Compliant Leverage Mechanism Design for MEMs Application

Compliant Leverage Mechanism Design for MEMS Applications  by Xiao-Ping Susan Su

B.S. (Tsinghua University) 1991 M.S. (University of California, Davis) 1997

A dissertation submitted in partial satisfaction of the requirements for the degree of  Doctor of Philosophy

in Engineering-Mechanical Engineering in the GRADUATE DIVISION of the UNIVERSITY OF CALIFORNIA, BERKELEY

Committee in charge: Professor Alice M. Agogino, Chair  Professor Dennis K. Lieu Professor Tsu-Jae King

Spring 2001

COMPLIANT LEVERAGE MECHANISM DESIGN FOR MEMS APPLICATIONS

Copyright © 2001  by Xiao-Ping Susan Su

ABSTRACT Compliant Leverage Mechanism Design for MEMS Applications by Xiao-Ping Susan Su Doctor of Philosophy in Mechanical Engineering University of California, Berkeley Professor Alice M. Agogino, Chair

Compliant microleverage mechanisms, including single- and multistage, can be used in micro-electro-mechanical system (MEMS) to transfer an input force/displacement to an output to achieve mechanical and/or geometry advantages. This thesis presents the original systematical study on this mechanism with primary focus on the design theory and synthesis issues of the mechanism. Starting from the basic nomenclature, definition and classification of the mechanism, an extensive first-order analytical model and a second-order refined one are  built for the single-stage microleverage mechanism, the basic element of all microleverage mechanisms. The amplification factor depends not only on the ideal leverage ratio ( L/l), the geometry of the lever, but also the axial and bending spring constant of the output system. Good agreement is obtained between the results of secondorder analytical modeling and those of FEM simulation with SUGAR, a MEMS simulation tool. The compliance match theory developed applies to the two-stage microleverage mechanism. The axial spring constant of the lever stage close to the output needs to be in

i

a specific region (Region II and III) in order for the entire mechanism to effectively amplify force. The increased resistance of a microlever to rotational or axial displacement when the output and pivot are at the different side of the lever arm leads to lower  amplification factor. The maximum amplification factor of a multistage microlever was derived in terms of the given output system and minimum flexure beam dimension. The design of  microleverage mechanism in a resonant accelerometer is illustrated at each step to present the theory. The leverage mechanism for displacement amplification is analyzed with application in a disk-drive suspension and a micro-valve. Experimental verification of the analytical equations and SUGAR simulation was carried out at both the micro- and macro-scale. A 1S-2D (first stage first kind lever with output and pivot at same side, second stage second kind with pivot and output at different side) type of mechanism was fabricated by the SOI-MEMS process for inertial force amplification in a resonant accelerometer. A macro-scale aluminum model was built and  the testing results agree qualitatively with the analytical and SUGAR predictions.

 _______________________________   ________________________ _______  Professor Alice M. Agogino Chair of the Committee

ii

To My Family:

Dr. Henry S. Yang  Jenny Su Yang  Rachel Su Yang

iii

ACKNOWLEDGMENTS I would like to extend my heartfelt thanks to my research advisor Prof. Alice Agogino for her encouragement, guidance, and advice for my research. Without her  encouragement and support, this research would have been given up in many times and  this thesis would never come out. Many thanks also go out to my thesis committee, Professors Agogino, Dennis Lieu, and Tsu-Jae King, for helping me put this document together.

During the journey, Prof. Roger Howe, Prof. Dennis Lieu, Prof. Liwei Lin have given valuable insight to and constructive advice on the research and the dissertation writing. During my graduate study here at Berkeley, Prof. Hedrick, Prof. Pello, Prof. Kazerooni all have helped me in different avenues.

This research initially followed the work by Dr. Trey Roessig. Thanks go to Dr. Timothy Brosnihan for processing the SOI resonant accelerometer, and Ms. Ningning Zhou for many discussions in SUGAR simulation. Ms. Jocelyn Lee showed me how to  bond the chips. Many friends I met here at Berkeley have helped me in many ways. There are too many to mention here.

Special thanks are due to my husband, Dr. Henry S. Yang, for building the aluminum model and a lot of tedious work in polishing the papers that came out of this research. My mother-in-law has helped taking care of my children during my Ph.D. study.

iv

I also would like to acknowledge the financial support of a GSR by Prof. Howe through a M3S DARPA contract 1-442427-25316(F30602-97-2-0266) (97-98), a NSF graduate research fellowship (96-99) and a Department GSI support (2000).

To anyone who is interested in reading this thesis, I would like to share the great secret of my life, which is the Lord Jesus. My strength is from Him, who died and raised  again. Everything I did is not on my own strength and my own ability, but by Him. He is faithful and has plans for each individual, no matter how common we are. And He can always uses our life, our brokenness to do something beautiful. Good and bad things are all from Him. He loves to work with us in difficult situations. And the Words said “With Him, you can do anything”. He said: “ In good time, I am with you. In bad times, I am carrying you through.” and “From glory to glory, I am changing you”. The great secret of  success, prosperity, happiness and joy in life is to love the Lord and to do His will.

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TABLE OF CONTENTS Abstract Dedication ................................................ ................................................... ................. iii

................................................. .................................. ................................. ..................... ..... iv Acknowledgments ................................ Table of Contents ................................................ ....................................................... vi List of Figures ...................................................... ............................................................................................................. ....................................................... xi List of Tables ........................................................ ............................................................................................................. ..................................................... xix Nomenclature ..............................................................................................................xx

................................................. .................................. ............................. ............ 1 Chapter 1 Introduction ................................ Chapter 2 Compliant Microleverage Mechanism ….............................. 7 2.1

Leverage Mechanism and Governing Laws ...............................................7

2.2

Compliant Leverage Mechanism ................................................................8

2.3

Classification of Compliant Leverage Mechanisms Mechanisms ...................................9

2.4

Design of Microleverage Mechanism and Its Challenge ..........................11

2.5

FEM Simulation ........................................................................................13

................................................. .............................. ............. 16 Chapter 3 Flexure Pivot Design ................................ 3.1

Flexure Pivot Design .................................................................................16

3.2

Different Models of Flexure Pivot ............................................................19

3.3

Pivot and Output System Configurations .................................................2 1

vi

Chapter 4 Single-stage Microleverage Mechanism ...............................27 4.1

Basic Structure of a Single-stage Single-stage Microleverage Mechanism ..................27

4.2

Amplification Factor of Single-Stage Microleverage Mechanism ...........30 4.2.1

Second-kind single-stage microleverage mechanism ...................30

4.2.2

First-kind microleverage mechanism ............................................ 34

4.3

Amplification Coefficient, A* ...................................................................35

4.4

Effect of Output System and Pivot Spring Constants on the Amplification Factor ................................................................................36

4.5

Second-order Refined Analytical Model ..... .............................................41

4.6

Spring Constant Calculation .....................................................................46

4.7

Strain Energy Analysis and Mechanical Efficiency .................................47

4.8

Mechanical Advantage and Geometry Geometry Advantage ...................................50

4.9

Single-Stage Microleverage Mechanism Design in a Resonant Output Accelerometer ...............................................................50 4.9.1

Resonant output accelerometer (RXL) .........................................50

4.9.2

Optimization of single-stage microleverage mechanism in the resonant accelerometer ....................................56

4.10

Other Design Issues of Microleverage Mechanism ..................................70 4.10.1 Beam column strength ................................................. .................70 4.10.2 Effect of horizontal force ..............................................................71

Chapter 5 Two-stage Microleverage Mechanism .................................73 5.1

Two-Stage Microleverage Microleverage Mechanism Structure .....................................74

5.2

Two-Stage Microleverage Mechanism Mechanism Amplification Amplification Factor ..................76 vii

5.3

Compliance Relationship Relationship Between Between Adjacent Levers ...............................80

5.4

Different Configurations Configurations of Two-Stage Leverage Mechanism Mechanism ................87

5.5

Two-Stage Microleverage Mechanism Design and o ptimization in the Resonant Accelerometer .....................................................................92

Chapter 6 Multiple-stage Microleverage Mechanism ........................109 6.1

Analysis of Multi-stage Microleverage Mechanism ...............................110

6.2

Maximum Amplification Factor of Multistage Lever for a Given Output System ..............................................................................113

6.3

Multi-stage Microleverage Mechanism Design in the Resonant Accelerometer ..............................................................116 6.3.1

Estimation of maximum achievable amplification factor ...........117

6.3.2

Feasibility analysis of a three-stage microleverage mechanism ..................................................................................118

6.4

Compliance in Microleverage Mechanism .............................................120

Chapter 7 SOI-MEMS Fabricated Resonant Accelerometer ............122 7.1

SOI-MEMS Process Run ........................................................................122

7.2

SOI-MEMS Fabricated Resonant Accelerometer ...................................124

7.3

Two-stage Microleverage Mechanism Optimization ..............................129

7.4

DETF Natural Frequency ........................................................................139

7.5

Sensitivity Analysis of the Resonant Accelerometer ..............................145

7.6

Optimization of The DETF Resonator ....................................................147

7.7

Testing of the SOI-MEMS Fabricated RXL ...........................................149

viii

7.8

7.7.1

Experiment Testing Set-ups................................................ Set-ups ................................................ .........149

7.7.2

Testing of the DETF Resonator ..................................................150

7.7.3

Testing of the Accelerometer ........................................... ...........152

Future Work on Resonant Accelerometer ...............................................1 54

Chapter 8 Experimental Verification with Macro Model ..................156 8.1

Building the Macro Aluminum Aluminum Model ................................................. ..156

8.2

Experimental Verification with the Macro Model ..................................160

Chapter 9 Microleverage Mechanism for Displacement Amplification 165 9.1

Different Configurations of Leverage Mechanism for Displacement Amplification .........................................................................................164

9.2

Amplification Factor of Leverage Mechanism for Displacement Amplification ..........................................................................................169

9.3

Displacement Amplification Leverage Mechanism in a Silicon Disk-Drive Suspension ....................................................... ...............................................................................................172 ........................................172

9.4

Application of Leverage Mechanism for Displacement Amplification in a Silicon Microvalve...................................................................................180

Chapter 10 Conclusions and Contributions ..........................................191

ix

................................................. ................................. .................................. ............................ ........... 195 Bibliography ................................. Appendix

................................. .................................................. ................................. ................................. ............................... .............. 202

A.

ABAQUS Input File for Single-stage Microlever Simulation ................206

B.

 Netlist Files for Different Different Pivot Design ............................................ .....209

C.

Mathematica File For Analytical Analysis ............................................ .210

D.

 Netlist File for Single-Stage Microleverage Mechanism in Resonant Accelerometer .........................................................................................212

E.

 Netlist File for Two-stage Microleverage Mechanism in Resonant Accelerometer .........................................................................................213

F.

 Netlist File for Calculating Resonator Frequency ..................................214

G.

 Netlist File for Simulation of the DETF Resonator ................................215

H.

 Netlist File for Simulation of the 1S-2D type Two-stage Leverage Mechanism in the Macro-model ……………………………………….216 ……………………………………….216

I.

 Netlist File for Simulation of the 1S-2S type Two-stage Leverage Mechanism in the Macro-model ……………………………………….217 ……………………………………….217

J.

 Netlist File for Simulation of the Two-stage Leverage Mechanism in the Disk-drive suspension …………………………… ………………………………………………….218 …………………….218

x

LIST OF FIGURES Chapter 2

Fig. 2.1

Mesh generated by ABAQUS for the leverage mechanism.

Fig. 2.2

Mechanism deflection simulated by ABAQUS.

Chapter 3 Fig. 3.1

The general flexure hinge model.

Fig. 3.2

A flexure pivot fabricated by the SOI-MEMS technology.

Fig. 3.3

A flexure beam pivot fabricated by the SOI-MEMS technology.

Fig. 3.4

Different pivot models: (a) vertical pivot, (b) horizontal pivot, (c) combined pivot.

Fig. 3.5

Comparison of the amplification factors of three models as a function of  the pivot beam length

Fig. 3.6

Effect of pivot beam with different angles on the amplification factor 

Fig. 3.7

A second-kind second-kind leverage mechanism with pivot and the output system system (a) on the same side, and (b) at different sides of the lever arm.

Fig. 3.8

The shape of a deflected pivot located on the opposite side of the lever arm. arm.

Fig 3.9

The deflection of pivot and connection beams of a 2D microlever.

xi

Chapter 4

Fig. 4.1

Schematic of three kinds of microleverage mechanisms (a) first kind, (b) second kind, (c) third kind.

Fig. 4.2

(a) A second-kind microleverage mechanism before and after loading; (b) Model of the second-kind microlever under loading.

Fig. 4.3

Amplification factor as a function function of output system axial spring constant for a series of (a) pivot lengths and (b) pivot widths at a fixed output system rotational spring constant k θ m o of 2 x 10-7 Nm; (c) output system bending spring constant.

Fig. 4.4

Amplification factor as a function of output system beanding spring spring constant for a series of output system axial spring constant.

Fig. 4.5

Free-body diagram of a second kind micro-leverage micro-leverage mechanism under  loading.

Fig. 4.6

A layout of the resonant-output microaccelerometer with a proof-mass, two resonators, and four symmetrical single-stage second-kind microlevers.

Fig. 4.7

(a) Schematic of DETF as the output system in the microaccelerometer; (b) simulated compound spring connecting seven springs.

Fig. 4.8

The node information for SUGAR simulation of single-stage microleverage mechanism in the resonant accelerometer.

xii

Fig. 4.9

Effect of tuning fork beam width w f  on (a) the amplification factor and  amplification coefficient; (b) the ratios of strain energies consumed at the tuning fork and at the pivot to the input energy.

Fig.4.10

Effect of T-F joint beam width w j on (a) the amplification factor and  amplification coefficient; (b) the ratios of strain energies consumed at the tuning fork and at the pivot to the input energy.

Fig. 4.11 Effect of pivot beam width w p on (a) the amplification factor and  amplification coefficient; (b) the ratios of strain energies consumed at the tuning fork and at the pivot to the input energy. Fig. 4.12 Amplification factor as a function of tuning-fork and joint beam width (tuning fork beam has the same width as that of joint beam) at various pivot beam width, 0.5 – 5 μm. Fig. 4.13

Effect of T-F beam length l f  on (a) the amplification factor and amplification coefficient; (b) the ratios of strain energies consumed at the tuning fork and at the pivot to the input energy.

Fig. 4.14

Effect of T-F joint beam length l j on (a) the amplification factor and  amplification coefficient; (b) the ratios of strain energies consumed at the tuning fork and at the pivot to the input energy.

xiii

Fig. 4.15 Effect of pivot beam length l p on (a) the amplification factor and  amplification coefficient; (b) the ratios of strain energies consumed at the tuning fork and at the pivot to the input energy. Fig. 4.16

The effect of lever ratio, L/l, on the amplification factor for a series of beam widths ( all beams are assumed to have same width).

Fig. 4.17

The effect of the distance between pivot and tuning fork, l, on the amplification factor for a fixed level ratio  L/l of 21.

Fig. 4.18

The effect of lever arm width on the amplification factor by SUGAR  simulation.

Chapter 5 Fig. 5.1

A schematic of a two-stage two-stage microleverage microleverage mechanism consisting of a firstkind lever as the first stage and a 2nd-kind as the second stage.

Fig. 5.2

Regime classification of a typical plot of the amplification factor as a function of the axial spring constant.

Fig. 5.3

Input axial spring spring constant of the overall first stage calculated by different different methods.

Fig. 5.4

Comparison of the amplification factor of different configuration.

Fig. 5.5

(a) A 1S-1D type two-stage microleverage mechanism for RXL. (b) A 1S-2D type two-stage microleverage mechan ism for RXL.

xiv

(c) A 1S-2D type two-stage microleverage mechanism for RXL. (d) A 1S-2S type two-stage microleverage microleverage mechanism for RXL. (e) A 2S-2D type two-stage microleverage mechanism for RXL. (f) A 2S-2S type two-stage microleverage mecha nism for RXL. Fig. 5.6

(a) Amplification factors,  A1,  A2, and  A  A; and (b) first-stage spring constant k  and second-stage amplification factor as a function of the width of first-stage  pivot, tuning fork and connection beam width (A series of beam width of the second stage were selected to see their effect on the second-stage amplification and total amplification factors).

Fig. 5.7

Amplification factors,  A1,  A2, and  A as a function of (a) T-F width; (b) T-F connection beam width; (c) lever 1 pivot width; (d) lever 1 pivot and  connection width.

Fig. 5.8

Amplification factors, A1,  A2, and  A  A as a function of (a) T-F beam length; (b) connection beam length; (c) first-stage lever pivot length.

Fig. 5.9

Comparison of SUGAR result with the analytical result.

Fig. 5.10

Effect of (a) the second-stage second-stage connection width; (b) (b) the second-stage second-stage pivot width on Amplification factors, A1, A2, and  A  A.

Fig. 5.11 Effect of (a) The second stage connection length; (b)The second stage pivot length on Amplification factors, A1, A2, and  A  A.

xv

Chapter 7 Fig. 7.1

Cross section schematic of the SOI-MEMS process run.

Fig. 7.2

(a) SEM of the resonant resonant accelerometer with two-stage microleverage microleverage mechanism fabricated by SOI-MEMS run. (b) Layout Design of the Resonant Accelerometer 

Fig. 7.3

SEM of the on-chip trans-resistance amplifier.

Fig. 7.4

Schematic of the trans-resistance amplifier amplifier circuit including (a) biasing supply, (b) amplifier, and (c) output stage.

Fig. 7.5

SEM micrograph showing the connection of the lever to proofmass.

Fig. 7.6

SEM of the lateral anchor by the SOI-MEMS process.

Fig. 7.7

Schematic of a two-stage microleverage mechanism for the resonant accelerometer and layout for SOI integrated run.

Fig. 7.8

SEM of the two-stage microleverage mechanism fabricated by SOI-MEMS.

Fig. 7.9

The Amplificaiton Amplificaiton factor as a function of (a) the width and (b) the length of  the tuning fork.

Fig. 7.10

The amplification factors as a function of (a) the width and (b) (b) length of the first stage pivot.

Fig. 7.11

The amplification factors as a function of (a) the width and (b) (b) length of the connection beam 2 between the two lever stages.

xvi

Fig. 7.12

The amplification amplification factors as a function of (a) the width and (b) length of the second stage pivot.

Fig. 7.13

SEM of the SOI-MEMS fabricated DETF resonator.

Fig. 7.14 SUGAR simulation of the the DETF resonator with with comb-drive comb-drive mass. mass. Fig. 7.15 SUGAR simulation of electrical electrical tuning of DETF resonator frequency. Fig. 7.16 Sensitivity as a function function of the DETF beam width. Fig. 7.17 Sensitivity as a function function of the DETF beam length. Fig. 7.18 SOI-MEMS RXL Post-Process Steps. Fig. 7.19

Testing apparatus for the SOI-MEMS resonant accelerometer.

Fig. 7.20

Off-chip unit gain buffer schematic.

Fig. 7.21 Plots of magnitude and phase of the DETF resonator. resonator. Fig. 7.22

Schematic

of

a

two-Axis

resonant

accelerometer

with

Microleverage mechanism (a) 1S-2S, (b) 2S-2S.

Chapter 8 Fig. 8.1

Aluminum backing plate for anchoring the leverage mechanism.

Fig. 8.2

Scaled-up two-stage aluminum leverage mechanism.

