation for a linear spring

Similar relationships do also apply for rotary (or torsion) springs, as the one

sketched in Fig. 1.2 (a). In this case, a torque is applied to a central shaft.

The applied torque has to overcome the torsion spring elastic resistance, and

the relationship between the torque and the shaft’s angular deflection can be

written as:

The compliance-based equation is of the form: