displacement-amplifying compliant mechanisms for sensor

DISPLACEMENT-AMPLIFYING COMPLIANT MECHANISMS FOR SENSOR APPLICATIONS

A Thesis

Submitted for the degree of

Master of Science (Engineering) IN THE FACULTY OF ENGINEERING

By

Girish Krishnan

DEPARTMENT OF MECHANICAL ENGINEERING INDIAN INSTITUTE OF SCIENCE

BANGALORE-560012 INDIA

DECEMBER 2006

Dedicated

To

My Parents

i

TABLE OF CONTENTS

ABSTRACT___________________________________________________________ v

AKNOWLEDGEMENT_________________________________________________ vi

LIST OF FIGURES ___________________________________________________viii

LIST OF TABLES ____________________________________________________ xiv

1. INTRODUCTION___________________________________________________ 1.1

1.1 Displacement amplifying Compliant Mechanisms __________________ 1.1

1.2 Evaluation and selection of DaCM topologies ______________________ 1.2

1.3 Topology Optimization for DaCMs_______________________________ 1.3

1.4 High-Sensitivity Micro-g Resolution Accelerometers ________________ 1.4

1.5 A Minute Force sensor Using a DaCM ____________________________ 1.5

1.6 Organization of the thesis_______________________________________ 1.6

2. LITERATURE REVIEW ____________________________________________ 2.1

2.1 Displacement amplifying Compliant Mechanisms (DaCMs) 2.1

2.1.1 Compliant mechanisms _________________________________________ 2.1

2.1.2 Displacement amplifying compliant mechanisms (DaCMs) _____________ 2.3

2.1.3 Optimal Design of Compliant Mechanisms__________________________ 2.4

2.1.4 Use of DaCMs for sensor applications _____________________________ 2.5

2.2 Micro-g Accelerometers 2.6

2.2.1 Introduction to Accelerometers ___________________________________ 2.6

2.2.2 Micro-g Accelerometers _______________________________________ 2.10

2.2.3 Force feedback in Accelerometers (Bao, 2000)______________________ 2.11

2.2.4 Noise in Accelerometers _______________________________________ 2.14

2.2.5 Evolution of high-resolution, high sensitivity accelerometers __________ 2.15

2.2.6 Electronic Circuitry for capacitance detection_______________________ 2.25

2.2.7 Need for mechanical amplification for accelerometers ________________ 2.27

ii

2.3 Force sensor for micro-manipulation of Cells 2.27 2.3.1 Introduction to force sensors for micro-manipulation _________________ 2.27

2.3.2 Use of DaCMs as force sensors __________________________________ 2.29

2.4 Closure 2.29

3. OBJECTIVE COMPARISON OF VARIOUS DACMS FOR SENSOR APPLICATIONS _____________________________________________________ 3.1

3.1 Introduction ___________________________________________________ 3.1

3.2 DaCMs for Sensor applications ___________________________________ 3.2

3.3 Spring-mass-lever Model of a sensor with a DaCM___________________ 3.3

3.4 Objective comparison of some DaCMs _____________________________ 3.8

3.4.1 Comparison criteria ____________________________________________ 3.8

3.4.2 Specification for Analysis ______________________________________ 3.11

3.4.3 Observations and insights ______________________________________ 3.12

3.4.4 Figure of merit _______________________________________________ 3.18

3.4.5 Selection vs. Optimization ______________________________________ 3.19

3.5 Closure ______________________________________________________ 3.23

4. DESIGN OF A MICRO-G ACCELEROEMTER WITH A DACM __________ 4.1

4.1 Introduction__________________________________________________ 4.1

4.1.1 General layout of an accelerometer with a DaCM _________________ 4.1

4.2 Capacitance detection__________________________________________ 4.3

4.3 Selection of a DaCM for an accelerometer_________________________ 4.7

4.3.1 Selection of a DaCM for Chea et al.’s (2004) micro-g accelerometer __ 4.8

4.3.2 Selection of a DaCM for a bulk-micromachined accelerometer for the

DRIE process ____________________________________________________ 4.12

4.4 Design of an accelerometer with a DaCM for a given chip area ______ 4.15

4.4.1 Suspension stiffness ( sk ) ___________________________________ 4.15

4.4.2 Proof mass ( M ) __________________________________________ 4.15

4.4.3 Size of the mechanism ( mechl )________________________________ 4.16

iii

4.4.4 Optimization of the accelerometer for a given chip-area ___________ 4.17

4.5 Design of sense combs and external suspension____________________ 4.18

4.5.1 Cross-axis Sensitivities _____________________________________ 4.18

4.5.2 Design of the sense combs __________________________________ 4.20

4.5.3 External sense-comb suspension______________________________ 4.24

4.6 Analysis of the accelerometer designs____________________________ 4.27

4.7 Closure _____________________________________________________ 4.28

5. SYSTEM LEVEL SIMULATION OF A MICRO-G BULK-MICROMACHINED ACCELEROMETER __________________________________________________ 5.1

5.1 Introduction__________________________________________________ 5.1

5.2 Mechanical components: Mode-Summation Method ________________ 5.2

5.2.1 Calculating the Damping Coefficients __________________________ 5.6

5.2.2 Mechanical noise in the system _______________________________ 5.7

5.3 Capacitance detection circuit____________________________________ 5.8

5.4 Closed-loop response _________________________________________ 5.10

5.4.1 Feedback combs __________________________________________ 5.10

5.4.2 Design of the PID controller (Kraft, 1997)______________________ 5.13

5.5 Closure _____________________________________________________ 5.22

6. TOPOLOGY OPTIMIZATION OF DACMS FOR SENSORS______________ 6.1

6.1 Introduction__________________________________________________ 6.1

6.2 Topology optimization for sensor applications _____________________ 6.2

6.2.1 Objective functions and constraints used for topology optimization ___ 6.3

6.3 Optimality Criterion with non-linear constraints ___________________ 6.7

6.3.1 Sensitivity analysis for the Objective functions and constraints ______ 6.9

6.4 Numerical Examples__________________________________________ 6.12

6.4.1 Topology optimization of DaCMs with constraints on cross-axis

displacement and natural frequency ___________________________________ 6.14

iv

6.4.2 Topology optimization of DaCMs for accelerometer applications____ 6.17

6.5 Closure _____________________________________________________ 6.23

7. A DISPLACEMENT-AMPLIFYING COMPLIANT MECHANISM AS A MECHANICAL FORCE SENSOR ______________________________________ 7.1

7.1 Introduction__________________________________________________ 7.1

7.2 Use of DaCMs as force sensors __________________________________ 7.2

7.3 Vision based force sensing in micromanipulation of cells_____________ 7.3

7.4 Force sensor for laparoscopic surgery ____________________________ 7.5 7.4.1 Topology optimization of DaCMs for force sensor application _________ 7.6

7.4.2 Fabrication Process of the mechanism ____________________________ 7.8

7.4.3 FEM Analysis of the mechanism using COMSOL__________________ 7.10

7.4.4 Displacement Sensing Technique (Hall-effect Sensor) ______________ 7.11

7.5 Experimental set-up to calibrate the force sensor and the DaCM _____ 7.13 7.5.1 Force required to rupture an inflated balloon ______________________ 7.14

7.6 Closure _____________________________________________________ 7.16

8. CONCLUSIONS AND FUTURE WORK _______________________________ 8.1

8.1 Summary ____________________________________________________ 8.1

8.2 Contributions_________________________________________________ 8.2

8.3 Future Work _________________________________________________ 8.3

A. EFFECT OF FABRICATION LIMITATIONS ON THE RESOLUTION OF AN ACCELEROMETER _________________________________________________ A.1

B DRIE WITH SOI PROCESS FOR FABRICATING THE ACCELEROMETER WITH A DACM ______________________________________________________B.1

R REFERENCES ____________________________________________________ R.1

v

ABSTRACT The thesis deals with Displacement-amplifying Compliant Mechanisms (DaCMs),

which use the input force applied at a point to a give amplified output displacement at

another point with a single elastic continuum. We developed a spring-mass-lever

model to capture the static and dynamic behavior of DaCMs. We used this model for

evaluating the topologies of DaCMs for sensor applications based on several criteria,

and used a combined figure of merit for selection. When none of the DaCM

topologies in the database are able to meet all the requirements of a new sensor, we

synthesize a new DaCM using topology optimization. This involves nonlinear

constraints that were linearized to incorporate them into the optimality criteria

method, which is used to solve the topology optimization problem. Two applications

of DaCMs, namely, a bulk-micromachined high-resolution accelerometer and a

minute mechanical force sensor are pursued in this work.

(i) The addition of a DaCM to a micromachined accelerometer increases the

sensitivity along the intended axis by an order of magnitude or more. But it has the

undesirable side-effect of increasing the cross-axis sensitivity. We overcame this

problem by a structural modification, and topology optimization that includes a

constraint on the cross-axis sensitivity. We optimized a DaCM along with the

accelerometer’s proof-mass and suspension for a silicon-on-insulator wafer-based

process. The system-level simulation, including the electronic circuitry in the force-

balance mode with mechanical and electronic noise, gives the resolution and

percentage cross-axis sensitivity for the designed accelerometers as 20 µg (0.03%), 40

µg (0.01%), and 70 µg (0.005%).

(ii) A novel DaCM-based mechanical force sensor of size 4 cm × 4 cm × 0.04

cm is designed and fabricated with spring steel foils using wire-cut electric-discharge-

machining. A Hall-effect sensor is included to measure the output displacement with a

sensitivity of 324 mV/N up to 1 N. The calibrated sensor is used to measure the force

needed to rupture a balloon. This sensor, when fabricated at the micron scale, has the

potential to measure the force in intra-plasmic live-cell injection.

vi

ACKNOWLEDGMENTS

“Education is what survives when what has been learned has been forgotten”. Echoing

these words of B. F. Skinner, the famous American psychologist, I would like to thank a

few special people who have educated me through two glorious years of my life.

One thing that is going to remain with me throughout is the experience of my

interaction with Prof. G. K. Ananthasuresh. His steadfast belief in my abilities and

constant encouragement has been instrumental in the completion of this work. The

experience of having worked with him has put me on a road in achieving all the

objectives that I had sought for before joining the Institute. He taught me the importance

of scientific reasoning for tackling any problem, technical or otherwise. I particularly

admire his presentation and writing skills and greatly appreciate his efforts to inculcate

the same in us. I consider myself extremely fortunate to go down in the books of being

one of his first students in India.

I am really grateful to all the faculty members at the Department of Mechanical

Engineering, Indian Institute of Science, for their inspiring lectures that grounded in me

the necessary fundamentals to begin research. I thank Prof. J. H. Arakeri, Chairman,

Department of Mechanical Engineering for providing timely facilities. I am also grateful

to the ex-chairman, Prof. S. K. Biswas for having introduced me to Prof. Suresh. I

express my gratitude to Prof. R. Narasimhan, Prof. C. S. Jog, Prof. A. Chatterjee, Dr. V.

R. Sonti, and Prof. B. Gurumoorthy for their guidance. I also thank Prof. Navakantha

Bhat, Department of Electrical Communications and Engineering, for his guidance in my

work.

Significant part of my experience in the past two years has resulted due to fruitful

interactions with all my colleagues in the ‘Compliant, Small and Bio-systems Lab’. I am

greatly indebted to each one of them for helping me in my work and also providing

constant encouragement when I needed it the most. I would like to reserve my special

thanks to Anup, Balaji, Dinesh, Pradeep, Manjunath, and Meena for helping me with

analysis and fabrication during various stages of the project. I have greatly benefited from

vii

the interactions with my lab mates, Narayan Reddy, Sourav, Sivanagendra, Aravind,

Jiten, Kiran, Vikhram, Manish, Rajesh, Siddharth, and others. I am very fortunate to be in

such a strong social circle, making them like a second family to me. I have also greatly

benefited from the interactions with Chaitanya and other students in the National MEMS

Design Center (NMDC). As a part of my work, I was fortunate to have fruitful

interactions with the MEMS group in LEOS (ISRO). I thank J. John, I. Saha and Dr. K.

Kanankaraju for support and valuable suggestions. I also thank all the workshop

technicians, especially Raja and Govindaraju, who have provided timely assistance with

workshop facilities.

Last, but not the least, I thank my parents Smt. Girija Krishnan and Sri K. R.

Krishnan, to whom I dedicate this thesis. Their constant encouragement and blessings

have kept the journey hassle free. I thank all my friends, whose names have been missed

out due to space constraints, for the constant support and timely help extended to me. I

thank God Almighty of making His presence felt through all the people who have guided,

helped and encouraged me all through my life.

viii

LIST OF FIGURES 2.1 Design domain and problem specification for a compliant mechanism

problem ……………………………………………………………………….

2.5

2.2 Principle of operation of an accelerometer…………………………………… 2.8

2.3 (a)The frequency response curve for an accelerometer (b) Resolution vs

Bandwidth required for the accelerometer ……………………………………

2.10

2.4 An open loop accelerometer system………………………………………….. 2.11

2.5 System representation of a closed loop accelerometer……………………….. 2.12

2.6 Lumped characterization of the mechanical noise……………………………. 2.14

2.7 Bulk micromachined piezo-resistive accelerometer by Roylance and

Angell………………………………………………………………………….

2.16

2.8 Capacitive Z-axis accelerometer with quad beam configuration…………....... 2.17

2.9 Capacitive torsional silicon accelerometer……………………………………. 2.18

2.10 Surface Micromachined Z-axis accelerometer by Lu et al. (1995)………........ 2.20

2.11 ADXL 50 surface micromachined lateral accelerometer…………………….. 2.22

2.12 A high sensitivity z -axis accelerometer using combined bulk and surface

micromachining by Chae et al. (2004)…………………………………….......

2.23

2.13 A high sensitivity In-plane accelerometer using combined bulk and surface

micromachining by Chae et al. (2004)………………………………………...

2.24

2.14 A DRIE in-plane accelerometer………………………………………………. 2.25

2.15 Using compliant mechanisms to measure force……………………………… 2.28

3.1 An accelerometer (sensor) with a DaCM and a suspension at the sensing

port…………………………………………………………………………….

3.4

3.2 Spring-mass-lever model of a sensor combined with (a) an inverting DaCM

and (b) a non-inverting DaCM………………………………………………..

3.5

ix

3.3 Proposed method to calculate the lumped stiffness of a DaCM (a) input

stiffness cik (b) output stiffness cok …………………………………………..

3.6

3.4 Nine symmetric DaCMs labeled M1-M9 along with their deformed

configurations of the right half………………………………………………..

3.10

3.5 Performance of DaCMs M1-M9 based on six criteria of comparison. The

units of dU (unloaded output displacement) are m/N, of stress are in MPa,

while others are dimensionless. Cross-axis stiffness shown is the lateral-axis

stiffness normalized divided by the stiffness in the desired direction.

Frequencies are divided by 500 Hz……………………………………….......

3.13

3.6 A plot of the geometric amplification with respect to the characteristic length

of the mechanism…………………………………………………...................

3.13

3.7 A plot of the maximum stress with respect to the characteristic length of M1-

M8……………………………………………………………………………..

3.15

3.8 (a) A plot of the stiffness ( /N m ) with respect to the characteristic length

(i.e., size) of M1-M8. Comparison of the FEM and analytical formula (Eq

3.8) capturing the variation of the stiffness (b) cik and (c) cok with respect to

the length of the mechanism mechl ……………………………………………..

3.16

3.9 A plot of the natural frequency of the mechanisms with respect to the

characteristic length of M1-M8………………………………………………..

3.17

3.10 A plot of the net geometric advantage ( NA ) with respect to the sensor

stiffness ( sk )…………………………………………………………………..

3.17

3.11 (a) Geometric amplification (GA ) (b) Maximum stress (c) Stiffness of all the

mechanisms vs. applied force based on geometrically nonlinear

analysis…………………………………………………………………………

3.22

3.12 A plot of the maximum force that the mechanism can handle before

failure………………………………………………………………………….

3.22

4.1 Layout of an accelerometer with a DaCM……………………………………. 4.3

x

4.2 (a) A single capacitance and (b) a differential capacitance arrangement (c)

Circuit representation of the differential capacitance…………………………

4.4

4.3 Differential Capacitance arrangement a) The positive and the negative static

combs are on the same side b) The positive and the negative static combs are

on either sides of the comb mass………………………………………………

4.6

4.4 Arrangement of both the configurations through out the sense-comb mass….. 4.6

4.5 Optimization of the mechanisms for Chae et al.’s (2004) accelerometer…….. 4.9

4.6 Deformed configurations of (a) M1 and (b) M2 that are combined with the

accelerometer of Chae et al. (2004)…………………………………………...

4.12

4.7 Proof-mass and the suspension of the accelerometer…………………………. 4.16

4.8 The optimized mechanism size and suspension-length of accelerometer with

DaCMs (a) M1 (b) M2 and (c) M3…………………………………................

4.20

4.9 Cross-axis sensitivity (a) The displacement of the output of the mechanism

M1 with the sense-combs in the desired direction and (b) displacement of the

output of the mechanism M1 in the cross-axis

direction……………………………………………………………………….

4.20

4.10 The sense capacitance at the output of the DaCM……………………………. 4.22

4.11 Effect of the sense-comb size and mass on the (a) Cross-axis sensitivity

without external comb suspension (b) Natural frequency (c) cross-axis

sensitivity with external suspension and (d) Sensitivity of the capacitance

detection circuit………………………………………………………………..

4.24

4.12 Sense combs with the external suspension. posl indicates the position of the

suspension along the comb holder.....................................................................

4.25

4.13 Effect of the external suspension length 2suspl on the cross-axis sensitivities

and resolution of accelerometers with DaCMs (a) M1 (b) M2 and (c)

M3……………………………………………………………………………..

4.26

4.14 A plot of cross-axis sensitivity with respect to the position of the external

suspension posl ………………………………………………………………...

4.27

4.15 Accelerometer designs with DaCMs (a) M1 (b) M2 and (c) M8…………….. 4.29

xi

5.1 A conventional accelerometer represented as a spring-mass-dashpot system

and its equivalent in Simulink ………………………………………………..

5.4

5.2 A state space (Simulink) representation of the mechanical structure using

modal-summation method…………………………………………………….

5.6

5.3 Block diagram of the capacitance detection circuit (Boser et al., 1994)…... 5.9

5.4 Simulink representation of the electronic capacitive sensing circuit………. 5.9

5.5 The feedback combs with dc bias…………………………………………….. 5.11

5.6 Simplified system design of the closed loop accelerometer………………….. 5.14

5.7 Mathematical model of the analog closed loop accelerometer……………….. 5.14

5.8 Bode plot of the open loop transfer function for design M1 for increasing

integral gain ik …………………………………………………………………

5.15

5.9 Bode plot of the closed loop transfer function for increasing proportional

gain pk …………………………………………………………………………

5.16

5.10 Complete system representation of the accelerometer and the electronics....... 5.17

5.11 Closed loop response of accelerometer M1 .(a) A step signal of 40 gµ at 0.1

s (b) Output at the PID controller (c) Displacement of the sense-combs……...

5.19

5.12 Closed loop response of accelerometer M2 .(a) A step signal of 20 gµ at 0.1

s (b) Output at the PID controller (c) Displacement of the sense-combs……...

5.21

5.13 Closed loop response of accelerometer M8. (a) A step signal of 60 gµ at

0.1s (b) Output at the PID controller (c) Displacement of the sense-combs…..

5.22

6.1 Ground structure made of frame elements used for topology optimization.

The black rectangular boxes stand for fixed supports while the arrows show

the input and the output points……………………………………….

6.4

6.2 Design domain for generating complaint mechanisms with cross-axis

displacement constraint……………………………………………………..

6.5

6.3 Schematic of the optimality criteria with non-linear constraints…………... 6.13

6.4 (a) Ground structure used for optimization for the formulation given by Eq.

xii

6.25 (b) Symmetric half of the optimized topology (c) Deformed plot of the

mechanism………………………………………………………….................

6.15

6.5 A plot of the (a) objective function and (b) constraint history during

optimization for the Case 1……………………………………………………

6.16

6.6 (a) Ground structure used for optimization for case (2) formulation (b)

Optimized topology (c) Deformed plot of the optimized topology……….......

6.17

6.7 (a) Ground structure used for optimization for case (3b) formulation (b)

Optimized topology (c) Deformed plot of the optimized topology……….......

6.19

6.8 (a) Symmetric half of the DaCM obtained from the topology optimization (b)

Modified topology after optimization of the in-plane width to comply with

the fabrication process (c) Complete topology………………………………..

6.21

6.9 (a) Ground structure used for optimization for formulation given by Eq. 6.29

(b) Optimized topology (c) Deformed plot of the optimized topology………..

6.22

6.10 Optimized mechanism in conjunction with a proof-mass and suspensions....... 6.24

7.1 Size optimization of the mechanisms for the force sensor application……….. 7.4

7.2 Vision based force sensing of cells…………………………………………… 7.5

7.3 Topology of a DaCM. (a) Ground structure made of grillages (b) Optimized

topology……………………………………………………….........................

7.7

7.4 Shape and size optimization of the DaCM. (a) Skeletal topology from

topology optimization (b) Final mechanism from shape and size

optimization……………………………………………………………………

7.8

7.5 SOLID WORKS model of the mechanism showing the dimensions in mm…. 7.9

7.6 The Spring steel sheet shown along with the fabricated mechanism…………. 7.10

7.7 Wire cut EDM machining the mechanism……………………………………. 7.10

7.8 Force vs. Displacement relation for the mechanism by FE analysis using

COMSOL software……………………………………………………………

7.11

7.9 Analysis in COMSOL indicating the maximum stress of 700 Mpa………….. 7.12

7.10 Hall Effect Sensor surface mounted on a PCB……………………………….. 7.13

7.11 Experimental setup for calibrating the sensor with a DaCM…………………. 7.13

7.12 Magnified view of the sensor and the magnet……………………………....... 7.14

xiii

7.13 Experimental Calibration of the Sensor………………………………………. 7.15

7.14 Figure showing the initial and final output reading before and after the

rupture of the balloon………………………………………………………….

7.15

A.1 An accelerometer proof-mass with suspension, DaCM and the comb-

drives…………………………………………………………………………..

A.1

B.1 Etching of proof-mass and suspension on the structural silicon layer of the

SOI wafer using DRIE. (a) SOI Wafer(b) Metallization (c) Etching a trench

on the substrate (d) DRIE etch of the DaCM and the suspension……………..

B.2

B.2 Pyrex glass used as the base for the accelerometer (a) Pyrex Glass (b)

Aluminium Sputtering…………………………………………………………

B.3

B.3 Anodic bonding and defining the proof-mass (a) Bonding of the structural

layer on glass wafer (b) DRIE etch of the base layer to define the proof-mass

(c) Selective etch of the oxide…………………………………………………

B.4

xiv

LIST OF TABLES

2.1 Various types of accelerometers and their sensing techniques ……………….. 2.6

2.2 A comparison of various capacitive accelerometers discussed in the section

above……………………………………………………………………………

2.26

3.1 Summary of the effect of scaling only the length dimension of the

mechanisms on various attributes………………………………………………

3.20

4.1 Net amplifications of the mechanisms optimized for the Chae et al. (2004)

accelerometer…………………………………………………………………..

4.10

4.2 Weights associated with the four criteria to choose a DaCM for Chae et al.

(2004) accelerometer…………………………………………………………...

4.10

4.3 Specifications of a high-resolution accelerometer by Chae et al. (2004)……… 4.11

4.4 Specifications of the M2 and M8 that are added to the accelerometer of Table

4.2………………………………………………………………………………

4.11

4.5 Performance of the accelerometer by Chae et al. (2004) of Table 4.3

integrated with M2 and M8…………………………………………………….

4.12

4.6 Net amplifications of the mechanisms optimized for the DRIE with SOI

accelerometer…………………………………………………………………..

4.14

4.7 Weights associated with the four criteria to choose a DaCM for the DRIE

with SOI accelerometer………………………………………………………...

4.14

4.8 Analysis of accelerometers with DaCMs M1, M2 and M8……………………. 4.28

5.1 Details of the feedback combs for the three designs discussed in Chapter 4…. 5.12

5.2 Specifications of the three designs of the accelerometer……………………… 5.16

2.14 Weights associated with a quantity for the computation of the overall figure of

merit for a forced sensor application…………………………………………..

7.5

A.1 A comparison of the sensitivity and resolution of various accelerometers with

different fabrication limitations………………………………………………..

A.2

1

Chapter 1

1INTRODUCTION

Summary This thesis is concerned with compliant mechanisms that use the force applied at one

point to give amplified displacement at another point of an elastic continuum. The single-

piece elastic body that constitutes a compliant mechanism thus acts like a lever but with

some stiffness. Two applications of such a Displacement-amplifying Compliant

Mechanism (DaCM), namely, a high-resolution micromachined accelerometer and a

minute mechanical force sensor are considered in this work. Comprehensive analytical

and computational modeling,, systematic design including topology optimization, and

finally testing of two devices comprise the thesis. The necessary background to all these

aspects is provided in this chapter.

1.1 Displacement amplifying Compliant Mechanisms

Compliant mechanisms (Howell, 2001) are becoming increasingly popular due to their

inherent advantages over traditional mechanisms and their amenability to small sized

systems in the rapidly growing areas in micro- and nano-systems. Some of the prominent

compliant mechanisms in use today are the Displacement-Amplifying Complaint

Mechanisms (DaCM) for piezo-actuators, gripper mechanisms for micromanipulation

and some other items of daily use such as compliant clips, clamps, pliers, staplers, etc,

which are further discussed in Chapter 2. The DaCMs constitute a class of compliant

mechanisms that, as suggested by their name, amplify the applied displacement due to

their inherent geometry. These mechanisms are equivalent to mechanical levers but they

do not have joints and thus are the natural choice for amplifying displacement in smart

systems (Robbins et al., 1991) and micromachined devices (Kota et al., 2000).

Chapter 1: Introduction 2

Many DaCMs have been reported in the literature for smart and micro actuator

applications. This effort considers DaCMs for sensor applications where the mechanical

sensitivity of any displacement-transduced sensor can be increased. But unlike in a

mechanical lever with joints, there is stiffness associated with DaCMs. This adds to the

sensor’s stiffness and alters its behavior. This is studied in detail in Chapter 3 where a

reduced spring-mass-lever model is proposed to understand the effect of coupling a

DaCM with a sensor. This model helps in objective evaluation and selection of DaCM

topologies for a given application.

1.2 Evaluation and selection of DaCM topologies

The literature shows that DaCMs are usually used with piezoelectric and electrostatic

actuators. As discussed in Chapter 2, while there are some differences in the

characteristics of a DaCM for actuator and sensor applications there are also some

similarities. Therefore, it is sensible to consider existing DaCMs that are used in actuators

and see if one or more of them meet the requirements of a sensor application. It is indeed

a common practice in the design of rigid-body mechanisms to choose from the existing

configurations from a catalog (Sclater and Chirnois, 1991; Artobolevski, 1939).

Therefore, towards the goal of creating a catalog of DaCMs, we identify eight topologies

for which the necessary details were reported in the literature.

The topology of a DaCM is the most important factor that decides its

performance. In order to find the most suitable topology for a given sensor application,

we modify the shape and size of all DaCM topologies with or without using optimization.

This is followed by a comparison of all the entries in the catalog using several criteria

that are relevant for the sensor. We present a figure of merit using which the best DaCM

topology can be selected. This figure of merit is a weighted average of the normalized

values of the criteria considered. The weights are decided by the relative importance of

different criteria for a given sensor. It is possible that none of the entries in the catalog

satisfy, as can be seen in Chapters 3 and 7, all the requirements of a sensor. This is not

unexpected because the DaCMs in the catalog would have accounted for only a few

criteria. Therefore, in addition to developing a systematic procedure to select from among


existing DaCMs, we propose a topology optimization method to synthesize new DaCMs.

We are thus able to add to the catalog, which brings the number of DaCMs in it to 12.

1.3 Topology Optimization for DaCMs

Topology optimization is a method that operates on a fixed mesh of finite elements and

defines a design variable, which is associated with each element in the mesh. The

optimization algorithm determines the value of the design variables, which defines the

optimal topology for a given set of loading conditions, an objective function and a few

constraints. In the two applications considered in this work, we show that existing

DaCMs do not meet the requirements imposed by all the constraints. In particular, the

cross-axis sensitivity and the natural frequency of the DaCMs in the catalog will be

shown to be far from the desired values. Therefore, in this effort we incorporate

constraints on the frequency and the cross-axis sensitivity in topology optimization.

These constraints, unlike volume constraint, are nonlinear in terms of the design

variables. We address this issue by linearizing the constraints in each iterative step of the

optimality criteria method. When it is difficult to include all the constraints in topology

optimization, we do shape and size optimization to account of those constraints that are

not included in the topology optimization step. The next two subsections give an

overview of the work done on the two applications considered in this work.

1.4 High-Sensitivity Micro-g Resolution Accelerometers

Micromachined accelerometers have recently gained a lot of importance in inertial

sensing, air bag deployment in automobiles, vibration sensing in machine tools etc.

(Yazdi et al., 1998). The advent of micro-fabrication has paved the way for

miniaturization of these sensors, thus enabling batch fabrication. Developing high

sensitivity micro-g resolution capacitive accelerometers within the limits of

microfabrication is the main challenge faced by researchers in the field. We provide an

overview of high-resolution accelerometers in Chapter 2. Most of them require

sophisticated microfabrication. For example, Chae et al. (2004) use a combination of

surface and bulk micromachining to achieve 10 gµ resolution. We demonstrate in this

work that similar resolution is possible with a simpler process by using a DaCM in the


force re-balance mode. We also demonstrate (as discussed in Chapter 4) that the

sensitivity of Chae et al.’s (2004) accelerometer can be increased by at least three times

by adding a DaCM whose topology is selected from the catalog of nine DaCMs. The

addition of DaCM necessitated a structural modification to retain the cross-axis

sensitivity at the same level as that of Chae et al. (2004). For the simple silicon-on-

insulator process developed in this work, however, the DaCMs in the catalog are found to

lack the required cross-axis sensitivity. Therefore, we use topology optimization to obtain

the design that best meets the cross-axis sensitivity and resolution requirements. Using

system-level simulations of three new accelerometer designs, in conjunction with the

electronic circuitry in the force re-balance mode including mechanical and electronic

noise, give the resolution and percentage cross-axis sensitivity as 20 gµ (0.03%), 40 gµ

(0.01%), and 70 gµ (0.005%).

1.5 A Minute Force sensor using a DaCM

Force sensors are integral parts of many integrated systems and testing and

characterization setups. A DaCM with its amplified output displacement as a measure of

force could serve as a sensitive mechanical force sensor. We use a DaCM to develop a

force sensor of mm–cm size for a resolution of the order of 30 mN . We find that none of

the DaCMs in the catalog have sufficient flexibility for this application for the chosen

process of wire-cut electro-discharge machining (EDM) of foils of spring steel. We

obtain a new DaCM using topology optimization and subsequently modify it using shape

and size optimization. This design in conjunction with Hall-effect sensor, as shown in

Chapter 7, gives a resolution of 30 mN .

The rest of the thesis is organized as follows.

1.6 Organization of the thesis

• Chapter 2: Literature review – This chapter gives a brief review of displacement-

amplifying compliant mechanisms (DaCMs), accelerometers, and force sensors to

identify the need for displacement amplification in sensors.

• Chapter 3: Objective comparison of various DaCMs for sensor applications −

This chapter gives a basic understanding of a DaCM using simplified models


along with identifying the important criteria for sensors and proposing a method

based on all these criteria for choosing a DaCM for specific applications.

• Chapter 4: Layout and design of an accelerometer with a DaCM − This chapter,

with various examples, demonstrates that the sensitivity of an in-plane

accelerometer can be increased by adding a DaCM. A micro-g resolution

accelerometer is then designed and its size and suspension are optimized to meet

the specifications.