Fig. 8.3

A photograph of the macro aluminum leverage mechanism.

Fig. 8.4

Experimental set-up for macro-model verification.

xvii

two-stage

Fig. 8.5

Measured amplification amplification factor as a function of the input load for two-stage aluminum microlevers: 1S-2S and 1S-2D.

Chapter 9

Fig. 9.1

Two kinds of leverage mechanisms for displacement amplification.

Fig. 9.2

Test structure of two-stage microleverage microleverage mechanism for displacement amplification.

Fig. 9.3

(a) PZT-actuated silicon suspension (Chen and Horowitz). (b) Schematic of  the silicon suspension design (Chen and Horowitz).

Fig. 9.4

Two-stage leverage mechanism design in the disk drive suspension.

Fig. 9.5

Node information of the two-stage mechanism in the disk drive suspension

Fig. 9.6

(a) The amplification factor as a function of the 2 pivot width.

nd 

nd 

(b) The amplification factor as a function of the 2 pivot length. Fig. 9.7

st

(a) The effect of 1 pivot width on the amplification factor. st

(b) The effect of 1 pivot width on the amplification factor. Fig. 9.8

 Z -direction -direction vibration mode displayed by SUGAR.

Fig. 9.9

Y -direction -direction vibration mode displayed by SUGAR.

Fig. 9.10

(a) Schematic of a silicon microvalve with single-stage leverage mechanism (Williams 1999); (b) Two-stage microlever in the silicon microvalve.

xviii

Fig. 9.11 (a) Effect Effect of first-stage pivot width change on the the amplification factor. (b) Effect of first-stage pivot length change on the amplification factor. Fig. 9.12

(a) Effect Effect of connection beam width change on the amplification factor. (b) Effect of connection beam length change on the amplification factor.

Fig. 9.13

(a) Effect of second-stage pivot width change on the amplification factor. (b) Effect of second-stage pivot width change on the amplification factor.

Fig. 9.14 A three-stage leverage mechanism mechanism in the silicon microvalve. st

Fig. 9.15 (a) Amplification factor change with the 1 pivot width. st

(b) Amplification factor change with the 1 pivot length. Fig. 9.16

nd 

(a) Amplification factor change with 2 pivot width. (b) Amplification factor change with 2

Fig. 9.17

nd 

pivot length. rd 

(a) Amplification factor change with with the 3 pivot width. rd 

(b) Amplification factor change with 3 pivot length.

xix

List of Tables

Chapter 3

Table 3.1

SUGAR Simulation Results of Different Pivot Models.

Table 3.2

Comparison of SUGAR results on Microlevers with Pivot and Output System on (a) the Same Side and (b) Different Sides of the Lever Arm.

Chapter 5

Table 5.1

Comparison of Different Configurations of Two-Stage Mechanisms in the resonant accelerometer 

Table 5.2

Beam Deflections and Amplification Amplification Factor of Various Types of 2-stage Microleverage Mechanisms.

Chapter 9

Table 9.1

Different Configurations of Two-stage Microleverage Mechanism for  Displacement Amplification.

xx

NOMENCLATURE th

 Ai

force amplification factor of the i -stage microlever 

 A*

Amplification Coefficient

 Am a x

maximum achievable force amplification factor of a c ompound microlever 

 E 

Young’s modulus of the microleverage mechan ism material

e

the mechanical efficiency of a microleverage mechanism, as defined by the ratio of output energy to input energy

ei

the efficiency of  i

th

microlever stage, as defined by the ratio of output

energy to input energy for the stage F in

input force

F out

output force

F i

Axial force at the input of i microlever stage

k vvc, i

vertical (axial) spring constant of the beam connecting the (i-1) to the i

th

th

th

microlever stage

k vv I  , i

th

input vertical (axial) spring constant of the i microlever stage connected  together with all the upstream lever stages and the output system

k vvo

the vertical (axial) spring constants of the external output system

xxi

th

k vvo, i

vertical (axial) spring constant at the output of the i microlever stage

k vvp, i

vertical (axial) spring constant of the pivot for the i microlever stage

k θ m c, i

bending spring constant of the beam connecting the (i-1)

th

th

to the i

th

microlever stage

k θ m I  , i

input bending spring constant of the i

th

microlever stage connected 

together with all the upstream lever stages and the output system

k θ m o

the bending spring constants of the external output system

k θ m o, i

output system bending spring constant of the i microlever stage

k θ m p, i

bending spring constant of the pivot of the i microlever stage

lc , i

the length of connection beam of the i microlever stage

l p , i

the length of pivot beam of the i microlever stage

li

length between the pivot and output for the i lever arm

 Li

length between the input and output for the i lever arm

th

th

th

th

th

th

th

wc , i

the width of connection beam of the i microlever stage

w p , i

the width of pivot beam of the i microlever stage

δ  o

vertical (axial) displacement at the external output system

th

xxii

th

δ    I , i

vertical (axial) displacement at the input of i microlever stage

k h h o, k hhp

the horizontal spring constants at output and pivot, respectively, when loaded with a horizontal force

k h m o, k hmp

the horizontal spring constants at output and pivot, respectively, when loaded with a bending moment

k vvo, k vvp

the vertical (axial) spring constants of the output system and the pivot, respectively, when loaded with a vertical force

k θ h o , k θ h p

rotational spring constant of the output system and the pivot, respectively, when loaded with a horizontal force

k θ m o , k θ m p

rotational spring constant of the output system and the pivot, respectively, when loaded with a bending moment

 M o, M   p

 bending moments at the output and pivot

U in

work performed by the input force (static loading)

U vo, U θ o

the tensile/compression strain energy and moment-bending strain energy, respectively, consumed at the output system

U vp, U θ  p

the tensile/compression strain energy and moment-bending strain energy, respectively, consumed at the pivot

t c , t  f , t  p

the thickness of all flexure beams and the lever arm

xxiii

θ 

the rotation angle of the lever arm under loading

θ mo, θ h o

the rotation angle of the output caused by bending moment and horizontal force

θ mp, θ h p

the rotation angle of the pivot caused by bending moment and horizontal force

xxiv

CHAPTER 1 INTRODUCTION

MEMS is a rapidly-growing field which mainly involves the fabrication of microscale mechanical devices with integrated circuit process technology. Micro-motors as small as the diameter of a human hair have been fabricated with silicon and their  functionality demonstrated. Many other micro-devices such as integrated sensors and  actuators were reported to have been successfully fabricated with thin film deposition and  etching methods. Machines are built at micro-scale, with components controlled by different actuation methods. Micro-mechanisms--including gears, four-bar linkages, and  micro-robotics--could be all integrated together. MEMS devices ranging from automobile crash sensors (inertial sensors), to drug delivery systems (microfluidics), to hard-drive suspension arms will impact our everyday life in the very near future. MEMS also brings many challenges to the fundamental science of many disciplines because different  prevalent forces and governing equations will come into play when the dimension changes from macro-scale to micro-scale. The length scale is a fundamental quantity that dictates the type of forces governing physical phenomena. With body forces scaled to the third power of the length scale, size effects can be seen in most physical phenomena at the micro-scale. With surface forces depending on the first or second power of the characteristic length, surface effects also become dominant in the micro-world. Many  phenomena at the micro-scale are different from those at macro-scale. For example,

-1-

mixing of two different fluids at the micro-scale takes much longer since diffusion rates are different from that at the macro-scale. The frictional force is a function of the contact surface at the microscale instead of the traditional frictional law f  law f = = μ  N .  N . Deviations from conventional thinking and understanding are commonly encountered at the micro-world, creating new scientific frontiers as well as new technologies. Mechanisms that are well understood in the macro-world become the subject of research in the micro-scale. This thesis focuses on the systematic study of the lever in micro-scale design method and  several design applications. Leverage mechanisms (Mach, 1960 and Ohanian, 1991), including single-stage and multi-stage, are very useful in MEMS in order to transfer an input to an output to achieve mechanical and/or geometrical advantage. Similar to its wide application at the macro-world,

micro-leverage

mechanisms

can

be

used

to

transfer

an

input

force/displacement to an output, either amplifying them or changing the force directions (Sniegowski, 1995; Keller, 1995; Lin, 1993; etc). The design of micro-leverage mechanisms is different from that of the conventional levers in the macro-world. With fabrication technology constraints, the microleverage mechanism is mainly formed by co planar flexures, with one end of the flexure beam anchored to the substrate as a pseudo pivot. Mechanical transformation in a microleverage mechanism is achieved by elastic deformation of its component flexure beams. This group of mechanism is the so-called  compliant mechanism. Compliant mechanisms transmit force and/or motion through flexibility and  undergo elastic deformation. Salanon (1989) defined a complaint mechanism as “a mechanism that gains all or part of its mobility from the relative flexibility of its -2-

members” (Salamon 1989 and Howell 1991). Compliant mechanisms have many advantages compared with the conventional rigid-body mechanisms. They require fewer   parts, no n o or minimal assembly, experience ex perience less wear and noise, less problems associated  with friction and backlash. For applications in MEMS planar microstructures, compliant mechanisms provide a new group of micro-devices such as compliant four-bar linkages and micro-grippers. Compared with the earlier micro-mechanical structures such as cantilevers and diaphragms, this new group of micro-devices can perform more types of  mechanical tasks than cantilevers and diaphragms could do, e.g., tools for microfabrication, nanofabrication, microsurgery and nanoprobing analysis systems. In microsurgery where accuracy is required in microscopic and very sensitive operations, the micro-mechanisms can function as precise tweezers and knives. In microanalysis and  nanoanalysis systems, the micro-mechanisms can be used as positioning tools or for   precision probing of a surface (Larsen, 1997 ). With the advancement of micro-fabrication technology, both compliant mechanisms and compliant microstructures can be designed  to have desired material properties such as specific thermal expansion coefficients, negative Poisson’s ratio and piezoelectric behavior (Kikuchi 1998 and Larsen 1997). Compliant microleverage mechanisms can be widely used as mechanical amplifiers in micro-grippers, micro-actuators, micro-fluidics and micro-sensors, or wherever a force or  displacement transformation is needed. There are several papers in the open literature on compliant microleverage mechanisms including both single-stage and multiple-stage compound microlevers designed to amplify either force or displacement (Sniegowski, 1995; Keller, 1995; Lin, 1993; etc). Snegowski and Smith (1995) presented a leverage mechanism that converts a

-3-

short-displacement, high-force electrostatic actuation to a long-displacement, mediumforce output. Keller (1995-1997) designed a lever structure with an amplification factor  of 27 to transfer a thermal actuation to the motion of a micro-tweezer tip. Roessig (19951998) used leverage mechanisms to increase the sensitivity in a resonant accelerometer   by amplifying the inertial force with both asymmetrical and symmetrical structures. Wad  et al . (1996) at the Jet Propulsion Laboratory developed a two-stage mechanical flexure amplifier with pairs of piezoelectric stacks for vibration suppression. Other leverage mechanisms have been designed to amplify stroke of piezoelectric stacks (Main and  Garcia, 1997). William (1999) used a mechanical lever in the operation of a micro-valve. The analysis and synthesis of compliant mechanisms can be very difficult and  time consuming due to their kinematic complexity and deflection non-linearity. Analysis of these mechanisms is typically accomplished by finite element methods. Synthesis of  compliant mechanisms, i.e., obtaining a suitable topology, shape and size of the mechanism for performing a specified mechanical operation, has caught the attention of  many researchers (Kota et al. 1994). From the literature, there are two major approaches: the kinematics synthesis approach and the continuum synthesis approach. The kinematics synthesis approach is also called the pseudo-rigid-body model approach (Howell, 1994) where flexure mechanisms are designed by creating a kinematic model of the basic mechanism configuration, then replacing the mechanism hinge joint by flexure hinges. This method normally requires an initial mechanism geometry configuration. The continuum synthesis approach uses the topology optimization technique for mechanical structures. Ananthasuresh et al. (1994) presents a homogenization method where the structure optimization approach is used.

-4-

Although compliant leverage mechanisms have great potential application in MEMS, the theory and design issues are not well understood and documented. Designers usually take a trial-and-error, intuitive and iterative approach. Most of the devices are custom-designed for particular applications on an ad-hoc basis. The basic kinematic mechanism configurations are generated based on intuition and experience, or by replacing conventional joints in a rigid-link mechanism with flexure hinges. The flexure designs are then refined through a trial-and-error or finite element methods. Although good accuracy is required, the amplification factor of a leverage mechanism can only be roughly estimated in most cases. In this dissertation, the design of compliant microleverage mechanism are systematically studied and presented with a direct analytical method approach and finite element method approach, followed by both micro- and macro-scale experimental verifications and design applications. Chapter 2 introduces the basic structure, classification and nomenclature of  microleverage mechanisms. Chapter 3 mainly discusses different flexure pivot designs in microleverage mechanisms. Analysis of a single-stage micro-leverage mechanism with a first-order model and a second-order refined model are presented in Chapter 4 to derive the equations for calculating the amplification factor. Related design issues are also discussed. An energy analysis and the concept of mechanical and geometrical advantages of a single-stage microleverage mechanism are also introduced. The design method  introduced for a single-stage micro-leverage mechanism lays the foundation for not only the single-stage microleverage mechanism design, but also the two-stage and multiplestage microleverage mechanism design. -5-

Based on the theories found for single-stage, the design of two-stage microleverage mechanism is presented in Chapter 5. Multiple-stage microleverage mechanism designs are discussed in Chapter 6. The maximum achievable amplification factor for a specific output system is calculated in terms of the output system spring constant and the lever ratio. The design of a microleverage mechanism in a resonantoutput micro-accelerometer is presented throughout this dissertation to illustrate the design issues and study the effect of lever ratio and the dimensions of pivot, connection  beams, and tuning fork beams on the amplification factor. A resonant accelerometer with a two-stage microleverage mechanism is designed and fabricated with the SOI-MEMS  processing. Chapter 7 focuses on several important issues concerning the SOI-MEMS fabricated resonant accelerometer. These issues include frequency and sensitivity calculations, the effect of thickness changes on the sensitivity and amplification factors of the mechanism, testing procedures and the analysis of experimental results. Chapter 8  presents the experimental verification of the design theory on a macro-scale aluminum model and the scaling-factor issues. Chapter 9 presents the design of microleverage mechanisms for displacement amplification with two application cases, one is for a diskdrive suspension and the other for a microvalve. Finally, the principal findings and the key contributions of this research work are summarized in Chapter 10.

-6-

CHAPTER 2 COMPLIANT MICROLEVERAGE MECHANISMS

In the World Book Encyclopedia A605, it is stated that the lever was first described by Greek scientist and philosopher Archimedes who once said, “Give me a long enough lever and a place to stand, I can move the earth”. Others claim that it is the Greek philosopher Aristotle who first described lever theory in his book titled  “Mechanics” (Mach, 1960; Beckwith et al .). .). Regardlessly, all agree that the applications of levers can be found widely in industry and nature. While levers are found in automobiles, bikes, weight machines, etc., many parts in human body (e.g., arm and jaw) also function as leverage mechanisms. It is a simple machine making many job functions “easier” to perform, transferring an input to an output to achieve mechanical and/or  geometrical advantages. It can change the force direction, e.g., e.g., from pushes to pulls, and  also amplify force or displacement.

2.1

Leverage Mechanism and the Governing Laws A mechanism is defined as “a device for the coupling and transforming of 

energies” (Hall, 1953). Alternatively, Sandor and Erdman (1984) define the mechanism as “a mechanical device that has the purpose of transferring motion and/or force from a

-7-

source to an output.” When a lever is used to transfer an input motion and/or force to an output one, it is called a leverage mechanism. The governing laws in a leverage mechanism are force and moment balance and  energy conservation. In an ideal situation (a rigid lever arm without bending and a perfect  pivot with free rotation or rigid support), the moment with respect to the pivot point should be balanced,  F in Lin =  F out Lout, while the lever is statically balanced. The mechanical work done by the input force should equal to that done by the output force,  F in δ in =  F out δ out if no strain energy is consumed at the pivot or by bending of any flexure beam. For a compliant mechanism the input energy should be equal to the output energy plus the elastic bending energies of the flexible components. When a lever is used  to perform a function, a trade-off is made between the force and displacement. For  example, to lift an object to a certain height, a greater force is needed to move a shorter  distance while a smaller force is needed to travel a longer distance.

2.2

Compliant Leverage Mechanisms Similar to its wide range of applications in the macro-world, levers have been

widely used in microelectromechanical systems, and are called microleverage mechanisms. With current microfabrication technology constraints, a microleverage mechanism is typically formed by planar flexures, achieving mechanical transformation through the elastic bending of its members. This group of microleverage microleverage mechanisms is also called compliant microleverage mechanisms.

-8-

Compliant microleverage mechanisms can transfer an input (e.g., force or  displacement) to an output to achieve mechanical advantage and/or geometrical advantage in MEMS, such as changing force directions between pushes and pulls and  amplifying force or displacement. For example, in the case of an actuator a leverage mechanism can attenuate the force and the displacement generated by the original actuation mechanism to a desired output. Other applications include magnifying inertial forces in inertial sensors such as micro-accelerometers and gyroscopes to increase sensitivity, amplifying the tensile force in a tensile testing machine, and tuning a microresonator. Similar to the role of the operational amplifier (common-source amplifier and  multi-stage amplifier) in microelectronics, microleverage mechanisms (single- and  multiple-stage) are mechanical amplifiers in MEMS.

2.3

Classifications of Compliant Leverage Mechanisms A compliant leverage mechanism consists of four major parts: lever arm (rigid 

 part), pivot, the input and output systems. Depending on the different positions of the  pivot, the input and an d the output system on the lever arm, the leverage leve rage mechanisms can be classified into three kinds. The first-kind is defined by the pivot lying between the input and the output. When the output lies in the middle of the pivot and the input, it is called a second-kind lever. When two lever arms of a second-kind lever join together at a hinge, it is called a double second-kind lever. A third-kind lever is defined by the input placed   between the pivot and the output. The third-kind lever is used to amplify displacement. A second-kind lever is mainly for force amplification. A first-kind lever can amply either 

-9-

force or displacement depending on the distance between the input and the pivot and the distance between the output and the pivot. When the arms of two third-kind levers join together as in the case of a pair of tweezers, the leverage mechanism is called a double third-kind lever.

For each kind of compliant leverage mechanism, there are two subgroups: (i) one group with the output system and the pivot beam on the same side of a lever arm (designated as “S” group); and (ii) the other group with the output system and the pivot  beam on the different sides of a lever arm (designed as “D” group). A single-stage leverage mechanism can be identified by a code starting with a number (1, 2 or 3) corresponding to the lever kind and followed by a letter (S or D) specifying the subgroup, e.g., 1S, 1D, 2S, 2D, 3S, 3D.

A compound leverage mechanism or multiple-stage leverage mechanism is formed by stacking multiple stages of lever together. Different kinds of levers can be stacked together. If individual single-stage levers are properly configured, their  amplification factors will be multiplied. In a compound leverage mechanism, each lever  stage needs to be identified for reference. There are major two kinds of classifications: downstream classification (from output to input) and upstream classification (from input to output). In the downstream classification, the one connecting to the output system is called the first-stage, then the second-stage, until the input system. In the upstream classification, the one connected to the input system is called the first-stage, and the one connected to the first-stage is called the second-stage, and so on till the output system. In this thesis, the downstream classification is used for the force amplification leverage

- 10 -

mechanism. The upstream classification method is used for the displacement amplification leverage mechanism.