• Chapter 5: System-level simulation of the accelerometer − In this chapter the

system-level modeling of the accelerometer that combines the mechanical

structure and the electronic circuits is presented.

• Chapter 6: Optimization of DaCMs for sensor applications − In this chapter,

topology optimization is used as a tool to synthesize DaCMs specific to sensor

applications.

• Chapter 7: Force sensor with a DaCM − This chapter introduces DaCMs for

miniature force sensor applications. A fabricated cm-scale mechanism is also

presented and characterized for use as a mechanical force sensor.

• Chapter 8: Conclusions and future work − This chapter summarizes the work of

the thesis and identifies the scope for future work in terms of techniques and

applications.

• Appendix A: Effect of fabrication limitations on the resolution of accelerometer –

This chapter demonstrates the dependence of the accelerometer’s resolution on

the limitations of the process used for fabrication.

• Appendix B: A DRIE on SoI process to fabricate an accelerometer with a DaCM

– This describes the process proposed to fabricate the accelerometer with a

DaCM. Various steps involved in fabrication as well as the mask layouts are

presented here.

2.1

Chapter 2

2Literature review

Summary In this chapter we provide an overview of some of the important contributions to the field

of complaint mechanisms and sensors, accelerometers in particular, that are relevant to

this thesis. The chapter begins with a review of displacement-amplification compliant

mechanisms (DaCMs) and their applications. Since our aim is to improve the sensitivity

of sensors with DaCMs, we review accelerometers and probe the benefit of using DaCMs

to enhance their sensitivity. Finally, we discuss another interdisciplinary research area,

micromanipulation of biological cells, wherein DaCMs could act as sensors. The aim of

this chapter is to prepare the groundwork for improvements and innovation presented in

the later chapters.

2.1 Displacement amplifying Compliant Mechanisms (DaCMs)

2.1.1 Compliant mechanisms

Compliant mechanisms are mechanisms that transmit motion, force or energy by elastic

deflection of flexural members instead of movable joints (Howell, 2001). Most of the

conventional mechanisms have rigid links and joints to transmit motion and force, and a

spring to store and release elastic energy as needed. In other words, movable parts and

parts storing elastic energy are separated in conventional mechanisms. In compliant

mechanisms, on the other hand, the parts that deform also store elastic energy thus

eliminating the need for a separate spring. They have some obvious advantages over

conventional mechanisms (Howell, 2001). They do not have joints and thus most of them

are available in one piece, which saves on assembly costs. Furthermore, the absence of

Chapter 2: Literature Review 2.2

backlash, friction and wear associated with joints in conventional mechanisms are almost

negligible in compliant designs. Compliant mechanisms are useful in micromachined

applications where fabricating the joints is difficult and failure is often associated with

friction and wear in the joints.

The many advantages associated with compliant mechanisms notwithstanding,

there are a number of difficulties associated with their design. Traditional kinematics

itself is quite insufficient and it usually has to be combined with elastic deformation

theory. As compliant mechanisms undergo large displacements, geometric nonlinear

effects are to be included in the elastic analysis. Stress concentration effects have to be

considered in thin and narrow regions. Howell and Midha (1994) have presented a

pseudo-rigid body model for designing compliant mechanisms with small-length flexural

pivots. Ananthasuresh (1994) and others have synthesized compliant mechanisms via

topology optimization. Subsequent efforts have made use of geometric nonlinearity in

finite elements for topology optimization to synthesize large-deflection compliant

mechanisms (Saxena and Ananthasuresh, 2001; Pedersen and Sigmund, 2001). In this

thesis, topology optimization is used for generating compliant displacement-amplification

mechanisms (DaCMs) for sensor applications.

A number of working prototypes of compliant mechanisms have been

systematically synthesized and demonstrated in the past decade. A compliant one-piece

stapler demonstrated by Ananthasuresh and Saggere (1995) eliminated the hinge and a

spring of a conventional stapler. Several compliant grippers have been designed by using

topology optimization (Frecker et al., 1997; Pedersen et al., 2001). Displacement-

amplifying compliant mechanisms have been designed by topology optimization as well

as using conventional linkage designs (Saxena and Ananthasuresh, 2000; Canfield and

Frecker, 2000; Maddisetty and Frecker, 2002; Lobontiu and Garcia, 2003) for piezo-

electric and electrostatic actuation. Various other mechanisms such as bistable

mechanisms, a micromachined AND gate (Saxena and Ananthasuresh, 2000), bicycle

clutches (Howell, 2001), frequency multiplier, and path generating devices (Mankame,

2004) have been designed and demonstrated by various research groups. Overall,

compliant mechanisms have aroused a lot of interest in the past decade and there is a


constant effort to make most mechanisms compliant when it is beneficial. A notable

recent real-life application is a compliant wheel (Michelin, 2006).

2.1.2 Displacement amplifying compliant mechanisms (DaCMs)

DaCMs are compliant equivalent of displacement-amplifying levers. But they do not

transfer the entire energy available to them at the input to the output because some of it is

stored as the elastic strain energy within the mechanism. They have a unique advantage

of having higher amplification factors for a given area of a fixed aspect ratio than

traditional levers since they make use of a 2-D topology to achieve the amplification.

DaCMs were first in use to amplify the output of piezo-electric stacks. The piezo-

electric material has become increasingly popular in positioning devices due to its

accuracy and ease of use (Robbins et al., 1991). These stacks generate high forces but

small displacements of around 10 mµ for a 1 cm stack. Piling on more stacks gives more

deflection, but affects the mechanical robustness of the stack at the same time. So,

displacement amplification is the only viable solution to get high output displacement.

A lot of work has gone into generating DaCMs for piezo-electric amplification.

The earliest mechanisms used for this purpose were elliptical in shape (Du et al., 2000).

Most others were inspired by conventional mechanisms such as the four-bar mechanism

wherein the joints were replaced by flexural hinges (Lobontiu and Garcia, 2003). High

stresses associated with these flexural hinges gradually led to the paradigm of distribution

of compliance (Ananthasuresh, 1994). Hetrick and Kota (1999) and others used straight

beams arranged in the form of a triangle as building blocks and combined two or three

such building blocks to develop a basic topology which was then optimized for the size

and shape of the final mechanism. Saxena and Ananthasuresh (2000) used topology

optimization to develop high-amplification mechanisms. Canfield and Frecker (2000) and

Maddisetti and Frecker (2002) have also used similar methods to synthesize amplification

mechanisms which were optimized both for static and dynamic loads with piezoelectric

actuation. Du and Lau (2000) synthesized elliptic amplifiers for ink-jet print head

actuators using continuum element optimization under dynamic conditions. Silva et al. ()

have used topology optimization for designing composite materials with prescribed

piezo-electric and mechanical properties.


All the aforementioned mechanisms claim to be optimum for applications for

which they have been optimized for in terms of the area occupied and the conditions of

loading. As a number of such mechanisms have been synthesized, a catalog of these

mechanisms can be created so that one can chose a DaCM that best meets the

specification of a new application. To this end, it is required to evaluate the performance

of these mechanisms under given geometric and manufacturing constraints for various

quantities suitable for a required application. Such an effort is presented in Chapter 3.

2.1.3 Optimal Design of Compliant Mechanisms

Topology optimization is one of the systematic methods for synthesizing compliant

mechanisms with distributed compliance (Anantahsuresh, 1994; Yin and Ananthasuresh,

2003). This method operates on a fixed finite element mesh of either continuum or

discrete elements to optimally distribute material in the designable region and thus

defining a topology. In other words, each element is associated with a design variable that

defines the element size or its contribution to the entire topology. The converged

optimization result is supposed to drive the value of all the design variables either close

to the lower and upper limits thus defining a definite topology. This method of topology

optimization, also known as the homogenization-based methods was introduced by

Bendsoe and Kikuchi (1984) for designing topologies with maximum stiffness having a

finite volume of material. This method was adapted by Ananthasuresh (1994), Frecker et

al. (1997), Sigmund (1997), Saxena (2000) and others for generating topologies which

have maximum displacement at a desired point. DaCMs and compliant grippers have

been designed using this approach.

The objective function that needs to be minimized for designing compliant

mechanisms from topology optimization is usually based on a trade off between

flexibility at a particular point to achieve deformation and stiffness to support the external

load. This can be effectively captured by the following formulation.

Minimize MSE/SE

where TdMSE dσ ε= Ω∫ and 1

2TSE dσ ε= Ω∫ (2.1)


The symbols used above are explained next. Referring to Fig. 2.1, SE is the strain energy

f the elastic continuum under the applied load inF ; MSE is the mutual strain energy due

to applied and unit dummy loads, inF and dF ; ε and σ are the strain and the stress fields

due to the load inF applied at point 1P ; and dε is the strain field due to the unit dummy

load dF applied at point 2P . It should be noted that MSE is numerically equal to the

displacement of point 2P in the direction of dF due to the applied load, inF . Saxena and

Ananthasuresh (2000) proposed an optimality property that emerges from this objective

function and any general objective function of the type ( ) ( )f MSE g SE+ and

( ) / ( )f MSE g SE . The ground structure on which the optimization is performed can be

made of frame elements or continuum finite elements. The continuum finite elements in

2D usually yield undesirable checker board patterns (Bendsoe and Sigmund, 2003) and

hinged regions unless special measures are taken. Furthermore, image processing is

necessary to get the final topology that can be fabricated. Beam structures yield

topologies with distributed compliance but have limited design space because of the fixed

orientation of the beams in the ground structure.

Figure 2.1 Design domain and problem specification for a compliant mechanism problem. A large deflection at 2P is desired due to the applied load at 1P .

2.1.4 Use of DaCMs for sensor applications

Just as the limited stroke of piezo-actuators can be amplified by DaCMs, the sensitivities

of sensors which rely on displacement-transduction can also be increased by adding a

in inF and u

d outF and u

P1

P2

Ω


DaCM to it. Because amplification of displacement in sensors by adding a DaCM has not

been attempted before, a review of displacement-transduced sensors is called for. Thus

we investigate the literature of high-resolution micro-g accelerometers and mechanical

force sensors and probe the benefits of adding DaCMs to enhance their sensitivity.

2.2 Micro-g Accelerometers

2.2.1 Introduction to Accelerometers

An accelerometer senses the acceleration of a moving body on which it is mounted.

Micromachined accelerometers are widely employed in automobiles for air-bag

deployment, biomedical applications for activity monitoring, vibration sensing in

machine tools, micro-gravity measurement in space, inertial navigation systems and other

consumer applications (Yazdi et al., 1998). Various types of accelerometers are listed in

Table 1.1.

Table 1.1 Various types of accelerometers and their sensing techniques Type of sensing, Description Range of sensing

Applications Advantages Disadvantages

1) Piezo-resistive (0.001-50 g) Bending of suspension beams due to proof- mass displacement produces strain in them, which is measured as a change in the resistance of a piezoresistive device placed at the support end of the beam. Change in resistance can be measured by using a wheatstone bridge configuration.

Air bag deployment in automobiles; High g accelerometers are used for impact testing etc.

a) Simplicity in structure and fabrication. b) Simple readout circuitry generating a low impedance voltage.

a) Large temperature sensitivity. b) Small overall sensitivity of 1-2 mV/g and thus requires a huge proof mass.

2) Capacitive (2µg- several g’s) Displacement of a mass due to applied acceleration leads to a change in capacitance between two plates, one fixed and the other one attached to the moving mass. The capacitance change can be due to a change in the overlapping area or due to a change in the gap.

Air bag deployment in automobiles, Inertial navigation, micro gravity detection,

High sensitivity, good dc response, good noise performance, low drift, low power dissipation, low temperature sensitivity, amenability for feedback.

Parasitic capacitance, electromagnetic interference.

3) Tunneling (10 ng- 5 g) This accelerometer consists of a proof-mass with a sharp tip separated from a bottom electrode. As the tip is brought sufficiently close to its counter-electrode (within a few Å) a tunneling current is established. This tunneling current can be a measure for acceleration. This is also operated in the closed loop mode.

Inertial navigation, zero-gravity measurement in space.

High sensitivity, excellent resolution, linearity can be maintained by operating in the closed loop mode.

Difficult to fabricate sharp tips; nonlinear variation of tunneling current with distance; high noise levels


4) Piezoelectric (High g’s) In this type, the sensing element is a crystal which has the property of emitting a charge when subjected to a compressive force. In the accelerometer, this crystal is bonded to a mass such that when the accelerometer is subjected to a gravity force, the mass compresses the crystal which emits a signal. This signal can be related to the imposed 'g' force.

Vibration sensors; active vibration control.

High bandwidth; simple in operation and can be incorporated in any application; amenable for actuation and thus vibration control.

Low resolution; low sensitivity.

5) Optical (Nano-g to several g’s) In this type of an accelerometer, the light from an Light emitting Diode (LED) source is projected onto a movable membrane and the reflected light from it has a lower intensity. The intensity loss is a measure of the acceleration. Similarly, a change in wavelength, polarization as well as diffraction could be used.

Inertial navigation, detecting small vibrations in machines.

High resolution, low noise, can operate in places where other principles fail (e.g., when electromagnetic fields are present, we cannot use capacitance sensors.)

Large weight, complex fabrication process and detection circuitry.

6) Resonant (5 – 5000 g’s) This type of acceleration measurement makes use of the shift in the natural frequency of the structure with applied acceleration.

Vibration sensors in machine tools.

High dynamic range, sensitivity, bandwidth, adaptable to digital circuits.

Leakage of signal, high noise levels.

7) Thermal (0.5 mg to several g’s) It consists of a substrate which is sealed with air or any other gas with a heater exactly at the middle. Two thermocouples are present at either ends of the substrate. When the acceleration is applied, the hot air around the heater is pushed to one of the ends by denser cold air thus giving a temperature difference in the thermocouples which is proportional to the acceleration.

Inclination sensing (Dual axis), automotive, electronic and gaming applications.

Low cost, low noise, less drift, simple signal conditioning circuitry.

Low resolution, batch fabrication can be difficult, temperature dependency.

8) Electromagnetic (0-50g) It consists of two coils, one on the proof-mass and the other on the substrate. A square pulse is given to the primary and a voltage is induced in the secondary which is proportional to the distance between the proof-mass and the substrate

Air bag deployment.

Good linearity, simple signal conditioning.

High power, low resolution, low sensitivity.

In most applications, the measured acceleration is used to determine displacement

by integrating the acceleration twice with respect to time and using a frame of reference.

Integration, being a summing operation, reduces the effect of noise and thus smoothens

the signal. Furthermore, acceleration can be measured without an external reference

system, while velocity and displacement cannot. For acceleration to be sensed, most


accelerometers convert it to displacement and then transduce it to a voltage so that it can

be signal-conditioned and worked upon further. Most accelerometers have a proof mass

which experiences an inertial force in the opposite direction of the acceleration that is to

be measured. The deflection of the proof mass caused by this force is determined by the

suspension stiffness and is converted to a voltage using some principle of transduction

(see Table 1.1). For frequencies of the input acceleration sufficiently smaller than the

natural frequency of the device, the deflection is linear with acceleration.

Figure 2.2 Principle of operation of an accelerometer

A lumped spring mass damper model schematically represents an accelerometer

as shown in Fig. 2.2. The acceleration to be sensed manifests as a force on the mass (due

to D’ Alembert’s force). The input acceleration is conveyed to the mass by base

excitation. The equations below show the derivation of the sensitivity of the accelerometer

using the fundamentals of mechanics and vibration theory (Thomson and Dahleh, 1997).