A lever is used to trade a force with distance, or vice versa, to make a job function “easier” to do. The L/l  The L/l value value is called the lever ratio where L where L is the distance between the input and the pivot and  l  between the output and the pivot. More terminology for  compliant leverage mechanisms will be presented later, such as amplification factor, mechanical advantage and geometry advantage, and mechanical efficiency.

2.4

Design

of

Microleverage

Mechanisms

and

Their

Challenges A microleverage mechanism differs in many aspects from that in the macro world. A pivot structure in the macro world can be formed by a pin-joint or bearing which permits free rotation and rigid support. Limited by current micro-fabrication technology, it is very difficult to achieve free rotation and rigid support in a microleverage mechanism. These elements are too complicated to be fabricated by surface micromachining. Some integrated fabrication processes, such as the SUMMIT  process

(The

Sandia

Ultra-planar

Multi-level

MEMS

Technology,

http://www.mdl.sandia.gov/Micromachine) and MCNC (Markus and Koester, 1994, http://www.mcnc.org/), provide multiple polysilicon layers that can make pin-joints or   bearings. However, at a pin-joint, the gap between the rotational and the fixed parts makes the mechanism behavior uncertain. Stiction force prevalent in the micro-world   prevents a bearing pivot from functioning properly. Therefore, the most commonly used  - 11 -

 pivot in MEMS is a flexure beam with one end anchored to the substrate. The geometry of the beam is designed to have a relatively small rotational spring constant allowing easy rotation. In a leverage mechanism, the lever arm needs to be kept rigid. The geometry and dimension of the flexure beams can be varied to have different spring constants.

In most cases, even for a single-stage leverage mechanism, the magnification factor is a rough estimation. Many design issues and trade-offs are not well understood. For the compound leverage mechanisms, the geometry of each component is usually determined with a trial-and-error approach. The overall amplification factor has been obtained by testing and/or estimation. In many applications, the amplification factor  needs to be accurately known. Taking a micro-tweezer for example, the force and  displacement generated at the tip of the micro-tweezer need to be exactly known for   picking up an object. Sensitivity of the resonant accelerometer, which is a precision instrument, needs to be precisely specified. In other cases where the input and output information is specified, a microleverage mechanism needs to be implanted to transfer  the input system to the output system. It is imperative that an accurate expression of the amplification factor of the mechanism and a thorough understanding of the governing theory of the microleverage mechanism be developed.

There are many design issues associated with microleverage mechanisms. The amplification factor depends on not only the geometry of the mechanism, but also the input and output system spring constants. In most cases, elastic deformation in the mechanism is small and the mechanism can be assumed to be operating in the linearelasticity regime. If the deflection is large enough to introduce non-linearity, the analysis

- 12 -

could become very complicated. The performance of the mechanism may also depend on the magnitude of the input force.

Many factors influence the amplification factor, e.g., the pivot geometry, and how the leverage mechanism is connected to the input and output system. The stiffness of the output system plays an important role in a leverage mechanism. In a multiple-stage microleverage mechanism, the spring constant of the previous stage must be high enough to allow the next stage to have an amplification effect. It is similar to the design of an opamp in analog circuits where the input and output resistance of different stages should  match.

Designing a microleverage mechanism for a specific application can be a complex  problem. In both single- and multiple-stage compound co mpound microleverage mechanism design, the challenging aspect is the conflicting stiffness requirement. The mechanism needs to  be stiff enough to transfer the input force, yet soft enough to deflect. de flect. These issues will be addressed in detail in later Chapters.

2.5

FEM Simulation For a single microlever, the magnification factor may be found from an analytical

solution. As more and more stages are added, the analytical approach becomes more and  more difficult and numerical methods are needed. While the analytical solution is based  on many assumptions, the finite element method (FEM) requires fewer assumptions and  is a more rigorous approach.

- 13 -

There are several software packages available for the FEM, such as ABAQUS (Hibbitt, 1994) and SUGAR (Clark, 1998). ABAQUS, together with PATRAN is a  powerful tool for FEM simulation. It can be used to simulate fracture and PZT structures and to calculate resonant frequencies. It works for both linear and nonlinear elastic beam  bending. Besides using PATRAN as a mesh generator, ABAQUS also has its own prior  mesh generator and post analysis tool. However, ABAQUS requires substantial set up and it is difficult to interface the mesh generation for geometry variations. The results from ABAQUS are unnecessarily detailed for the analysis of microleverage mechanisms.

SUGAR is a MEMS CAD tool developed at U.C. Berkeley (Clark, Zhou and  Pister, 1998). It needs an input of a net-list file specifying the information of each connecting point as a node, which is similar to SPICE simulations of circuit. The netlist file is easy to change for geometry variations. SUGAR only simulates small deformation within linear elasticity which is the case for most microleverage mechanism designs. When a compliant mechanism undergoes large deflection and non-linearity is introduced, then a nonlinear FEM simulation such as ABAQUS must be used.

Both ABAQUS and SUGAR are used for the single-stage microleverage mechanism simulation. The results are compared with each other and also with the analytical result. Figure 2.1 shows the mesh of the mechanism generated by ABAQUS. Figure 2.2 shows the deflected mechanism model. The ABAQUS input file for the simulation is attached in Appendix A. The geometry of the structure are as follows: horizontal beam length 210 μm, width 20 μm; middle beam length 24 μm, width 2 μm. The left beam functions as the pivot with length 6 μm width 2 μm. Distance between the

- 14 -

left and the middle beam is 10 μm. With a input force 1.25e-9N, the average reaction force at the middle beam is 2.37e-8N from ABAQUS. From SUGAR the output force is 2.58e-8N and the amplification factor is 20. Results from ABAQUS and SUGAR are very close, with a discrepancy of 5%. Since in most of the application, the leverage mechanism deformations are in elastic regime, SUGAR is used in the microleverage mechanism simulation and is found to be quick and accurate. There are some discrepancies between the results from the 1st-order analytical model and the SUGAR  simulation. With a second-order analytical model, the analytical result and the SUGAR  simulation agree very well.

Fig. 2.1 Mesh generated by ABAQUS for the leverage mechanism. mechanism.

Fig. 2.2 Leverage mechanism mechanism deflection simulated by ABAQUS.

- 15 -

CHAPTER 3 FLEXURE PIVOT DESIGN

An important aspect in the compliant microleverage mechanism is to replace the conventional pivot with a flexure hinge since a rotational pivot requires a more complex fabrication process. In a microleverage mechanism, the pivot is usually formed by a simple flexure beam with one end anchored to the substrate. In order to minimize stress concentrations at the corners where the pivot beam joins the lever arm, the joint usually has a rounded corner. Pivot design and optimization in microleverage mechanisms are  presented in this Chapter.

3.1

Flexure Pivot Design The classical paper on flexure pivot design is entitled “How to design flexure

hinges” by Paros and Weisbord (1965). The paper states that “while simple in shape and  operation, flexures are mathematically complex”. They extensively studied the design of  the flexure hinges in several configurations and developed many approximate formulas. A general flexure hinge model is shown in Figure 3.1. The most common type of flexure hinge is basically a mechanical member which is compliant in bending about one axis but rigid about the cross axis. The term “compliance” is referred to as the reciprocal of the stiffness or the spring constant. In microleverage mechanisms, the flexure pivot is a very important component.

The general flexure hinge model shown in Figure 3.1 can also minimize the stress concentrations. The flexure rigidity K  rigidity  K   b (corresponding to a bending spring constant) and  - 16 -

the tensional rigidity K s (corresponding to axial spring constant) are approximated by the following formulas (Her and Chang, 1994):

 K b =

 K  s =

 M  δθ 

 F  x δ  x

=

=

2 Eb t 5 / 2

(3.1)

1/ 2

9π  r 

 Eb

(3.2)

π (r / t )1 / 2 − 2.57

Where subscripts “b “b” and “s” denote bending and stretching, respectively. Both the analytical model and ABAQUS simulation of the hinges with different geometries are  presented by Her and Chang (1994).

r  Fx

Fx

 b t

h Fig. 3.1

The generalized flexure hinge model.

Figure 3.2 and 3.3 show SEM of the pivots in a two-stage microleverage mechanism, which is fabricated with a Silicon-On-Insulator MEMS process, and is further described in Chapter 7. The first pivot is a generalized flexure hinge which serves as the first-stage pivot in the two-stage microleverage mechanism, Fig. 3.2. The other is a narrow beam which is the second-stage pivot in the two-stage microleverage mechanism, Fig. 3.3.

- 17 -

Fig. 3.2 A flexure pivot fabricated by SOI-MEMS technology.

Fig. 3.3 A flexure beam pivot in the second-stage microlever in the resonant Accelerometer fabricated by SOI-MEMS technology.

- 18 -

3.2

Different Models of Flexure Pivot In microleverage mechanisms, most of the flexure pivots are formed by narrow

 beams. Figure 3.4 shows several prototypes with a flexure beam as the output: (a) the vertical model (model I), (b) the horizontal model (model II) and (c) the combined model (model III). The amplification factors of these three configurations are analyzed with SUGAR. A simple beam is used as the output and the output force is calculated as the axial tensile or compressive force.

Table 3.1 lists the displacements and also the forces for these three models. The netlist files for geometry information are attached in Appendix B. The following  parameters are used for the simulation: beam thickness, t , of 2 μm; modulus of elasticity  E =  E  = 1.65 x 1011 N/m2; lever arm length L length  L = 200 μm; distance between pivot and output system l = l  = 10 μm; input force, F in, of 1.5 x 10-9 N; pivot beam and output beam width

w p = wo = 2 μm; pivot beam and output beam length l  p = l o = 60 μm. Compared with model II, model I has much greater amplification factor (See Table 3.1). Model III has an amplification factor in-between those of Model I and Model II. Figure 3.5 shows the comparison of the amplification factor of the each model as a function of the pivot beam length. Model I has the highest amplification factor compared  to the other two and is not sensitive to variations of the pivot beam length. Model I will  be used for all designs presented later. The vertical pivot is found to be the most effective for different pivot beam designs. The horizontal displacement and the rotation angle are slightly different when the pivot and output beams are at the same or different sides.