The movement of the base is denoted by x and that of the mass is denoted by y . The net

extension of the spring and the damper will thus be z y x= − . The equation of motion is

then given by

( - ) ( - ) 0my k y x d y x+ + =&& & & (2.2)

For base excitations of the form sin( )x X tω ψ= + where ψ is the phase difference

between the input and output signals. By substituting this in the above equation, we get 2 2[(- ) sin( ) cos( )] sin( )m k t d t Z mX tω ω ω ω ω ω ψ+ + = + (2.3)

By considering only the amplitudes in the solution of the above equations, we get

x

ma

yz y x= −

m m

k ka


2

2 2 2

( ) ( )Z mX k m d

ωω ω

=− +

(2.4)

Now, by defining terms of the form 2 and where 4n cc

k d d kmm d

ω ξ= = = , we

recast Eq. 2.4 as 2

2

22

2

(1 ) (2 )

n

n n

ZX

ωω

ω ωξω ω

=

− +

(2.5)

When operated at frequencies much less than the resonance frequency, i.e., when

nω ω<< , the denominator approaches unity. We then get 2 2 / nZ Xω ω= . But

2 X Aω = (equal to acceleration) gives

2 n

AZω

= (2.6)

By observing the above formulae, it is evident that the natural frequency has to be low for

high sensitivity. But this effectively reduces the bandwidth, which is dependent on both

the fundamental frequency and the damping factor. This is shown in Fig. 2.3a where the

frequency response is shown. For larger damping factors, the straight line representing

the operating range starts rising well before the resonance condition is encountered. Thus

greater the sharpness at resonance, the greater the available operating range for the

frequency of the accelerometer (see Fig. 2.3). The sharpness at resonance is denoted by

the quality factor Q that is given by 1/ 2Q ξ= .

Figure 2.3b shows the various resolutions and the corresponding bandwidth

requirements for various applications. It can be seen that space and inertial navigation

applications


(a)

(b) Figure 2.3 a) The frequency response curve for an accelerometer. b) Resolution vs

bandwidths required for various applications.

2.2.2 Micro-g Accelerometers

Micro-g accelerometers are high-resolution accelerometers which can detect minute

changes in acceleration which is of the order of micro-g. To obtain such a fine resolution,

the mechanical and electronic components of the accelerometer should be highly

/X A Amplitude Ratio

Working range

Frequency ratio / nω ω 1

Bandwidth in Hz

Ran

ge in

g


sensitive. From the previous section, it can be seen that sensitivity of the mechanical

components can be increased by increasing the proof-mass dimensions and reducing the

suspension’s stiffness, thus decreasing the bandwidth of the system. Obtaining a thick

proof-mass and a compliant suspension requires involved fabrication which is discussed

in section 2.3.5.

To increase the dynamic range, the accelerometer is operated in the force-feedback

mode wherein the measured signal is fed back to the proof-mass to bring it back to its

mean-position. The resolution of the system is then dependent upon the noise in the

system which occurs both in the mechanical components in terms of the Brownian noise

and in the electronic circuit in terms of dc offsets, 1/f noise and Johnson’s noise. The

mechanical noise is further elaborated in section 2.3.4

2.2.3 Force feedback in Accelerometers (Bao, 2000)

To increase the sensitivity of capacitive accelerometers, we need to have small gaps

because capacitance increases nonlinearly with change in small gaps. But this

nonlinearity also limits the useful dynamic range of the accelerometer. So, inertial grade

accelerometers need to be operated in the closed-loop form to make the response linear,

and increase the dynamic range and the bandwidth.

Figure 2.4 An open loop accelerometer system From the model shown in Fig. 2.4, we get

/outV Apma k= (2.7)

where A is the Op-Amp amplification, p is the gain of the capacitance measurement

circuit, M is the inertial mass and k is the spring constant of the suspension and a is the

constant acceleration applied to the mass. For open-loop accelerometers, we see that p

F Ma= Spring, Mass and Damper. At steady state, 1/ k

Change in capacitance proportional to displacement given by a gain factor p

Amplification A

Output

outApMaV

k=


need not be linear since the capacitance for large enough displacements is nonlinear with

respect to displacement. Furthermore, we have seen earlier that there is a trade-off

between the bandwidth and the sensitivity.

If the same system is made to operate in the closed-loop mode, we get the model

shown in Fig. 2.5.

Figure 2.5 System representation of a closed loop accelerometer In the above model the displacement of the proof-mass is given by

( - ) /x Ma Apqx k= (2.8)

where M is the proof-mass, a the applied acceleration to be detected, p the gain

obtained by conversion of displacement to capacitance change , A the amplification of

the Op-Amp, q the gain of the electrostatic feed-back force, and k the stiffness of the

accelerometer suspension. The displacement of the proof mass is converted to an

electronic signal whose value is proportional to the rate of change of capacitance and is

amplified to obtain the read-out voltage given by

( - ) /outV Ap Ma Apqx k= (2.9)

and the feed back force for small displacements is proportional to the output voltage and

can be given by

( - ) /fF Apq Ma Apqx k= (2.10)

But from the Fig. 2.5, fF Apqx= . By equating the two expressions we get

/( )x Ma Apq k= + (2.11)

Spring-mass-damper system with a steady state gain of 1/ k

Change of capacitance. Gain p

A

Electrostatic feedback q

+ –

outV Apqx=

/x F k=m fF F F= −

fF Apqx=

mF Ma=


We note that the effective stiffness term has increased. Here, A , p and q are the

electronic stiffness parameters. By defining a quantity /Apq kβ = as the ratio between

the electronic and the mechanical stiffness parameters. The displacement x is now given

by

/ (1 )x Ma k β= + (2.12)

The value of β could be made much greater than unity, i.e. 1β >> , by increasing values

of A and p , which are electronic parameters. The displacement of the mass is decreased

thus leading to linear change in capacitance, which also makes the feedback voltage

linear.

The readout from the capacitance measurement circuit then becomes

(1 ) (1 )outpMa MaV

k qβ

β β= =

+ + (2.13)

Using the approximation of 1β >> , the output voltage becomes

outMaVq

= (2.14)

It is thus shown that for large values of β , the read-out voltage is independent of

stiffness of the accelerometer suspension. This is advantageous since the readout will

now no longer be sensitive to small changes in stiffness that occurs due to the

manufacturing process.

In the closed-loop mode the equation of motion for the spring-mass-damper

system representing the accelerometer is given by

(1 ) 0mx cx k xβ+ + + =&& & (2.15)

The natural frequency of such a system is now given by

n = (1 ) / = (1+ )ω β ω β+nff k m . (2.16)

So, the natural frequency has increased by a factor (1+ )β . The damping ratio now

becomes

0 / / 2 (1 ) / (1 ) cC C C k mξ β ξ β= = + = + (2.17)


Thus, it has decreased by a factor 1/ (1 )β+ . The bandwidth is proportional to the

natural frequency and is inversely related to the damping. Thus, force feedback increases

the bandwidth.

2.2.4 Noise in Accelerometers

To probe the possible noise sources in a mechanical system, we make use of the

Fluctuation-Dissipation theorem (Gabrielson, 1993), which states that if there is a

mechanism for dissipation in a system, then there will also be a component of fluctuation

directly related to the dissipation. This is because any random motion generated within

the system decays if there is an energy dissipating mechanism. This might lead to the

temperature of the system becoming less than that of the surroundings. To account for

this there is an associated fluctuating force, which acts as noise. In a spring-mass-damper

model (see Fig. 2.6) the energy is dissipated through the damper and hence there should

be a component of the force at the input due to fluctuations.

By using the Nyquist criterion, we can determine the spectral densities of the

fluctuating force as shown below.

4 f BF K DT= (2.18)

f = Fluctuating force in / = Boltzmann's constant in - /

= Damping coefficient in - / = absolute temperature in

B

F N HzK N m KD N s mT K

Figure 2.6 Lumped characterization of the mechanical noise

Thus, any complex mechanical system can be analyzed for thermo-mechanical noise by

adding a force-generator along with a damper. If the frequency components in the noise

4f BF K DT=

M

D

K


signal are within the operating range, then we can express the displacement of the mass to

be

Bn

4K DTX = in m/ Hz

K (2.19)

The detected displacement of the proof mass due to a signal expressed in the spectral

density form, can be given by

ss 2

0

AX in m/ Hzω

= (2.20)

Now, the signal to noise ratio ( SNR ) is as shown below. 2 2 2 2

20 0

4 4 4

s s s s

n B B B

X KA MA MA QSNRX K DT K DT K Tω ω

⎛ ⎞= = = =⎜ ⎟

⎝ ⎠ (2.21)

From the above equations, it can be inferred that increasing the mass and the quality

factor along with decreasing natural frequency can reduce the noise considerably. But

this limits the bandwidth of the system. In general we can conclude that increasing the

sensitivity as well as decreasing the noise in an accelerometer involves decreasing the

natural frequency, thus limiting the range of operating frequencies. Next, we discuss how

noise is reduced and the sensitivity is increased in practical realization of high-resolution

accelerometers.

2.2.5 Evolution of high-resolution, high sensitivity accelerometers

It was mentioned in the previous section that high-resolution accelerometers require a

large proof mass and flexible suspensions which in turn depended on the effectiveness of

the fabrication processes. We shall chronologically illustrate the developments in the

design and fabrication innovations that lead to the high-sensitivity accelerometers and

make a comparison of these in terms of their sensitivities and resolution.

2.2.5.1 Bulk micromachined piezo-resistive accelerometer ( Roylance and Angell, 1979)

It appears that the first bulk-micromachined accelerometer was made by Roylance and

Angell (1979). It was a z -axis accelerometer consisting of a cantilever support holding a


huge proof- mass. Piezo-resistors are embedded at the support end of the cantilever beam

where the maximum strain is experienced. Figure 2.7 shows a schematic of this

accelerometer.

Figure 2.7 Bulk micromachined piezo-resistive accelerometer by Roylance and Angell

By using the linear beam theory for the suspension, the stiffness can be obtained as

2 2

12(4 6 3 )

bst

b b m b m

EIKl l l l l

=+ +

(2.22)

where bI is the area moment of inertia of the beam, E is the elastic modulus and ml and

bl are as shown in Fig. 2.7.

Further improvement on the structure could have two or more beams on either

side of the mass to reduce the cross-axis sensitivity. A similar structure could be used for

capacitive sensing (Kuehnel and Sherman, 1994). Two electrodes could be patterned on

glass slabs and wafer-bonded on the top and the bottom of the mass. Since small gaps are

required for larger sensitivities, the distance between the electrode and the mass could be

small. Through slits on the wafer reduce damping that arise due to small gaps below the

proof-mass. Wet etching in silicon <110> wafers is employed to obtain straight vertical

side-walls on the wafer.

Piezo resistors

Mass

Cantilever suspension

ml

bl


Figure 2.8 Capacitive Z-axis accelerometer with quad beam configuration

Fabrication: The silicon wafers of (110) orientation are taken and wet-etched to

determine the suspensions as well as the slits. The entire substrate thickness is used for

the mass to increase its value. Finally, electrodes are deposited on the glass plates and

they are wafer-bonded to the wet-etched silicon wafer so that differential capacitance can

be sensed. The electrodes can be used for both sensing as well as actuation in a closed

loop mode. The major processes required for this fabrication is wet etching as well as

wafer bonding. Since these are simple processes these types of accelerometers are

popular and are batch fabricated.

Advantages:

a) Simple design and fabrication process.

b) High base-capacitance of close to 20 pF almost eliminating the parasitic

capacitance.

c) Low cross-axis sensitivity.

d) Ability to incorporate slits that reduce damping.

Disadvantages:

a) Low stiffness for inertial sensing is tough to obtain.

Electrodes

Beams

Sli


b) Small gaps below the proof-mass are also not possible for inertial grade

sensitivity.

c) Temperature-sensitivity is high if there is a mismatch in the coefficient of

expansion of the glass plate and the silicon, which are bonded together.

2.2.5.2. A capacitive accelerometer using Pressure sensor fabrication technology - Rudolf et al. (1983)

Figure 2.9 Capacitive torsional silicon accelerometer This is also a z -axis accelerometer but uses a torsional suspension. Just as in the

previous case, the electrodes are deposited on the glass plates and the glass plates are

wafer bonded to the silicon chip. If d , h and L are the width, thickness and the length

of the torsion bars, respectively, the torque required to twist the bar by unit radian is

given by the formula 32

3T Gdh

Lα= (2.23)

The torque is related to the acceleration by the formula T a r dm= ∫ where r is the

distance of the elemental mass dm from the rotational axis. The moving plate rotates

about the torsional beams and thus creates a capacitance change between the upper and

lower electrodes.

Mass

Suspension

Electrodes


Fabrication: This particular structure uses the pressure sensor technology. A slightly p-

doped silicon (100) wafer is back-etched up to a certain depth and then electrochemical

etch-stop is used to thin the membrane further down. The etch-stop is obtained by doping

(n-type) of the wafer before the process, which defines the suspension. Then, electrodes

are deposited on the glass plates and wafer-bonded onto silicon and diced. The entire

device is vacuum-packaged.

Advantages:

a) A very thin suspension is obtained which can reduce the stiffness of the device.

b) Fabrication is easy and its repeatability is good due to the use of the

electrochemical etch stop.

c) Vacuum-packaging enables the detection of even the slightest of motions and thus

increases the resolution of the device.

Disadvantages:

a) The entire device is of uniform thickness, thus the proof-mass is reduced.

b) Vacuum-packaging reduces the damping factor and thus even small vibrations do

not die out.

2.2.5.3 Surface Micromachined Accelerometers.

In the mid-90’s when the field of microelectromechanical systems (MEMS) was catching

the researchers’ interest, there was a constant effort to integrate electronics as well as the

inertial sensing components in the same chip. Thus, the CMOS processes to fabricate

surface micromachined accelerometers started to gain predominance because they were

amenable for monolithic integration of electronics on the same chip. The monolithic

integration had the advantage of good capacitance resolution and low parasitics.

Furthermore, the power consumed by the entire device is low. Surface micromachining

on the other hand, could give small gaps and thus larger sensitivity than the bulk

micromachined accelerometers. However, these accelerometers could not be made to be

of inertial-grade quality because of their poor sensitivity due to small size of the proof-

mass. Furthermore, small stiffness and gaps decrease the pull-in voltage, which reduce

the sense voltage that can be applied for sensing. These accelerometers are best suited

for batch-fabrication and for applications such as air-bag deployment in automobiles.


Surface micromachined accelerometers can be both of z -axis as well as x y−

axis. The z -axis accelerometers have high sense capacitances, but suffer from squeeze-

film damping. The x y− accelerometers suffer from low damping but require a number of

comb drives to increase the base-capacitance. Fringing fields contribute to the majority of

the capacitance in the x y− axis accelerometers. Some of the popular x y− as well as z -

axis accelerometers are discussed below.

2.2.5.3a A Z-axis surface micromachined accelerometer with integrated CMOS circuitry - Lu et al. (1995) The structure consists of a rectangular mass with L-shaped beams in a pin-wheel

arrangement as shown in Fig. 2.10. The L-shaped beams give a very low stiffness in the

z -direction and low cross-axis sensitivity. If bl is the length of the beam and bI its area

moment of inertia, the stiffness of the entire structure with four beams is given by:

3

24 bstr

b

EIKl

= (2.24)

Fabrication: This structure is fabricated by surface micromachining. First, an oxide

layer of thickness equal to the desired gap between the electrodes is deposited. It is then

patterned to define the anchors of the suspension. Then a polysilicon layer of the desired

thickness is deposited and patterned to shape the mass and the suspension. Etch-holes,

which help decrease damping, are then made. The sacrificial oxide is then etched away to

release the proof-mass and the suspension.

Figure 2.10 Surface Micromachined z-axis accelerometer by Lu et al. (1995)

Proof mass

Suspension


Advantages

a) High sense-capacitance.

b) Low stiffness leading to high sensitivity.

Disadvantages

a) Fabrication process needs to take care of stiction.

b) Low stiffness leads to a low pull-in voltage.

c) Low stiffness may also lead to sagging of the mass

d) Small gaps lead to large squeeze-film damping.