- 19 -

Table 3. 1 SUGAR Simulation Results Results of Different Different Pivot Models.

Model I

Model II

Model III

 Node 3

X Displacement Y Displacement Rotational Angle

1.6556e-11 -2.6905e-13 5.5145e-07

2.6395e-13 -7.878e-11 7.9263e-06

2.0073e-10 -1.3215e-10 1.3324e-05

 Node 4

X Displacement Y Displacement Rotational Angle

1.6556e-11 2.8268e-12 5.523e-07

2.6615e-13 4.8811e-13 7.9273e-07

2.0073e-10 1.0925e-12 1.3324e-05

 Node 5

X Displacement Y Displacement Rotational Angle

1.6556e-11 1.1556e-10 5.6934e-07

2.6615e-13 1.5882e-09 7.9443e-06

2.0073e-10 2.6682e-09 1.3341e-05

 Node 2

X Displacement Y Displacement Rotational Angle

-

-

1.1055e-13 -1.3167e-10 -1.607e-06

20.7

3.6

8.0

Amplification Factor

Table 3.2.

Comparison of SUGAR Results with the Model I Pivot and Output System on (a) the Same Side and (b) Opposite Sides of the Lever Arm

 Node 3

 Node 4

 Node 5

X Displ isplac acem emen entt Y Displacement Rotational Angle X Displ isplac acem emen entt Y displacement Rotational Angle X Displ isplac acem emen entt Y Displacement Rotational Angle

Amplification Factor

- 20 -

Same Side 1.65 1.6556 56ee-11 11 -2.6905e-13 5.5145e-07

Different Sides 1.27 1.271e 1e--14 -2.5857e-13 5.3048e-07

1.65 1.6556 56ee-11 11 2.8268e-12 5.523e-07

1.28 1.2858 58ee-14 14 2.7221e-12 5.3134e-07

1.65 1.6556 56ee-11 11 1.1556e-10 5.6934e-07

1.28 1.2858 58ee-14 14 1.1126e-10 5.4838e-07

20.6

19.8

3.3

Pivot and Output System Configuration Figure 3.6 is an interesting plot which shows how the amplification factor varies

with the angle between the pivot beam and the lever arm. It is found that the amplification factor is maximum when the angle between the pivot beam and the lever   beam is at 90 degrees. At 270 degrees, the amplification factor is slightly smaller than that at 90 degrees. When the pivot is at the same or opposite side of the output system, the mechanism is deflected differently.

Sometimes, the micro-fabrication technology constraints will determine the leverage mechanism configuration, whether the pivot and the output system can be  placed on the same side or different sides of the lever arm, a rm, e.g., SOI-MEMS technology only allows an outside anchor. Figure 3.7 shows the Model I pivot where the pivot and  output system are on (a) the same side and (b) opposite sides of the lever arm. Table 3.2 lists the results from SUGAR simulation for a comparison between the two cases. The horizontal displacements are significantly different and the force direction changes. Also different are the rotation angles. Those differences cause slight amplification factor  change for the single-stage leverage mechanism.

Figure 3.8 shows the shape of the deflected pivot beam when the pivot and the output system are on different sides of the lever beam. Although the effect of the lever  configuration on the amplification factor of a single-stage microleverage mechanism is insignificant, the effect becomes much more pronounced in a two-stage microleverage

- 21 -

mechanism. For a second-kind lever, the 2 D microlever has a lower amplification factor  than 2S  2S , as explained in detail in later Chapters. The deflections of pivot and connection  beams are simulated with SUGAR and plotted in Fig. 3.9. For illustrative purposes, the scale of the vertical axis (location on a beam) is 106 times that of the horizontal axis (deflection). When the connection beam and the pivot are on different sides of the lever  arm, the shape of connection beam under loading is interesting, noting that they would  have been bent in the same fashion if they had been on the same side of the lever arm. Because of the increased resistance to rotation when the pivot and output connection  beam are on different sides of a lever arm, the amplification factor is reduced.

- 22 -

(a) Vertical ertical P ivot ivot 7 6

1

Output System 2

Pivot l 3

4

5

L

Input

(b) Horizontal Horizontal Pivot P ivot

7 6 Output System

1

l 2

4

3

5

L

Pivot

Input

(C) (C) Combined Combined Pivot 7 6 11 Output Sy System

1 2

3

Pivot l 4

5

L

Input

Fig. 3.4 Different Pivot Models. Models. - 23 -

25 Model I

20 2nd-Kind Microlever  Lever Ratio 21:1

A

15

All dimensions in microns L=200 l =10 lo =60 wo =wp =2

Model III

10 Model II

5

0 0

10

20

30

40

50

60

70

80

90

100

Length of Pivot Beam, micron

Fig. 3.5

Comparison of the amplification factors of three Models as a function of the  pivot beam length. 25 Model I

Model I 2nd-Kind Microlever  Pivot Model I, II, and and In-between Lever Ratio 21:1

20

  r   o    t   c   a    F 15   n   o    i    t   a   c    i    f    i    l 10   p   m    A

All dimension in microns L=200 l =10 lo =60 wo =wp =2

Model II

5

0 90

110

130

150

170

190

210

230

250

270

Angle Between Pivot and Lever Arm

Fig. 3. 6

Effect of pivot beam with different different angles on the amplification factor.

- 24 -

(a) Pivot at the same side sid e as output output

Output System

Pivot

Input

(b) Pivot at the different side sid e as output output

Output System

Input Pivot

Fig. 3.7

A second-kind leverage mechanism with the pivot and an d output system (connection beam) (a) on the same side and (b) different sides of the lever arm.

- 25 -

120 120 110 110   n   o100 100   r   c    i   m 90  ,   m 80   a   e    B    t 70   o   v    i    P 60   e    h 50    t   n   o 40   n   o    i    t 30   a   c   o 20    L

Second-Kind lever with single beam as output  All dimensions in microns L = 210 l = 10 lo = 100 lp = 120 wo = wp = 2

10 0 0

5

10

15

20

25

30

Horizontal Displacement, 1E-6 micron

Fig 3.8 The shape of a deflected pivot located on the opposite side of the lever arm.

Horizontal Displacement, Displacement, 1E-6 1E -6 micron -6 -3 0

24

Node 6 20 Connection Beam

16

12

8

4

Node 7 Lever Arm

0

θ (exaggerated)

6 Node 8

4

Pivot Beam

2 0

Node 9 -6 -3

0

Horizontal Displacement, 1E-6 micron

Fig 3.9 The deflection of pivot and connection beams of a 2D microlever.

- 26 -

CHAPTER 4 SINGLE-STAGE MICROLEVERAGE MECHANISMS

The single-stage microlever is the basic element in the microleverage mechanism. Analysis of the single-stage microlever provides insight and foundation for the design of   both single- and multi-stage microleverage mechanisms. This chapter presents the analysis of single-stage microleverage mechanisms. A first-order analytical model and a second-order refined analytical model are built to derive the amplification factor,  A, of  the first-kind and second-kind microleverage mechanisms. The amplification coefficient,  A*,

is introduced to study the effect of the pivot geometry and output system on the

amplification factor. Also presented are the strain energy analysis, mechanical efficiency, and mechanical and geometrical advantages of the mechanism. The design and  optimization of the single-stage microleverage mechanism in a resonant output accelerometer is presented in detail. Other design issues, such as the effect of horizontal force and buckling are also addressed.

4.1

Basic Structure of A Single-Stage Microleverage Mechanism The single-stage microleverage mechanism consists of four major parts: lever 

arm, pivot, the input system and output system, as schematically shown in Figure 4.1(a).

- 27 -

As a compliant mechanism consists of both rigid part and flexure members, the lever arm in a compliant microleverage mechanism is the rigid part while the pivot and the connections are the flexible elements.

According to the different positions of the pivot, the input and output systems in relation to a lever arm, a microleverage mechanism can be classified as one of the three kinds. The first kind is defined by the pivot lying between the input and the output as in shown in Fig. 4.1(a). When the output lies in-between the pivot and the input, the microlever is then called a second kind as shown in Figure 4.1(b). When two lever arms of a second-kind lever join together by a hinge (i.e., pivot), it is then called a double second-kind lever. A third kind shown in Fig. 4.1(c) is defined as a microlever with the input lying between the pivot and the output. Joining the pivots of two third-kind  microlevers makes a double third-kind lever, such as micro-tweezers. While first- and  second-kind microlevers can amplify force, a third kind is typically used to magnify displacement. A first kind microlever can either amplify force or displacement, depending on the respective distances between (i) the pivot and the input and (ii) the  pivot and output system. A first-kind microlever can also be used to change the force direction.

- 28 -

(a)First Kind

Anchor

4. Output System 3. Piv P ivot ot l

L 2. Lever Arm

1. Input System

(b) Second Kind Output

l L

Input

(c) Third Kind Output

l L

Input

Fig. 4. 1

Schematic of three kinds of micro-leverage mechanisms.

- 29 -

4.2 Amplification Factor of a Single-Stage Microleverage Mechanism The leverage factor, l/L, is the maximum mechanical or geometrical advantage that a leverage mechanism can achieve. For force amplification, the leverage factor can also be called the amplification factor,  A which is the output force v.s. the input force. The force amplification factor of a second-kind microlever is analyzed first. The analysis is similar for the other two kinds of microlever.