2.2.5.3b Surface micromachined lateral ( x y− ) commercial accelerometer (ADXL-50)

- (Kuehnel and Sherman, 1994)

This accelerometer shown in Fig. 2.11 is one of the earliest of the accelerometers

manufactured by Analog Devices Inc. It uses a surface micromachined proof-mass,

suspension, and comb-drives for lateral sensing. The accelerometer was incorporated

with a closed loop circuit. The suspension used in this consisted of a pair of simple

guided cantilevers on either side. If bl is the length of each beam, and bI is the area

moment of inertia of the cross section of the beam then the stiffness of the suspension is

given by

3

48 bstr

b

EIKl

= (2.25)

The fabrication process of this accelerometer is similar to any surface micromachining

process. Advantages of these types of accelerometers are that low stiffness is achievable

without the problems associated with sagging due to self-weight.

2.3.5.4 Micro-g resolution accelerometers fabricated by a combination of surface and

bulk micro machining- Chea et al. (2000, 2002, 2004)

It is evident from the previous examples that surface micromachining can provide low

gaps and sufficiently flexible suspensions. However, low mass decreases the overall

sensitivity and increases noise. Bulk micromachining on the other hand provides a huge

mass, a large capacitance area but has the drawback of stiff suspensions and large

minimum gaps. Then the gradual evolution for accelerometers towards high resolution

and sensitivity lead to the combination of the advantages of surface and bulk micro


machining processes where huge proof-mass was obtained by bulk micromachining while

small gaps and suspension stiffness were realized by surface micromachining.

Figure 2.11 ADXL 50 surface micromachined lateral accelerometer

2.2.5.4a A z -axis accelerometer using bulk and surface micromachining (Chae et al.,

2004)

The proof-mass of this accelerometer, which is shown in Fig. 2.12, has an area of 22000 2000 mµ× . The thickness of the proof mass is 450 mµ having a mass of 2.07 mg .

There are eight suspension springs each 700 mµ long, 3 mµ thick and 40 mµ wide

providing a stiffness of 14 /N m . The electrodes are at the top and bottom of the proof

mass maintaining a gap of 1.5 – 2 mµ . This gap is obtained by surface micromachining,

i.e., by depositing poly-silicon on sacrificial oxide layer. The electrode has considerable

number of damping holes to reduce mechanical noise. The electrode has vertical

stiffeners to decrease its deflection under the internal forces.

Moving combs

Static combs

Suspension

Proof mass


Figure 2.12 A high sensitivity z -axis accelerometer using combined bulk and surface micromachining by Chae et al. (2004) 2.2.5.4b An In-plane accelerometer using bulk and surface micromachining (Chae et al., 2004) The fabrication procedure for this accelerometer, which is shown in Fig. 2.13, is the same

as that of the above z -axis accelerometer shown in Fig. 2.12. Deep trenches are made on

the proof mass and the electrodes are deposited in these trenches. The gap, obtained by

sacrificial etching of the oxide, is around 2 mµ . The stiffness of the suspension is given

by 3 96 /stiff b bK EI l= (2.26)

where bl is the length of the suspension length and bI is the area moment of inertia of the

suspension springs. The capacitance obtained in the lateral case is slightly less than that

of the out-of-plane case. The sensitivities (relative change in capacitance per applied ‘g’

acceleration) of these accelerometers, expressed as the fractional change of the base

capacitance is reported to be 0.25, although calculations indicate that it can be as high as

five.


Figure 2.13. A high sensitivity In-plane accelerometer using combined bulk and surface micromachining by Chae et al. (2004) 2.2.5.5 Bulk micromachined accelerometer using high aspect ratio etching technique

(DRIE) - Chae et al. (2002)

The design of this accelerometer is very simple; it consists of a mass with comb drives on

both sides of the mass. A gap of 2 mµ is obtained within the comb drives, the aspect

ratio of 1:60 permits the thickness of the mass and the comb drives to be 120 mµ . To

overcome the disadvantage of large gaps obtained by traditional wet etching and wafer

bonding techniques, a high aspect ratio etching technique called the DRIE (Deep-

Reactive Ion Etching) has been used by Chae et al. (2002). This process is a combination

of alternate etching and deposition. The etching creates cavities into the wafer and

deposition protects the sidewalls from being etched away. This process produces deep

trenches with aspect ration of around 1:60. Thus, small but deep gaps between the fixed

and the movable plates can be realized in capacitive accelerometers.

Table 2.1 summarizes the physical and performance parameters of the high-

resolution accelerometers described above. The mass, stiffness and the base-capacitance

are determined by the physical dimensions. The natural frequency, sensitivity and the

resolution are decided by the overall performance of the accelerometer. It can be noticed

in Table 2.1 that less than 1 gµ resolution is possible but its bandwidth is decreased to

250 Hz . In fact, it is worth noting that large mass, low stiffness and relatively large base-

capacitance are important for high resolution. It is not only the mechanical structure that

is important to obtain high resolution; electronic circuitry also plays a significant role as

described next.


Figure 2.14. A DRIE in-plane accelerometer

2.2.6 Electronic circuitry for capacitance detection

As stated earlier the net sensitivity of an accelerometer is due to the combined

sensitivities of mechanical components and the electronic circuitry. The electronics

circuitry has to be sufficiently sensitive to detect small changes in capacitance, some of

them of the order of atto-Farads. The electronic circuitry consists of capacitance-

detection circuit, certain noise-cancellation circuits and a PID (Proportional Integral

Derivative) controller in case of closed-loop accelerometers (Senturia, 2000). Circuits

could be both analog and digital. The resolution of the electronic components is limited

by the noise in the circuit. Thus, most capacitance circuits need complicated noise-

cancellation techniques. Typically, a hybrid circuit with separate electronic and

mechanical chips would be able to resolve better than 10 parts per million.

The capacitance change which is triggered by the movement of the proof-mass is

converted to voltage by a high-frequency pulse applied to the static combs of the

electrode. The resulting voltage change is amplified by the Op-Amp. Various kinds of

sensing techniques such as correlated double sampling (CDS), chopper stabilization,

unity-gain buffer and switched capacitor methods differ in how non-idealities of the Op-

Amps such as input offset, 1/f noise, thermal noise, etc., are dealt with. The amplified

signal is then fed into a PID controller which is designed based on the desired dynamic

response of the system. The output of the PID controller is applied onto the static


electrodes of the sense capacitors and this acts as a feedback force equal and opposite to

the applied acceleration. The feedback voltage is then the read-out signal for the

accelerometer. For open-loop accelerometers, the read-out signal is the output of the Op-

Amp.

Table 2.1 A comparison of various capacitive accelerometers discussed in the section

above Accelerometer structure (with

reference)

A torsional capacitor (Rudolf ,1983)

(2.3.5.2)

Bulk micromachined capacitive accl. (Rudolf et al.

,1990) (2.3.5.1)

Surface micromachined accelerometers (Lu et al., 1994)

(2.3.5.3)

Accelerometers with

combination of bulk and

surface micron.

(Chae et al., 1997-2000)

(2.3.5.4)

Accelerometers with high aspect ratio

etching using DRIE

(Chae et al. , 2001)

(2.3.5.5)

Mass in kg

1e-9 4e-6 0.5e-9 2.76e-6 1.8216e-6

Stiffness N/m

0.158 572 0.94073 4.2570 90.4

Natural frequency in

Hz

2000 1835 7500 250 2140

Base Capacitance

in pF

1 12.3 0.88 8 32.1

Sensitivity in ( )/ /C C g∆

0.04 0.0148

0.0052 0.25

0.0249

Resolution (assuming a circuit with

10 ppm resolution)

250 gµ 650 gµ 1.9 mg

0.4 gµ 400 gµ

Digital accelerometers are much simpler. The change in capacitance produces a

voltage change and this signal is amplified just as in the analog circuits. The resulting

signal is sent into a comparator which triggers a constant value of the feedback voltage if

the detected signal is greater than the reference value. Out of all the digital methods, the

first-order sigma-delta is most widely used. Digital circuits are simple in design but

occupy a large chip space and suffer from instability problems such as limit cycles, which

are constant oscillations of the proof mass about the mean position.


There are various noise sources, which limit the resolution of the circuit. Out of

these, the white Johnson’s noise and the 1/f noise are predominant. Johnson’s noise is due

to the Brownian motion in the resistors and 1/f noise is usually in the amplifiers at semi-

conductor junctions. Furthermore, digital accelerometers suffer from quantization noise.

Most of these noise sources can be compensated or cancelled. For example, sampling at

high frequencies can reduce 1/f noise and nested loops cancel the offset-noise.

2.2.7 Need for mechanical amplification for accelerometers

The net resolution of an accelerometer depends upon the resolution of both electronic and

the mechanical components. As stated in the previous section, the resolution of both

mechanical and electronic components is limited by their respective noise floors. For

most accelerometers it is the electronic noise which predominates (Chae et al., 2004) and

this determines the accelerometer’s resolution. Since the acceleration signal is detected

first by the mechanical components, the minimum detectable acceleration depends upon

its sensitivity. This means that if a small acceleration can create a large mechanical

movement which produces a signal greater in magnitude than the electronic noise floor,

the net resolution of the accelerometer can be increased. Most of the accelerometers aim

to increase the mechanical sensitivity by decreasing the natural frequency of the

mechanical component. But this limits the bandwidth of the entire system. In this thesis,

we aim to improve the mechanical sensitivity of accelerometers by adding a DaCM.

2.3 Force sensor for micro-manipulation of Cells

In addition to micromachined accelerometers, force sensors are also investigated in this

thesis to illustrate the use of DaCMs in sensors. A brief background for a specific clan of

force sensors is provided below

2.3.1 Introduction to force sensors for micro-manipulation

With the increasing trend towards miniaturization, researchers face hard tasks of

manipulating micron-sized objects with special tools. Automating these micro-

manipulation operations in micro-scale requires precision sensors to measure forces.

These sensors can either be vision-based or otherwise. For mechanical operations such as

piercing a cell in single-cell studies or a human organ in laparoscopic surgery, it is useful


to incorporate force sensors which measure the forces exerted by the tool and thus aid in

feedback control of the entire process, with or without human interface. Force sensing is

also beneficial in bio-manipulation and other micro-manipulation operations where the

objects involved are fragile (Greminger and Nelson, 2004).

Force measurement at the microscale was conventionally done using laser-based

optical techniques and piezo-resistive sensors embedded in elastic structures, which

transmit force. These techniques required special elastic structures and perfect alignment

for accurate force estimation ( Greminger and Nelson, 2004). So, a new class of force

sensors were introduced whose deformation at a single or several points was used to get

back the applied force (Wang et al, 2001; Greminger and Nelson, 2004). This

deformation can be obtained either by vision using microscopes or by a non-contact

sensor such as a Hall-sensor.

Figure 2.15. Using compliant mechanisms to measure force

A lot of work has gone into developing elastic-force sensors for micro-

manipulation. The simplest form of such a force sensor is the Atomic Force Microscope

(AFM) in which a cantilever beam whose end deflection gives the force to be measured.

Then came the use of elastic mechanisms whose deformation-patterns were matched with

standard templates for force estimation (see Fig. 2.15 where a fixed-fixed beam is

shown). These methods were limited in their resolution and the direction of forces that

could be sensed. Wang et al. (2001) used nonlinear finite-element methods to derive

δ


forces applied to elastic mechanisms. This method requires effective noise reduction

algorithms for correct force estimation.

2.3.2 Use of DaCMs as force sensors

Single point forces could be sensed by a cantilever beam whose deflection at a point can

be tracked. In most of these configurations, the point of application of load would be the

same as the point of displacement measurement. For single-axis force sensors with non-

contact displacement-detection sensors, it would be easier for the point of displacement

detection to be decoupled from the point of application of load. It would also be essential

for the point of displacement detection to be insensitive to cross-axis loads. Furthermore,

points of load application need to be stiff and should not deform extensively. This might

limit the range of the forces that can be sensed. At the same time, the point of

displacement detection should be sensitive enough to detect small forces applied at the

force-application point. These motivate the need for investigating a DaCM for single-axis

force sensing.

2.4 Closure

In this work, we have laid the foundation for the future chapters of the thesis by

reviewing the literature on related topics. Effective gaps in the literature, especially in

conceptualization of DaCMs and compliant mechanisms in general for sensor

applications, were highlighted. It was pointed out further that the performance of

different DaCMs available in literature needed to be evaluated for sensor applications

through a simple and effective model. Literature survey of high sensitivity accelerometers

indicated a need to improve the mechanical sensitivity of accelerometers. Finally,

synthesizing new mechanisms for sensor applications via topology optimization needs to

incorporate objective functions and constraints, which are specific to these applications.

3.1

Chapter 3

3OBJECTIVE COMPARISON OF VARIOUS DaCMs

FOR SENSOR APPLICATIONS

Summary Displacement-amplifying compliant mechanisms (DaCMs) reported in literature are

mostly used for actuator applications. Here, we consider them for sensor applications

that rely on displacement measurement, and evaluate their topologies objectively. The

main goal is to increase the sensitivity under constraints imposed by several secondary

requirements and practical constraints. A spring-mass-lever model that effectively

captures the addition of a DaCM to a sensor is used in comparing nine DaCMs. It is

observed that they significantly differ in performance criteria such as geometric

advantage, stiffness, natural frequency, mode amplification, factor of safety against

failure, cross-axis stiffness, etc.; but none excel in all. Thus, a combined figure of merit is

proposed using which the most suitable DaCM could be selected for a sensor application.

Some other insights gained with the analysis presented here were the optimum size-scale

for a DaCM, the effect on its natural frequency, limits on its stiffness, and the working

range of the sensor.

3.1 Introduction

Displacement-amplifying Compliant Mechanisms (DaCM’s) have been used to amplify

the output of actuators as described in Section 2.1. Among the sensor applications, it is

worth noting that Su and Yang (2001), and Pedersen and Seshia (2005) have used Force-

amplifying Compliant Mechanisms (FaCMs) for increasing the sensitivity of resonant

accelerometers. On the other hand, the literature on DaCM’s suggests that sensor

applications remain largely unexplored. We propose that DaCMs have much use in

micro-machined high-resolution inertial-grade capacitive accelerometers and in vision-

based sensing of forces in the manipulation of single biological cells. This chapter aims at

Chapter 3: Objective Comparison of various DaCMs for Sensor Applications 3.2

an objective comparison of the topologies of available DaCMs for any given sensor

application. The criteria used for comparison would also help formulate an optimization

problem for designing DaCMs anew for sensor and actuator applications.

In the next section, we discuss how DaCMs for sensor applications resemble or

differ from those for actuator applications. We then present a spring-mass-lever model

that captures the essential behavior of a DaCM in conjunction with a given sensor. This is

followed by an objective comparison of nine DaCM topologies from which the most

suitable one can be chosen for a sensor application.

3.2 DaCMs for Sensor applications

Sensors that rely on a displacement-based transduction scheme benefit from a large

amplification ratio n , which is the ratio of output and input displacements of a DaCM, to

increase their sensitivity. Large n is necessary for actuators too. The earliest DaCMs

were developed for amplifying displacements of piezo-electric actuators using four-bar

linkages and making them compliant by replacing their joints by flexural hinges. Apart

from these, elastic elliptic amplification mechanisms were used to amplify the output

from piezo-electric stacks which were further optimized by making use of flexural hinges

(Lobonitiu and Garcia, 2003). On the other hand, topology optimization techniques have

been used to design DaCMs with distributed compliance (Saxena and Ananthasuresh,

2000; Yin and Ananthasuresh, 2003).

Dynamic characteristics of DaCMs are equally important for both sensors and

actuators. Canfield and Frecker (2000) and Maddisetty and Frecker (2002) synthesized

DaCMs for static and dynamic loads with piezoelectric actuation. Du et al. (2000)

synthesized elliptic amplifiers for ink-jet print-head actuators using continuum-element

optimization under dynamic conditions.

Bandwidth, working range, resolution and strength considerations are equally

important when DaCMs are combined with sensors or actuators while the natural

frequency plays a greater role in the case of actuators. Linearity of the output response is

more important for a sensor application than it is for an actuator. There are also some

other differences in using DaCMs in sensors or actuators.