4.2.1 Second-kind single-stage microleverage mechanism A second-kind microlever under loading is schematically shown in Fig. 4.2(a). When an input force is applied, the lever arm will be rotated by a small angle and  displaced by a small distance, as shown in Fig. 4.2(a). The system under loading can be modeled as shown in Fig. 4.2(b). The flexural pivot is modeled as a kinematic joint, revolute pair, represented by a torsional spring and a vertical spring. For this analysis, five assumptions are made: (1)

The elastic tension/compression or bending strain in the entire structure is within the regime of linear elasticity.

(2)

The lever arm is rigid and remains straight during loading. The total deflection can be represented by a vertical displacement δ (positive sign if  the same direction as the input force, negative sign otherwise) and a rotation angle θ about the pivot (positive sign if counterclockwise).

- 30 -

(a)

Output

Pivot

Connection Beam to Output System System

l θ+δ

δ

θ

θ

Connection Beam to Input Syst System

l 

L

(b)

θ δ

Fig. 4.2

(a) A second-kind microlever before and after loading; (b) Model of the second-kind microlever under loading.

- 31 -

(3)

The un-anchored ends of the pivot beam and the connection beam to the output system maintains the 90° orientation with respect to the lever arm after loading. Therefore, the pivot beam and the connection beam will  both be rotated by the same angle θ  at the respective beam end.

(4)

The vertical output force does not cause any horizontal displacement or  rotation of the output system.

(5)

The horizontal forces acting on the pivot beam and the output connection  beam are negligible when a vertical input force is applied. In addition, the vertical displacement at output caused by the bending moment is assumed  to be negligible.

Assumptions (3) to (5) imply that the rotation angle, θ , at the ends of the pivot beam and  the output connection beam are solely caused by the respective bending moments. The vertical displacement at the output system is only caused by the vertical output force. Applying the force and moment (with respect to the joint between the pivot beam and the lever arm) equilibrium condition to the lever arm leads to the following equations: F in

= k vvo (lθ  + δ ) + k vvp δ 

F in L

= k vvo (lθ  + δ ) l + k θ m oθ  + k θ mpθ 

(4.1)

(4.2)

where F in is the input force, k vvo the vertical (first subscript “ v” of the term k vvo) spring constant of the output (third subscript “o”) system under a vertical (second subscript “v”) force, k vvp the vertical spring constant of the pivot under a vertical force, k θ m o the

- 32 -

rotational spring constant of the output system when loaded with a bending moment,

k θ m p the rotational spring constant of the pivot when loaded with a bending moment, l the distance between the pivot and the output and  L  L the distance between the pivot and  the input. The above equations were solved with Mathematica 4.0 and the Mathematica file used is attached Appendix C. Solving equations (4.1) and (4.2) for 

θ 

and 

δ 

by

Mathematica, we have:

θ 

=

k vvo + k vvp  L − k vvo l

( k vvo + k vvp) ( k  m o + k  m p ) + θ 

δ 

=

θ 

k vvo k vvp l

2

k θ m o + k θ m p − k vvo l ( L − l )

( k vvo + k vvp ) ( k  m o + k  m p ) + θ 

θ 

2 k vvo k vvp l

F in

(4.3)

F in

(4.4)

Since the output vertical (i.e., axial) displacements caused by the internal bending moment and horizontal force are negligible, the output force is equal to the axial spring constant of the output system, k vvo, multiplied by the output axial displacement, δ + lθ .

1 F out = k vvo (δ  + lθ ) =

k vvp

( k  mo + k  m p ) + lL θ 

θ 

⎛  1 1  ⎞⎟ 2 ⎜ + ⎜ k vvo k vvp ⎟ ( k θ m o + k θ m p ) + l ⎝   ⎠

The amplification factor of the microlever is:

- 33 -

F in

(4.5)

1  A =

F out F in

=

For an ideal lever, k vvp

k vvp

( k  mo + k  m p ) + lL θ 

θ 

⎛  1 1  ⎞⎟ 2 ⎜ + ⎜ k vvo k vvp ⎟ ( k θ m o + k θ m p ) + l ⎝   ⎠

(4.6)

⇒ ∞ , k θ m p ⇒ 0, k θ m o ⇒ 0 and the amplification factor 

approaches the lever ratio:

 Ao

=

 L

(4.7)

l

4.2.2 First-kind microleverage mechanism To derive the amplification factor of a first-kind microlever, the locations of the  pivot and output system are exchanged with each other from those in a second-kind  microleverage mechanism. For the amplification factor, we can just exchange k vvp with

k vvo, k θ m o with k θ m p in equations (4.1) and (4.2), and obtain the following: 1  A =

k vvp

( k  m o + k  m p ) − lL θ 

θ 

⎛  1 1  ⎞⎟ 2 ⎜ ( ) + + + l k  k  m o m  p θ  θ  ⎜ ⎟ ⎝  k vvo k vvp ⎠

(4.8)

Using the same notation for  L and  l, the difference between the amplification factor of the first-kind and the second-kind microleverage mechanism is the sign of the “lL” term in the numerator. The first-kind microlever has a negative sign while the second-kind has a positive sign.

- 34 -

The first-kind and second-kind microleverage mechanisms are simulated with SUGAR for the purpose of comparison. It is found that the first kind had a slightly smaller amplification factor than the second kind for the same total length of the lever  arm,  L. This is because of the different lever ratios, i.e.,  L/l for the second kind and ( L  L l)/l for the first kind.

4.3

Amplification Coefficient, A* The expression for the amplification factor in equation (4.6) does not give very

much insight for the mechanism design. Another parameter, the amplification coefficient  A*, which is more directly related related to the design parameters, is defined as follows.

The amplification factor of a microlever cannot be larger than that of an ideal lever, L/l. Taking into account that l>k v v o) and small

k θ m p (k θ m p 1.52844 10E-7 , D -> 2.93688 10E-13 }} m = Kvf (L1 T - d)/Fin Out[10]= 20.1667

In[12]:= K=Fin/(L2 T - d) Out[12]= 48.0615 ***************************************************************

II. Second-Stage Microleverage Mechanism: L1 = 0.00001; L2 = 0.00021;Fin = 0.0000000015; Kvp = 100000; Ktp = 0.000000033; Kvf = 48.0615; Ktf = Ktp = 0.000000033;

 NSolve[{Kvf (L1 T - De) - Kvp De == Fin, Kvf L1^2 T + (Ktp + Ktf) T - Kvf DeL1 == Fin L2}, {T, D}] Out[4]= {{T -> 4.44881 10E-6 , De -> 6.37858 10E-15 }} t = 0.00000444; De = 0.00000000000000638;

- 211 -

In[9]:= m = Kvf (L1 (L1 t - De)/Fin Out[9]= 1.42242 In[10]K=Fin/(L2 t - De) Out[10]= 1.60876 *****************************************************************

III. Third-Stage Microleverage Mechanism: L1 = 0.00001; L2 = 0.00021;Fin = 0.0000000015; Kvp = 100000; Ktp = 0.000000033; Kvf = 1.60876; Ktf = 0.000000033;

 NSolve[{Kvf (L1 T - De) - Kvp De == Fin, Kvf L1^2 T + (Ktp + Ktf) T - Kvf DeL1 == Fin L2}, {T, De}] Out[4]= {{T -> 4.76112 10E-6 , De De -> -1.42338 10E-14 10E-14 }} In[5]:= t = 0.0000047; d = -0.000000000000014; In[7]:= m = Kvf (L1 t - d)/Fin Out[7]= 0.0504228

- 212 -

APPENDIX D Netlist Files For Resonant Accelerometer With Single-Stage Microleverage Mechanism A

2

1

8e- 6

90

16e- 6

b

2

3

4e4e- 6

- 90

12e12e- 6

b

3

4

200e 200e-- 6

- 90

2e2e- 6

b

4

5

20e20e- 6

- 90

12e12e- 6

b

5

6

20e20e- 6

0

20e- 6

b

6

7

24e24e- 6

- 90

2e2e- 6

b

7

8

10e10e- 6

180 180

20e20e- 6

b

8

9

6e6e- 6

90 90

2e2e- 6

a

9

10

4e- 6

90

8e- 6

b

7

11

200e200e- 6

0

20e20e- 6

b

5

12

20e20e- 6

180 180

20e20e- 6

b

12

13

24e24e- 6

- 90

2e2e- 6

b

13

14

10e10e- 6

0

20e20e- 6

b

14

15

6e- 6

90

2e- 6

a

15

16

4e- 6

90

8e- 6

b

13

17

200e200e- 6

180

20e- 6

f

11

1

1. 5e- 9

90 90

0

f

17

1

1. 5e- 9

90 90

0

- 213 -

APPENDIX E Netlist File for Resonant Accelerometer with Two-Stage Microleverage Mechanism a

2

1

8e8e- 6

90

16e16e- 6

b

2

3

4e4e- 6

- 90

12e12e- 6

b

3

4

100e 100e-- 6

- 90

8e8e- 6

b

4

5

20e20e- 6

- 90

12e12e- 6

b

5

6

20e20e- 6

0

20e20e- 6

b

6

7

6e- 6

- 90

4. 0e- 6

b

7

8

10e10e- 6

180

40e40e- 6

b

8

9

6e6e- 6

90

4e4e- 6

a

9 10

4e- 6

90

8e- 6

b

7 11

200e200e- 6

0

40e40e- 6

b

5 12

20e20e- 6

180

20e20e- 6

b

12 13

6e6e- 6

- 90

4e4e- 6

b

13 14

10e- 6

0

40e40e- 6

b

14 15

6e- 6

90

4e- 6

a

15 16

4e- 6

90

8e- 6

b

13 17

200e200e- 6

180

40e40e- 6

b

11 25

60e60e- 6

- 90

2e2e- 6

b

25 24

10e10e- 6

180

20e20e- 6

b

24 23

200e200e- 6

180

20e20e- 6

b

24 26

100e 100e-- 6

- 90

2e2e- 6

a

26 27

8e8e- 6

- 90

16e16e- 6

b

17 18

60e60e- 6

- 90

2e2e- 6

b

18 19

10e- 6

0

20e20e- 6

b

19 20

200e200e- 6

0

20e20e- 6

b

19 21

100e 100e-- 6

- 90

2e2e- 6

a

21 22

8e8e- 6

- 90

16e16e- 6

f

20 1

1. 5e- 9

90

0

f

23 1

1. 5e- 9

90

0

- 214 -

APPENDIX F Netlist File for Calculating Resonator Frequency

1. DETF with with Comb-drive as Center Attached mass mass a

2

1

8e8e- 6

90

16e16e- 6

b

2

3

100e 100e-- 6

- 90

2e2e- 6

b

3

4

100e 100e-- 6

- 90

2e2e- 6

b

3

6

20e20e- 6

0

4e4e- 6

b

6

7

60e60e- 6

90

12e12e- 6

b

6

8

60e60e- 6

- 90

12e12e- 6

a

4

5

8e8e- 6

- 90

16e16e- 6

2. DETF with Capacitive Tuning a

2

1

8e8e- 6

90

16e16e- 6

b

2

21

30e30e- 6

- 90

2e2e- 6

b

21

31

60e60e- 6

- 90

2e2e- 6

b

31

3

10e10e- 6

- 90

2e2e- 6

b

3

32

10e10e- 6

- 90

2e2e- 6

b

32

41

60e60e- 6

- 90

2e2e- 6

b

41

4

30e30e- 6

- 90

2e2e- 6

b

3

6

20e20e- 6

0

4e4e- 6

b

6

7

60e60e- 6

90

12e12e- 6

b

6

8

60e60e- 6

- 90

12e12e- 6

a

4

5

8e8e- 6

- 90

16e16e- 6

g

21

31 23

33 60e60e- 6

- 90 1e1e- 7 4e4e- 6 4e4e- 6 0

g

32

41 34

44 60e60e- 6

- 90 1e1e- 7 4e4e- 6 4e4e- 6 0

- 215 -

APPENDIX G Netlist File for Simulation of the DETF Resonator a1 a2demf p1 [ 2

1] 1]

[ l =5e- 6

b1 b2dem b2dem p1 [ 2

3]

a2 a2demf p1 [ 5

oz= oz =90

w=10e- 6

R= R=100]

[ l =100e100e- 6 oz=oz=- 90

w=2e- 6

R=1000] 1000]

6]

[ l =5e- 6

w=10e- 6

R= R=100]

b2 b2dem b2dem p1 [ 4

5]

[ l =100e100e- 6 oz=oz=- 90

w=2e- 6

R=1000] 1000]

b1c b2dem b2dem p1 [ 3

4]

[ l =10e10e- 6

oz=oz=- 90

w=2e- 6

R=500] 500]

b2c b2dem b2dem p1 [ 3

7]

[ l =20e- 6

oz= oz =0

w=2e- 6

R=500]

b3c b2dem b2dem p1 [ 4

8]

[ l =20e- 6

oz= oz =0

w=2e- 6

R=500]

b4c b2dem b2dem p1 [ 7

9]

[ l =5e- 6

oz=oz=- 90

w=2e- 6

R=1000] 1000]

b5c b2dem b2dem p1 [ 8

9]

[ l =5e- 6

oz= oz =90

w=2e- 6

R=1000] 1000]

b6c b2dem b2dem p1 [ 9

10] [ l =2e- 6

oz= oz =0

w=2e- 6

R=100]

b2m b2dem b2dem p1 [ 10 15] 15] [ l =2e- 6

oz=0 oz=0

w=2e- 6

R=100] 100]

oz= oz =90

w=12e- 6

R= R=100]

oz =- 90

w=12e- 6

R=100]

oz= oz =- 90

b3m b2dem b2dem p1 [ 10 16] [ l =60e- 6 b4m b2de c1

p1 [ 10 17] [ l =60e- 6

comb2d

[ 15 11] [ l =4e- 6

oz=0

N=18 gap=3e- 6]

a3 a2demf

p1 [ 11 12] [ l =5e- 6

oz= oz =0

w=10e- 6

c2

p1

comb2d

a4 a2demf

p1

[ 9 13] [ l =4e- 6 oz= oz =180

p1 [ 13 14] [ l =5e- 6 oz= oz =180

v1 vol c

*

[ 15 20] 20] [ V=10 sv=5]

v2 vol c

*

[ 9 20] 20] [ V=10 sv=5]

v3 vol c

*

[ 11 20] 20] [ V=0]

v4 vol c

*

[ 13 20] 20] [ V=0]

e1 egr egr oun ound

*

[ 20] [ ]

- 216 -

R= R=100]

N=18 gap=3e- 6] w=10e- 6

R=100]

APPENDIX H Netlist File for Simulation of the 1S-2D type Two-stage Leverage Mechanism in the Macro-model a b b b b b b b a b b b b b a b

2 2 3 4 5 6 7 8 9 8 5 12 13 14 15 14

1 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

8e8e- 3 4e4e- 3 85e85e- 3 20e 20e-- 3 20e20e- 3 22e22e- 3 23e23e- 3 22e 22e-- 3 4e4e- 3 221e221e- 3 20e- 3 22e22e- 3 23e23e- 3 22e22e- 3 4e- 3 221e221e- 3

90 - 90 - 90 - 90 0 - 90 0 90 90 0 180 - 90 180 90 90 180

16e16e- 3 20e20e- 3 3. 6e- 3 20e20e- 3 20e20e- 3 1. 8e- 3 40e40e- 3 1. 8e8e- 3 8e8e- 3 40e40e- 3 20e20e- 3 1. 8e8e- 3 40e40e- 3 1. 8e8e- 3 8e- 3 40e40e- 3

b b b b a

11 25 24 24 26

25 24 23 26 27

85e85e- 3 19e19e- 3 257e257e- 3 136e 136e-- 3 8e8e- 3

- 90 0 180 - 90 - 90

2e2e- 3 20e20e- 3 20e20e- 3 2e2e- 3 16e16e- 3

b b b b a

17 18 19 19 21

18 19 20 21 22

85e85e- 3 19e19e- 3 257e257e- 3 136e 136e-- 3 8e8e- 3

- 90 180 0 - 90 - 90

2e2e- 3 20e20e- 3 20e20e- 3 2e2e- 3 16e16e- 3

f f

20 23

1 1

1. 5e- 3 1. 5e- 3

90 90

0 0

- 217 -

APPENDIX I Netlist File for Simulation of the 1S-2S type Two-stage Leverage Mechanism in the Macro-model a b b b b b b b a b b b b b a b b b b b a b b b b a

2 2 3 4 5 6 7 8 9 8 5 12 13 14 15 14 11 25 24 24 26 17 18 19 19 21

1 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 25 24 23 26 27 18 19 20 21 22

8e8e- 3 4e4e- 3 85e85e- 3 20e 20e-- 3 20e20e- 3 22e22e- 3 23e23e- 3 22e 22e-- 3 4e4e- 3 221e221e- 3 20e- 3 22e22e- 3 23e23e- 3 22e22e- 3 4e- 3 221e221e- 3 85e85e- 3 19e19e- 3 257e257e- 3 136e136e- 3 8e8e- 3 85e85e- 3 19e19e- 3 257e257e- 3 136e136e- 3 8e8e- 3

90 - 90 - 90 - 90 0 - 90 0 90 90 0 180 - 90 180 90 90 180 - 90 0 180 90 - 90 - 90 180 0 90 - 90

f f

20 23

1 1

1. 5e- 3 1. 5e- 3

90 90

16e16e- 3 20e20e- 3 3. 6e- 3 20e20e- 3 20e20e- 3 1. 8e- 3 40e40e- 3 1. 8e8e- 3 8e8e- 3 40e40e- 3 20e20e- 3 1. 8e8e- 3 40e40e- 3 1. 8e8e- 3 8e- 3 40e40e- 3 2e2e- 3 20e20e- 3 20e20e- 3 2e- 3 16e16e- 3 2e2e- 3 20e20e- 3 20e20e- 3 2e- 3 16e16e- 3 0 0

- 218 -

APPENDIX J Netlist File for Simulation of the Two-stage Leverage Mechanism in the Disk-drive suspension a b

2 2

1 3

200e200e- 6 1000 1000ee- 6

0 180 180

1600e1600e- 6 20e20e- 6

b b b b b b b a b

3 4 4 6 7 8 9 10 8

4 5 6 7 8 9 10 11 12

100e100e- 6 400e400e- 6 500e500e- 6 4000e4000e- 6 400e400e- 6 500e500e- 6 1000e 1000e-- 6 200e200e- 6 5500e5500e- 6

90 0 90 0 90 0 - 90 - 90 180

100e100e- 6 20e20e- 6 100e100e- 6 100e100e- 6 20e20e- 6 100e100e- 6 20e20e- 6 1600e1600e- 6 100e100e- 6

b b b b b b b a b

3 20 20 19 18 14 15 16 14

20 21 19 18 14 15 16 17 13

100e100e- 6 400e400e- 6 500e500e- 6 4000e4000e- 6 400e400e- 6 500 500ee- 6 1000e1000e- 6 200e200e- 6 5500e5500e- 6

- 90 0 - 90 0 - 90 0 90 90 180

100e 100e-- 6 20e20e- 6 100e100e- 6 100e100e- 6 20e20e- 6 100e100e- 6 20e20e- 6 1600e1600e- 6 100e100e- 6

b

12

13

2000e2000e- 6

- 90

1000e1000e- 6

1. 5e- 4 1. 5e- 4

180 0

0 0

f f

5 21

1 1

- 219 -