Mechanical efficiency, which is defined as the ratio of the output work done to the

input work supplied, is of concern in the case of actuators. For example, Hetrick and Kota

(1999) used this as the primary criterion along with the amplification ratio for optimizing

the size and shape of a DaCM. Modification of the force-displacement characteristic, for

example to achieve a constant output force over a large displacement as done by Pedersen

et al. (2006), is not as important for sensors as it is for an actuator. Because of inherent

stiffness that most actuators have, their maximum stroke is often restricted by their

deliverable force capability. Therefore, a stiff DaCM is needed to limit the actuator’s

displacement. In contrast, the DaCM for a sensor should be compliant to give a large

stroke.

Sensors by their very definition need to be sensitive to only the desired signal and

should annul the undesirable disturbances. Therefore, the effects of noise assume more

significance in sensors than in actuators. Similarly, the cross-axis sensitivities will have a

pronounced effect in sensors and should be given due consideration in selecting or

designing a DaCM.

The complex geometry of a DaCM is not easily amenable to evaluate against all

the aforementioned criteria that are relevant for a sensor application. Therefore, a reduced

order model of a sensor combined with a DaCM is presented in the next section.

3.3 Spring-mass-lever Model of a sensor with a DaCM

Figure 3.1 shows an accelerometer which uses capacitive sensing technique (note that the

electrostatic combs are not shown). Because the capacitance depends on the

displacement, the input side of a DaCM is integrated with this sensor so that the output

side of the DaCM can be used as the sensing port for increased sensitivity. The figure also

shows a suspension for the sensing port in order to decrease the lateral-axis sensitivity.

Since our evaluation of DaCMs and their comparison is targeted for sensor applications,

we should first understand how sensors behave when coupled with a DaCM. A two

degree-of-freedom spring-mass-lever model, shown in Figs. 3.2 a-b, effectively captures

this behavior. The lumped parameters shown in Figs. 3.1 and 3.2 are explained below.

• The stiffness and the inertia of the sensor in the intended direction are denoted by sk

and sm . These can be determined with either an analytical lumped model using the


beam theory or with the relevant dominant mode from the modal analysis of the

sensor alone.

• The stiffness at the input side of the DaCM is denoted by cik and this can be

determined by applying a unit force at the input port of the DaCM and dividing it by

the displacement at the same point (Fig. 3.3a). This requires a finite element analysis

of the DaCM.

• The stiffness at the output port cok is calculated by applying a unit force outF at the

output port of the mechanism, finding the displacement at the same point and using

the formula in Fig. 3.3b. It is worth noting that the method of computing cok is

different from that of cik . The application of force at the output results in an output

displacement that is influenced by cik whereas the output side stiffness has no role

when only an input force is applied. This is because the output side is not anchored if

there is no output suspension.

• The external stiffness and inertia at the output of the mechanism are denoted by extk

and extm . It should be noted that most sensors would not require output spring

stiffness extk . A few sensors such as an accelerometer would.

It should be noticed that the geometric amplification factor n is schematically shown

with a lever and a four-bar linkage in Figs. 3.1a and 3.1b respectively, indicating whether

the DaCM reverses the direction of the applied displacement (called an inverter) or

maintains the direction of the input displacement (called a non-inverter).

Figure 3.1 An accelerometer (sensor) with a DaCM and a suspension at the sensing port

DaCM

extm

extk

( , , , , )ci co ci cik k m m n

sk

smSensor

External (suspension)


• By denoting the input and output forces by inF and outF , we can obtain the amplified

output displacement in either case as

( )2

2 2

( )( ) ( )

out s co ci in co

co s ci ext co ci s

F k k n k F nkx

k k k k n k k k+ +

=+ + + +

m (3.1)

where the negative sign refers to an inverter and positive to a non-inverter in this and

the next equation.

• By noting that the measured displacement when there is no DaCM is simply

3 /in sx F k= , the net amplification factor, NA , is given by

( )

2

22

3

( )

( ) ( )out s co ci in co s

co s ci ext co ci s in

F k k n k F nk kxNAx k k k k n k k k F

+ += =

+ + + +

m (3.2)

(a)

(b)

Figure 3.2 Spring-mass-lever model of a sensor combined with (a) an inverting DaCM and (b) a non-inverting DaCM

sk

cik cok

n

extk

1x

2x

2x cok

extk

n1x sk

cik


(a) (b)

Figure 3.3 Proposed method to calculate the lumped stiffness of a DaCM (a) input stiffness cik (b) output stiffness cok

The net amplification refers to the ratio of the displacements that will be transduced

with and without the DaCM.

• In sensor applications the inherent displacement amplification factor n is not the one

that matters but the net amplification factor, NA. To gain some insight and intuition, if

we assume that the output inertia and force are relatively negligible, i.e., by

substituting outF = 0, extk = 0, Eq. 3.2 simplifies to the form

( 1)( )

sci s

s ci

nk np k kk k p

> ⇒ < −+

(3.3)

Thus, we get an upper bound on the input-side stiffness for a minimum net

amplification ( NA ) of p . It can be seen that if <n p , cik is negative. This means

that NA must be smaller than n .

inci

in

Fkx

=2

out cico

ci out out

F kkk y F n

=−

outy

inx

outF

inyinF

outx

inx inF

outx

cik

cok

n

outF outy

ncik

cok iny


• The natural frequency of the system, which is useful in characterizing the dynamic

behaviour, is given as follows.

21 1 2

1 12 2

f λ λ λπ

= − − (3.4)

where

2

1

2

24( )( ) 4

co ext s ci co

s ci co s co ext

k k k k n km M

k k k k n k kMm Mm

λ

λ

⎛ ⎞+ + += +⎜ ⎟⎝ ⎠

+ += +

,

s ciM m m= + and ext com m m= + , cim and com are the inertia of the DaCM at the

input and the output sides respectively.

• In most applications the inertia of the DaCM would be very low when compared to

that of the sensor. In that case we can ignore the inertias of the DaCM, cim and com .

But when they cannot be ignored, we can obtain the approximations for these two

quantities, by applying a uniform body load1 to the entire mechanism in the FEA

model. The output displacement from the FEM model is equated with the value of the

output displacement obtained from Eq. 3.1 by substituting 0sk = , 0sm = , 0extk = ,

0extm = , in ciF m g= and out coF m g= . Another equation is obtained by evaluating the

first natural frequency ( bf ) of the mechanism from the FEA model and equating this

with Eq. 3.4 by substituting 0sk = , 0sm = , 0extk = and 0extm = . Eq. 3.6 and Eq. 3.7

are two non-linear equations which can be solved iteratively to get estimates of cim

and com .

( )2( )co co ci ci cob

co ci

m k n k m nky

k k+

=m

(3.6)

22 2 4( )( )1 12 2

co ci co co ci co ci cob

co ci co ci co ci

k k n k k k n k k kfm m m m m mπ⎛ ⎞ ⎛ ⎞+ +

= + − + −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

(3.7)

The lumped parameters defined in this section will be used to quantify various criteria that will be useful

in comparing different DaCMs, as described next.

1 This body load is the inertial force due to an assumed uniform acceleration on the FEA model.


3.4 Objective comparison of some DaCMs

3.4.1 Comparison criteria

As mentioned in Section 3.2, a number of DaCMs have been reported in literature. Each

of the DaCMs are claimed to be optimal for the application for which they have been

synthesized. It is their topologies that enable them to serve their main function of

amplifying an applied displacement. But no effort has so far gone into comparing these

topologies for the same application. With the aim of comparing DaCMs for sensor

applications, we set out five important criteria, each seemingly independent of the other.

• Net Amplification ( NA): Net amplification ( NA ) is given by Eq. 3.2. It is

dependent on the stiffnesses ( sk and extk ) and inertias ( sm and extm ) of the sensor

and external element. It signifies the net advantage in the sensitivity of the sensor

due to the addition of the DaCM. Between inherent amplification ( n ) and net

amplification ( NA) of a DaCM, the latter is more important for almost any sensor

application that has limited input force, which is directly related to the quantity to

be measured.

• Factor of safety against failure ( FS ): The maximum stress experienced by the

mechanism over the intended range of the sensor determines the factor of safety

against failure ( FS ). This depends upon the material chosen for the mechanism.

The optimal selection of the material is beyond the scope of the thesis. The mode

of failure depends on the material as well as the mechanism which are both

dictated by the application.

• Natural Frequency ( f ): The lowest natural frequency ( f ) is an important

criterion for most sensor applications as they operate in the dynamic regime. The

bandwidth and time-constant of the sensor system are significantly affected by the

natural frequency.

• The cross-axis stiffness ( crossk ): This is also an important criterion that decides the

sensitivity and resolution of a sensor with a DaCM. Usually, mechanisms that are

optimized for increased output displacement in one intended direction may make

the mechanism flexible in the perpendicular directions or in the angular modes.


Thus, to make effective use of a DaCM in practical situations, cross-axis stiffness

should be given due consideration.

• Unloaded Output Displacement ( dU ): Unloaded output displacement ( dU ) is

simply the output displacement under an applied input force at the respective

points when 0sk = . This quantity is a measure of the flexibility of the DaCM.

The mode shapes corresponding to the natural frequencies also matter. The mode-

magnification factor ( mn or mNA ) corresponding to the first natural frequency indicates

the amplification achieved at dynamic loads within the dynamic range of the mechanism.

But it was found that the mode-magnification factor ( mn or mNA ) was numerically close

to the static amplification ( n or NA ) for most of the mechanisms considered. Hence this

quantity is not considered for comparison.

Manufacturability ultimately decides the suitability of a DaCM for an application.

Minimum feature size, tolerances, the number of fixed supports and guides, etc., should

be given due attention in selecting a mechanism for an application. To see how these

criteria can be used, first we need to select some candidate DaCMs, as discussed next.

Some of the DaCMs found in the literature were mentioned in the first section.

These and others can be grouped into ones having flexural hinges and those having

distributed compliant members such as beams. The design insights of flexural hinges are

well known and they have recently been extensively analyzed (eg., Lobontiu and Garcia,

2003). In this study, however, we have chosen compliant mechanisms with distributed

compliance. Nine DaCMs are chosen here. The mechanisms labeled Mechanisms M1-M6

were optimized for static applications but M7 was optimized for dynamic behavior.

Mechanism M8 is a new design obtained using topology optimization that included

lateral cross-axis sensitivity requirement. Mechanism M9 is a continuum example and

was obtained by optimization for dynamic loads by Du et al. (2000). These are shown in

Fig. 3.4. Clearly, all these are quite different from each other. So, when it comes to

selecting one among them (or others that are not considered here), quantitative evaluation

becomes very important.


(a) M1 (Kota et al., 1999)

(b) M2 (Kota et al., 2000)

(c) M3 (Hetrick et al.

1999)

(d) M4 (Saxena and Suresh, 2000;

Ananthasuresh, 2005)

(e) M5 (variant of M4)

(f) M6 (Canfield and Frecker, 2000)

(g) M7 (Maddisetty and

Frecker, 2002)

(h) M8 (new)

(i) M9 (Du et al., 2000)

Figures 3.4 (a-i) Nine symmetric DaCMs labeled M1-M9 along with their deformed configurations of the right half.


Linear elastic analysis is sufficient for closed-loop sensors operating in a force re-balance

mode. That is, the sensing point is maintained stationary with a feedback force and the

actuation used for this is calibrated for estimating the measurand. Since the entire DaCM

actually remains undeformed in this mode, the instantaneous net amplification NA is

considered. An example of such a sensor is the accelerometer, which will be detailed in

the next chapter. When the sensor is operated in the open loop mode, such as the force

sensor, the inherent geometric amplification and the stiffness do not remain the same

throughout the range of the applied force. The maximum force that can be supplied to

each mechanism depends on its range of travel as well as the maximum stress

experienced by it. For both of these, geometric nonlinearity needs to be considered. It

should be noticed that some of the mechanisms are susceptible to contact. Therefore, the

force required to stress the mechanism to half its yield stress determines the maximum

force because of the impending self contact.

3.4.2 Specification for Analysis

Five criteria noted above and the inherent amplification ( n ) are computed for the nine

DaCMs shown in Fig. 3.4. In order to compare the topologies of these mechanisms on

equal grounds, all the mechanisms were modified to fit in an area of 3.125 610× 2mµ .

This area is motivated by micro-sensors applications where DaCMs have an important

role to play. All but the ninth mechanism were optimized by varying their shapes and

sizes. To calculate NA , a stiffness of 500 /N m is assumed for sk while sm , extk , extm

are assumed to be zero. The beam widths for each mechanism were optimized to obtain a

high net amplification ( NA ). This was done keeping in mind the fabrication constraints

for bulk-micromachining process limiting the minimum width to be 3 mµ . The overall

thickness of the mechanism is 25 mµ . There is no constraint on the maximum beam

thickness provided that the slender beam theory remains valid. That is, the largest cross-

section dimensions is less than 1/15th of the length of the beam.

The optimization of the beam widths for a high NA is important because the

comparison would then mean a comparison of the topologies alone. Furthermore, no

modification of the beam-widths or thickness will be able to make the mechanism any

better for the application.


The material is also fixed for all the mechanisms. Here, silicon is chosen as the

material. Its young’s modulus and density are 169 GPa and 2300 3/kg m respectively.

Mechanisms M1-M8 have slender beam segments and thus modeling them using

beam/frame finite elements is valid. Beam elements programmed in Matlab was used for

the analysis. The continuum topologies were analyzed using COMSOL (Mechanism M9),

a commercial FEA package. The load applied to the input is fixed at 400 Nµ for all the

mechanisms. Figure 3.5 shows the six criteria ( NA , n , cU , FS , f , and crossK ) for

DaCMs M1-M9, obtained from the linear elastic analysis. According to the requirements

of an application, we want to minimize some of these and maximize the others. The

figure of merit, proposed for the selection of the mechanism in the next section will

consider this issue. First, some general observations are discussed below.

3.4.3 Observations and insights

Linear elastic analysis

It is striking to see in Fig. 3.5 that no mechanism outperforms the others in all the criteria.

We recall that the area occupied by each of them is fixed. It is also observed that none of

the mechanisms taken from literature (mechanisms M1-M7, M9) were found to have

sufficient cross-axis sensitivities. Mechanism M8, which was obtained from topology

optimization with constraints on cross-axis stiffness, is found to give better cross-axis

sensitivity than that of all the other mechanisms. To understand the mechanisms better, it

needs to be seen as to how the geometric amplification, stiffness and maximum stress of

the mechanisms vary with respect to the area that they occupy. In Fig. 3.6, it is seen that

the geometric amplification of each mechanism increases. After a certain size2, it remains

constant. This is because axial stiffness becomes comparable to bending stiffness for

excessively slender beams. It can also be seen that to get the maximum geometric

advantage it is adequate to increase the size by a factor of two from the nominal

dimensions assumed at the outset. The proportions of all the mechanisms were

maintained while fitting the mechanisms to the assumed fixed area to get the most

geometric advantage in each case.

2 Only the lengths of the beams are scaled while keeping the widths and thickness the same in view of having lower stiffness

with increase in size.


Figure 3.5. Performance of DaCMs M1-M9 based on six criteria of comparison. The units of dU (unloaded output displacement) are m/N, of stress are in MPa, while others are dimensionless. Cross-axis stiffness shown is the lateral-axis stiffness normalized divided by the stiffness in the desired direction. Frequencies are divided by 500 Hz.

Figure 3.6 A plot of the geometric amplification with respect to the characteristic length of the mechanism


Figure 3.7 shows the variation of the stress with respect to increasing

mechanism’s size. Stiffness of the mechanisms decreases nonlinearly as shown in Figs

3.8 (a), (b) and (c). Assuming that the linear-beam theory is valid, the variation of the

input and the output stiffness ( cik and cok ) with the length of the mechanism ( mechl ) can

be approximated by the following equations below.

3 2cimech mech mech

a b ckl l l

= + + (3.8a)

1 1 13 2co

mech mech mech

a b ckl l l

= + + (3.8b)

In the above equations, the constants a , b , and c depend on the material properties and

fixed geometrical parameters such as the in-plane widths and out of plane thickness of the

mechanisms. Figures 3.8 (b) and (c) show that the variation of cik and cok with the size

of the mechanism mechl using finite-element beam elements and the Eqs 3.8 (a) and (b)

are in good agreement. Figure 3.9 shows the variation of the natural frequency with

respect to increasing size. Figure 3.10 shows that the net amplification ( NA ) approaches

the geometric advantage of the mechanism at high sensor stiffness values. These guide a

designer to adjust a mechanism’s width and thickness to fit in a given area or to decide

the size of the mechanism for a given sensor’s stiffness. The table below summarizes the

effect of scaling on various attributes.

Analysis with geometric nonlinearity

Geometrically nonlinear elastic analysis of the nine mechanisms revealed that the

inherent geometric amplification of most of the mechanisms did not differ much from the

linear counterpart for small forces (~100 Nµ ). For large forces the amplification reduces

and the stiffness increases for all the mechanisms. Figures 3.11a-c show the variation of

the inherent amplification, stress, and stiffness of the mechanisms with respect to the

applied force. Figure 3.12 shows the maximum force that a mechanism can withstand

before the maximum stress approaches the ultimate stress with a factor of safety of two.


Figure 3.7 A plot of the maximum stress with respect to the characteristic length of M1-

M8.

(a)


(b)

(c)

Figure 3.8 (a) A plot of the stiffness ( /N m ) with respect to the characteristic length (i.e., size) of M1-M8. Comparison of the FEM and analytical formula (Eq. 3.8) capturing the variation of the stiffness (b) cik and (c) cok with respect to the length of the mechanism mechl .


Figure 3.9 A plot of the natural frequency of the mechanisms with respect to the characteristic length of M1-M8

Figure 3.10 A plot of the net geometric advantage ( NA ) with respect to the sensor stiffness ( sk ).


3.4.4 Figure of merit

Figure 3.5 gives the relative comparison of nine mechanisms based on the six criteria.

However, for objective comparison and selection of a DaCM for a particular application,

a figure of merit that combines various performance parameters or attributes will be

useful. This is also useful in formulating an objective function for multi-criteria

optimization to synthesize new DaCMs.

In most cases it is impossible to achieve excellent performance for all the

attributes. Hence, there needs to be a trade-off among various conflicting attributes of a

given application. The most common practice to define such an objective function is to

normalize each of the attributes to a value between zero and one and get a weighted

average of all the attributes based on weights suitable for the application in mind as given

by Eq. 3.9.

1 1 2 21

1 2

....

n n

n

w w wPw w w

α α α+ + +=

+ + + (3.9)

1/

1 1 2 22

1 2

....

ss s sn n

n

w w wPw w w

α α α⎛ ⎞+ + += ⎜ ⎟+ + +⎝ ⎠

(3.10)

where iw s denote the weights assigned to the various attributes, iα s the normalized value

of the attributes and s the degree of compensation. Changing these weights assigned for

each attribute spans a Pareto-frontier (Pareto, 1971). But the inability of weighted

average to capture all the possible trade-offs or Pareto-points (Koski, 1985) prompted

researchers to define new ways to obtain such figures of merit. Maddulapalli et al. (2005)

proposed a normed measure while Wan and Krishnamurthy (2001) proposed an iterative

interactive procedure for the same. The most effective and simple procedure to define

such a multi-criteria objective function was proposed by Scott and Antonson (2005)

where they combined the weights along with a degree of compensation s , which depends

on these weights. One such measure is shown in Eq. 3.10. This measure was shown to

span a wider Pareto-frontier than a simple weighted average.

The attributes that define an objective function are in this case the six criteria,

which were proposed for comparison. Each of these criteria is normalized with respect to

its maximum value for all the mechanisms. This normalization is equivalent to having a

value of one for the maximum value of the attribute. Each attribute is then associated


with a weight, which defines its relative importance. The weights are chosen subjectively

to determine the importance of one criterion over the other. The figure of merit for the

mechanism is then given by a weighted sum for all the criteria. The weighted average is

more than sufficient if one particular criterion dominates for an application and thus has a

higher associated weight. In case all the criteria have to be given equal weightage, the

weighted average is insufficient for selection and thus better methods need to be used. In

this thesis, one or two criteria dominate over the rest.

3.4.5 Selection vs. Optimization

In this chapter, we have developed a catalog of DaCMs from literature. The catalog is

modest with only nine mechanisms. Considering that specialized mechanisms in catalogs

such as Sclater and Chironis (1991) are of the same order, this modest catalog is not too

small in comparison. The rapidly growing field of compliant mechanisms is sure to add

more to this catalog in due course of time. We have also proposed a procedure for

systematic selection of DaCMs, for a particular application, based on the figure of merit

introduced in the previous section. This is perhaps the first attempt to systematically

select a DaCM for a particular application form a catalog of such mechanisms. The

concept of selection from a catalog is not new for rigid-body mechanisms (Sclater and

Chirnois, 1991; Artobolevski, 1939) as the number of criteria for selection is limited and

straightforward. However, for compliant mechanisms, we have identified six criteria,

which are important for most applications. The dependency of one criterion over another

cannot be easily established. Additionally, different criteria can be important for different

applications. Hence, it is useful to propose an application-dependent figure of merit for

selection.

It should be noted, however, that this method of selection can be applied only if

there are sufficient entries in the catalog. If the number of mechanisms is few, then design

of new mechanisms, either by topology optimization or other rigid-body based methods,

are needed. Designing new mechanisms are also useful if none of the mechanisms in the

catalog fare well for a criterion. This was seen in Fig. 3.5 where none of the mechanisms

from literature had sufficient cross-axis stiffness. This is because all the mechanisms


from literature were optimized for applications where this criterion was not considered.

Thus, topology optimization with cross-axis stiffness constraints was performed (see

Chapter 6) to get mechanism M8. This mechanism has higher cross-axis stiffness than

any mechanism in the catalog. Topology optimization is also useful if none of the

mechanisms meet a rigid specification put forth by an application. It will be seen in

Chapter 7 that none of the mechanisms in the database can meet the specifications of the

vision-based force sensing in micromanipulation of cells.

This approach of selection, modification and optimization is followed for the two

applications considered in the thesis.

Table 3.1 Summary of the effect of scaling only the length dimension of the mechanisms on various attributes

Quantity Effect on scaling

Geometric

advantage ( n )

Remains constant if the

proportions are fixed (Fig. 3.6).

Input stiffness ( cik ) Varies nonlinearly vs. the length

scale (Fig. 3.8).

Stress Increases linearly vs. the length

scale (Fig. 3.7)

Inertia (M1 and m1) Increases linearly.

Natural frequency Decreases non-linearly (Fig. 3.9).


(a)

(b)


(c)

Figure 3.11 (a) Geometric amplification (GA ) (b) Maximum stress (c) Stiffness of all the

mechanisms vs. applied force based on geometrically nonlinear analysis

Figure 3.12. A plot of the maximum force that the mechanism can handle before failure.


3.5 Closure

In this chapter the similarities and differences between DaCMs for sensor and actuator

applications have been highlighted. A spring-mass-lever model has been proposed to

further understand the behavior of sensors coupled with a DaCM. Various DaCMs from

literature have been compared based on all the attributes that are important for sensor

applications. Since all these quantities are important to a relative degree based on the

desired application, a figure of merit is proposed for each mechanism. This figure of

merit concept is used in the next chapter to guide the selection of the DaCM for sensor

applications.

4.1

Chapter 4

4. DESIGN OF A MICRO-g ACCELEROEMTER WITH

A DaCM

Summary

This chapter deals with the first of the two case-studies involving the application of

DaCMs for sensors. First, to show a proof of concept, a DaCM is selected based on the

weighted criteria as introduced in Chapter 3. It is then coupled with a standard bulk

micro-machined micro-g accelerometer taken from the literature. The capabilities and

the limitations of a micro-fabrication process determine the improvement in sensitivity

achieved with a DaCM. For a chosen process, three new designs of accelerometers with

DaCMs are obtained and then are optimized for a given area on the accelerometer chip.

4.1 Introduction

The literature review on micro-g accelerometers discussed in Chapter 2 revealed the

potential of mechanical amplification in increasing the sensitivity of accelerometers

further. For this reason, DaCMs were investigated for sensor applications and a number

of selection criteria were proposed. In this chapter, we use the proposed selection

technique to design and optimize the topology and the size of the mechanism and the

suspension of the accelerometer for high sensitivity and low cross-axis stiffness.

4.1.1 General layout of an accelerometer with a DaCM

The main purpose of adding a DaCM to the accelerometer is to increase its sensitivity

without considerably affecting its natural frequency. The layout of an accelerometer with

a DaCM is given by Fig. 4.1. It consists of a proofmass and suspension along with a

Chapter 4: Design of a Micro-g Accelerometer with a DaCM 4.2

DaCM, sense-comb fingers, and an external suspension. We explain below how different

this layout is when compared to conventional accelerometers.

Accelerometers have sense-combs at the points of maximum displacement and

feedback combs at points that experience maximum inertial force. In conventional

accelerometers both sense-combs and feedback-combs are located on the proof-mass,

which is the point of maximum displacement as well as the point that experiences

maximum inertial force. However, by combining the accelerometer with a DaCM, the

maximum deflection is obtained at the output of the DaCM whereas the inertial force is

experienced by the proof-mass. Thus, sense-combs need to be located at the output of the

DaCM while the feedback-combs are to be located on the proof-mass. Since sufficient

combs need to be packed at the sensing port, considerable mass gets added there. This

mass experiences swaying moments when the acceleration is applied in the lateral cross-

axis direction. This causes considerable cross-axis deflection owing to the relatively

flexible regions at the output of the DaCM. To reduce this cross-axis deflection, an

external suspension is added at the sense-comb end. This external suspension stiffness

should be small enough to prevent large decrease in the sensitivity of the accelerometer.

The mass due to the comb drives at the output port of the DaCM accounts for an

additional deflection when subjected to acceleration. This makes it a two degree-of-

freedom system. When the accelerometer is operated in the force re-balance mode,

additional feedback-combs are required at the sensing port in addition to the proof-mass

end, to bring it completely to the stationary position. The additional deflection due to the

comb-drive mass might aid in the overall sensitivity if the DaCM is a non-inverter while

it may decrease the sensitivity if the DaCM is an inverter. A large additional mass due to

the sense-combs may also reduce the natural frequency of the device. These effects are

further detailed in this chapter. Keeping these in mind, it is advantageous to have a small

sense-comb mass, which in turn depends upon the capacitance detection method.


Figure 4.1 Layout of an accelerometer with a DaCM

4.2 Capacitance detection

Capacitive sensors are widely used in micro-machined sensor circuits because of their

high-sensitivity, good dc response, low drift, low temperature sensitivity, and simple

structure (Yazdi et al., 1998). A capacitor consists of two electrodes of opposite polarity

separated by a distance, thus storing the electrostatic energy. The capacitance of a

parallel-plate capacitor is given by Eq. 4.1.

0AC

dε

= (4.1)

Its capacitance is changed by change in the area of overlap A , permittivity of the

medium between the plates ε and the distance between the plates d as shown in Fig 4.2a

massb

2suspl

mechl

massl

suspl

2 mechl×

Sense combs and external suspension

DaCM

Proof mass with suspension

combl

Sense comb fingers

Feedback comb-fingers

bcl

DaCM


Figure 4.2 (a) A single capacitance and (b) a differential capacitance arrangement (c) Circuit representation of the differential capacitance

Single capacitor arrangements are seldom used for sensing displacement because

change in capacitance is small. Differential capacitor arrangements shown in Figs. 4.2b-c

are preferred because of their high sensitivity and enhanced range of linearity. It consists

of two variable capacitors; the variation is caused by the change in the sense-gap between

the static and the moving electrodes. A high-frequency pulse V of opposite phase is

applied to nodes 1 and 3 as shown in Fig. 4.2b. This means that at any cycle, if the

voltage at node 1 is V , the voltage of node 3 is then zero. The output is taken at node 2.

The output voltage is given by

( )0 2

0 2 12out

C C VV V

C C C−∆

= −−∆ + ∆

which simplifies to

1 2 1 2

0 1 2 0

2( )2out

C C C CV V VC C C C∆ + ∆ ∆ + ∆

= ≈−∆ −∆

= 1 2

2 s

C C VC

∆ + ∆ for 1 2 0C C∆ −∆ ≈ (4.2)

where 02sC C= is the sense capacitance.

+

−d x−

d x+ 0 2C C−∆

0 1C C+ ∆(b)

d

b h

V + V + V −

0 1C C+ ∆0 2C C−∆

1

2

3

outV

(a) (c)


Changes in capacitance 1C∆ and 2C∆ are obtained by a change in the distance between

the electrodes d . Referring to Fig. 4.2b, the output voltage is given by

0 0 0 0

2 20 02 2out

A A A Axd xd x d x d x d xV V V V V for x dA A d x d

d d

ε ε ε ε

ε ε

− −− + − += = = ≈ <<

− (4.3)

Differential capacitance can be realized in an accelerometer by having a moving

electrode between two oppositely charged stationary electrodes. Fig. 4.2b shows a z -axis

accelerometers introduced in Section 2.3.5 where the proof mass acts as the moving

electrode. However, in-plane accelerometers need comb-fingers to measure capacitance

change. In both cases, the distance between the fixed and the moving electrode changes.

Though this change is nonlinear, for small displacements differential capacitance setting

could provide good linearity. In comb drives, we explore two ways to get differential

capacitance.

Case (a): In this type of capacitance-detection arrangement (see Fig. 4.3a), there are two

stationary combs between two moving combs. This arrangement is compact and thus

more combs can be packed in a given area. But the alternative static combs, which are

positive and negative, need to be isolated from each other to prevent shorting. This is

possible in surface micromachining but not in most bulk micromachining processes.

Case (b): In this type (see Fig. 4.3b), the stationary negative combs are connected on one

side and the entire positive combs on the other. This means that two combs on either side

make one differential capacitance. Two combs on the same side are separated from each

other by a large distance so that the capacitance between the stationary comb of one pair

and the moving comb of the other pair is minimal. The disadvantage is that there will be a

capacitance change between two subsequent combs unless they are insulated.

The change in capacitance for the first case (Fig. 4.4a ) is given by

0 0 01 2 02 2

2

1 1

xxA A A ddC C C C

d x d x d x xd d

ε ε ε⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟

⎝ ⎠⎜ ⎟∆ = ∆ −∆ = − = =⎜ ⎟− + ⎛ ⎞ ⎛ ⎞− −⎜ ⎟⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠⎝ ⎠

(4.4)

For small displacements, i.e. x d<< , we get


0 0201 a

xx C xdC Cd C dx

d

⎛ ⎞⎜ ⎟ ⎛ ⎞∆⎝ ⎠ ≈ ∴ =⎜ ⎟⎛ ⎞ ⎝ ⎠− ⎜ ⎟⎝ ⎠

(4.5)

Figure 4.3 Differential Capacitance arrangement a) The positive and the negative static combs are on the same side b) The positive and the negative static combs are on either sides of the comb mass.

Figure 4.4 Arrangement of both the configurations through out the sense-comb mass.

In case (b ) (Fig. 4.3 (b)), the change in capacitance is given by

10 0 0 00 12 2

1 1

1

1 1

xxdA A A A dC C C

d x d x d x d x x xd d

ε ε ε ε⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠ ⎝ ⎠∆ = − − + = −− − + + ⎛ ⎞ ⎛ ⎞− −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

(4.6)

For small displacements, i.e. x d<< , we get

+ −+

Case ( a ) Case (b ) –

Moving combs

Static combs

++

d x−

−

+ d x+d x−

1d x−_

d x+

Case ( a ) Case (b )


10 0 0 12 2

1

1

1 1

xxd x xdC C C C

d dx xd d

⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠ ⎝ ⎠− ≈ −⎛ ⎞ ⎛ ⎞− −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

(4.7)

where 0 02 /C A dε= is the base capacitance between a pair of combs. And 1 0 12 /C A dε=

is the capacitance between the stationary comb of one pair and the moving comb of the

other pair which can be termed as the cross-capacitance. The ratio of the change in

capacitance in case (b ) to the base capacitance can then be given as 2 2

0 111 1

0 1 2 1

1

1 1( )

1 1b

x xC Cd dd d x d dC x

C C C d dd d

⎛ ⎞ ⎛ ⎞−− ⎜ ⎟ ⎜ ⎟⎛ ⎞ −∆ ⎝ ⎠⎝ ⎠= = =⎜ ⎟ +⎝ ⎠ + (4.8)

So, by comparing the two types of sensing, the ratio of the fractional change in

capacitance in case b to that in case ( a ) is given by

( )( )

0 1/

0 1

//

ba b

a

C C d drC C d

∆ −= =

∆

taking 1dd

α= , /11a brα

= − (4.9)

From the above equation it can be seen that for a large value of α , the loss in sensitivity

for case (b ) is minimized. However, this would result in large external sense comb-

holder, i.e., combl in Fig. 4.1. This results in large external sense-comb mass thus

worsening the cross-axis sensitivities as will be explained in the later sections. We

explain below, the process of systematic selection of a DaCM from a catalog, for the

accelerometer application.

4.3 Selection of a DaCM for an accelerometer

In this section, a DaCM is chosen for the accelerometer application from the catalog of

mechanisms introduced in Chapter 3. To select the appropriate DaCM, we use a figure of

merit for a mechanism based on the criteria relevant to the application. The figure of

merit was explained in Section 3.4.3. The criteria relevant to the accelerometer

application are net amplification ( NA ), cross-axis sensitivity ( crossk ), natural frequency


( f ) and maximum stress ( FS ). Of these criteria, NA and f depend upon the

suspension stiffness and inertia of the accelerometer. These are in turn dependent on the

fabrication process, which determines the geometrical parameters like the proof-mass’s

dimension, minimum in-plane width and out of plane thickness of the suspension.

Appendix I shows how restrictions on the feature sizes affect sensitivity and resolution of

accelerometers. In the following section, we select a DaCM for two accelerometer

applications with different fabrication requirements.

4.3.1 Selection of a DaCM for Chea et al.’s (2004) micro-g accelerometer

In this section, it will be shown that the sensitivity of a high-sensitivity in-plane

accelerometer can be improved by combining it with a DaCM. The accelerometer

consists of a 2 mm × 1 mm × 450 mµ proof-mass and eight suspension beams that are

750 mµ long and 3.5 mµ thick. The minimum feature size that can be realized by this

process is 3.5 mµ . The details of the mechanism are given in Table 4.3. This

accelerometer has been fabricated with a combination of surface- and bulk-

micromachining processes. Grooves along the proof-mass define capacitances with a

sense-gap of 1.1 mµ . The capacitance detection arrangement is of the type (a) shown in

the previous section with a sensitivity ( 0/C C∆ ) of around three.

The layout of the accelerometer with a DaCM is shown Fig. 4.1 with the blank

area representing the DaCM that is to be combined with the accelerometer. The sense

combs are placed at the output of the DaCM with an additional suspension as shown in

the Fig. 4.1. However, there needs to be a set of feedback combs at the proof-mass.

As mentioned in Chapter 3, the mechanisms obtained from literature are mainly

optimized for actuator applications and may not be suitable for sensor applications. For

the accelerometer application, it is the net amplification ( NA ) that determines the

sensitivity and thus the usefulness of the DaCM. So, these mechanisms need to be

optimized to obtain a high NA , adhering to the fabrication limitations. Since most of the

mechanisms were modeled using beam elements, optimization was performed with in-

plane beam widths ( iX ) as the design variables as shown in Fig. 4.5. It should be noted

that the topology and the area occupied by the mechanisms remain unchanged during


optimization. A comparison of optimized mechanisms would then be a comparison of

their topologies alone. The optimization formulation is stated as

Max NA ( iX ) given by Eq. 3.2

( )

2

2

( )

( ) ( )s co ci co s


m k k n k M nk kNA

k k k k n k k k F

+ +=

+ + + +

m

Subject to Equilibrium equations

And bounds max miniX X X< < (4.10)

To compute NA from Eq. 3.2 the following quantities were used.

M = Proof-mass = 4 mg

m = Sense-comb mass = 0.04 mg

sk = Sensor stiffness = 15 /N m

extk = External suspension stiffness = 0.01 /N m

minX = minimum beam width = 3.5 mµ

Figure 4.5 Optimization of the mechanisms for Chae et al.’s (2004) accelerometer

Since the primary aim of adding a DaCM is to increase the sensitivity, we exclude

for comparison, all mechanisms with NA below unity. Table 4.1 shows that five out of

the nine mechanisms have a value of NA greater than unity. These five mechanisms are

compared for various criteria shown in the Table 4.2. The values of each of the criteria

compared for a particular mechanism is normalized with respect to the maximum value

for that criterion. Each of the normalized values is then associated with a weight-factor

iXArea occupied by the mechanism is 2 mm × 2 mm


which defines the relative importance of the criteria for that application. A weighted sum

of all the criteria for a mechanism gives a figure of merit for the application as given by

Eq. 3.9. The higher the figure of merit, the better the mechanism.

Table 4.1 Net amplifications of the mechanisms optimized for the Chae et al. (2004) accelerometer

SI no. Mechanism no. Net Amplification

1 M1 1.68

2 M2 3.97

3 M3 0.93

4 M4 1.01

5 M5 0.35

6 M6 0.13

7 M7 2.28

8 M8 1.16

9 M9 0.04

Table 4.2 Weights associated with the four criteria to choose a DaCM for Chae et al. (2004) accelerometer

M1 M2 M4 M7 M8

nNA 0.42 1.00 0.25 0.57 0.29

nFS 1.00 0.09 0.31 0.11 0.43

nf 0.45 0.15 0.74 0.30 1.00

crossnK 0.00 0.00 0.01 0.00 1.00

Case 1 0.41 0.77 0.28 0.46 0.44

Case 2 0.34 0.63 0.30 0.40 0.57

Case 3 0.17 0.23 0.20 0.17 0.85

Weights nNA nFS nf crossnK

Case 1 0.75 0.05 0.10 0.10

Case 2 0.60 0.00 0.20 0.20

Case 3 0.20 0.00 0.20 0.60


It can be seen from Table 4.2 that M2 is the most preferable mechanism for the

application when higher sensitivity ( NA ) is important for the application while

mechanism M8 is preferable if cross-axis stiffness is deemed important. Furthermore, it is

shown in Table 4.5 that the addition of mechanism M2 with the accelerometer is able to

increase its sensitivity by more than a factor of three.

Table 4.3 Specifications of a high-resolution accelerometer by Chae et al. (2004)

Mass M 2000µ m long , 1000 µ m wide and

450 µ m thick weighing 2.07 mg

Stiffness sk Eight beams each 700µ m long,

40µ m wide and 3µ m thick. Spring

const. = 14.717 N/m

Natural

frequency

f 226.49 Hz

Sense gap g 1.1 µ m

Base capacitance 0C 8 pF

Change in

capacitance for

1g acceleration

0/C C∆ 3.91

Cross axis

sensitivity crossX 0.0011%

Table 4.4 Specifications of the M2 and M8 that are added to the accelerometer of Table 4.2

Mass, M 1.3483e-9 kg

Size of the mechanism 2000µ m × 2000µ m

Magnification -6.5 (M1), 5.8 (M2)

Thickness (out of plane) 3.5 µ m

Minimum width (in-plane) 3.5 µ m


Table 4.5 Performance of the accelerometer by Chae et al. (2004) of Table 4.3 integrated

with M2 and M8

Quantity M8 M2

Net amplification ( NA ) 1.165 3.97

Natural Frequency in Hz 255 230

Sensitivity per g 0/C C∆ 7.25 13.88

Cross axis sensitivity crossX 0.002% 0.05%

(a)

(b)

Figure 4.6. Deformed configurations of (a) M1 and (b) M2 that are combined with the accelerometer of Chae et al. (2004)

4.3.2 Selection of a DaCM for a bulk-micromachined accelerometer for the DRIE

process

The combination of surface- and bulk-micromachining as done by Chea et al. (2004) is

effective but complex. We have developed a relatively simple bulk-micromachining

process involving deep-reactive ion etching (DRIE) on silicon-on-insulator (SOI) wafer

(see Appendix II). An SOI wafer consists of a structural layer of 25 mµ thickness and a

base layer 250 mµ thickness. The structural layer is used to define the suspension and

the DaCM while the base layer is used to define the proof-mass. The minimum feature

size possible with this process is taken as 5 mµ . The details of the process are presented

in Appendix II. This process requires only DRIE process for defining both thin


suspensions as well as the thick proof mass. Furthermore, the tolerances and the

minimum dimensions are as high as 5 mµ as compared to 1 mµ in the process proposed

by Chae et al. (2004)

The layout of the accelerometer with a DaCM for the process is shown in Fig. 4.1

where the blank area represents the DaCM that is to be added. The sense capacitance

technique is of the type (b) where alternate moving electrodes are separated from each

other by a large distance. This type of sensing technique is chosen because two

subsequent static electrodes cannot be electrically isolated from each other with bulk-

micromachining alone.

Just as in the previous section, in-plane beam widths of the mechanism were

optimized for a high NA along with constraints on the fabrication process. The entire

mechanism is fitted into a rectangular box or area 2 mm × 2 mm . The formulated

optimization problem is shown below with reference to Fig. 4.5

Max NA ( iX ) given by Eq. 3.2

( )

2

2

( )

( ) ( )s co ci co s


m k k n k M nk kNA

k k k k n k k k F

+ +=

+ + + +

m

Subject to Equilibrium equations

And bounds max miniX X X< < (4.11)

To compute NA from Eq. 3.2 the following quantities were used.

M = Proof-mass = 4 mg

m = Sense-comb mass = 0.04 mg

sk = Sensor stiffness = 200 /N m

extk = External suspension stiffness = 1.5 /N m

minX = minimum beam width = 5 mµ

Table 4.6 shows that four out of nine mechanisms have a NA greater than unity.

Normalization and assigning of weights were performed to obtain a figure of merit for

each mechanism just as in the previous section. It is seen from Table 4.7 that mechanisms

M1, M2 and M8 can be chosen based on trade-offs between sensitivity ( NA ) and cross-

axis stiffness ( crossk ). In the next section, these mechanisms will be added on to a proof-


mass and suspension and optimized to obtain a high sensitivity for a given area of the

chip.

Table 4.6 Net amplifications of the mechanisms optimized for the DRIE with SOI accelerometer

SI no. Mechanism no. Net Amplification

1 M1 1.98

2 M2 4.25

3 M3 0.97

4 M4 0.56

5 M5 0.35

6 M6 0.17

7 M7 1.75

8 M8 1.20

9 M9 0.20

Table 4.7 Weights associated with the four criteria to choose a DaCM for the DRIE with SOI accelerometer.

M1 M2 M7 M8

nNA 0.47 1.00 0.41 0.28

nFS 1.00 0.09 0.12 0.44

nf 0.46 0.15 0.24 1.00

crossnK 0.00 0.00 0.00 1.00

Case 1 0.44 0.77 0.34 0.43

Case 2 0.37 0.63 0.29 0.57

Case 3 0.28 0.46 0.20 0.65

Weights nNA nFS nf crossnK

Case 1 0.75 0.05 0.10 0.10

Case 2 0.60 0.00 0.20 0.20

Case 3 0.20 0.00 0.20 0.60


4.4 Design of an accelerometer with a DaCM for a given chip area

From the previous section, it was found that mechanisms M1, M2 and M8 are suitable for

the proposed DRIE with SOI process. These mechanisms will be combined with a proof-

mass and suspension along with external combs and its suspension. The general layout of

such an accelerometer is shown in Fig. 4.1 where the blank region represents the DaCM

that is to be added. The combined accelerometer with a DaCM occupies certain area in

the chip. This area can be set based on fabrication requirements. Within a given chip area,

these is a need to optimize for the mechanism size, suspension stiffness and the proof-

mass size to obtain high sensitivity.

4.4.1 Suspension stiffness ( sk )

The suspension chosen for the accelerometer application has a folded-beam design as

shown in Fig. 4.7. One such suspension consists of two beams in series making its net

stiffness half the stiffness of a single beam. This kind of an arrangement is flexible in the

desired direction but stiff in the perpendicular in-plane direction. We use four such

beams, two on either side of the proof-mass as shown in Fig. 4.7. If suspl is the length of

the beam, suspb the in-plane width and 1h its thickness, the total stiffness, assuming

linear-beam theory, is given by 3

13

2 susps

susp

Eb hk

l= (4.12)

where E is the Young’s modulus of silicon. suspb and 1h are determined by the

fabrication procedure. In this case, suspb = 10 mµ and 1h = 25 mµ . The variable suspl

needs to be determined.

4.4.2 Proof mass ( M )

The proof mass M is determined by its geometrical quantities i.e., its length ( massl ),

width ( massb ) and thickness ( 2h ) shown in Fig. 4.7 and Fig. 4.1:

2mass massM l b h ρ= (4.13)


where ρ is the density of silicon. The thickness of the proof-mass is fixed by the

fabrication process to 250 mµ . The quantities massl and massb are determined by the total

area occupied by the mechanism. This is explained in the next section.

Figure 4.7 Proof-mass and the suspension of the accelerometer

4.4.3 Size of the mechanism ( mechl )

The attributes of the DaCM that are dependent on its size are its input stiffness cik , output

stiffness cok , input inertia im , output inertia om and inherent amplification n . The

method of extracting these parameters form the finite-element model of the mechanism

was discussed in Chapter 3.

The size of the mechanism is represented by the width of a rectangular box that

completely contains it. This is considered as the characteristic length, mechl , of the

mechanism. For all the mechanisms compared in Chapter 3, the length of the rectangular

box is twice the width as shown in Fig. 4.1. It was further shown in section 3 that cik and

cok can be approximated as a function of the characteristic length mechl as shown in Eq.

4.14

3 2cimech mech mech

a b ckl l l

= + + and 1 1 13 2co

mech mech mech

a b ckl l l

= + + (4.14)

where the constants a , b and c depend upon the Young’s modulus of the material, in-

plane widths of the beams that make up the mechanisms and out of plane thickness which

suspb

suspl Proof mass

massb

massl


are fixed by the fabrication process. It was further shown in Chapter 3 that the inherent

amplification does not vary with the size of the mechanism and can be considered

constant. The inertia parameters im and om can be considered negligible when compared

to the proof-mass M and sense-comb mass m .

The size of the mechanism mechl has to be optimized for a given area.

4.4.4 Optimization of the accelerometer for a given chip-area

The entire accelerometer with the DaCM can be made to fit into a rectangular area shown

in Fig. 4.1 with length 2 mechl and width ( mech massl l+ ). The total area is then given by

2 ( )mech mech massArea l l l= + (4.15)

For a given Area , massl can be determined from the above formula as

2( 2 )2

mechmass

mech

Area lll−

= (4.16)

The width of the proof-mass ( massb ) can be expressed in terms of the mechanism length

( mechl ) and suspension-length ( suspl ) with reference to Fig. 4.1 as

2( )mass mech suspb l l= − (4.17)

We have now expressed all the quantities needed for calculating output displacement

given by Eq. 3.1 in terms of two geometric parameters mechl and suspl . By substituting the

values of all these quantities from Eqs. 4.12-4.17 into Eq. 3.1 for cik , cok , M and sk ,

we get

( )3 2

1 223

2 3 31 12

3 3

2 ( 2 )( ) ( )

2 2( ) ( )

susp mechco ci mech susp ci

susp mech

susp suspco ci ext co ci

susp susp

Eb h Area lm k n k l l h nkl l

xEb h Eb h

k k k n k kl l

ρ−+ + −

=+ + + +

m

(4.18)

where cik and cok are given by Eq. 4.14

In the above formula, the values for the sense-comb mass m and external-suspension

stiffness extk are as used in Section 4.3.2. Figures 4.8a-c show the optimum value for the

suspension stiffness and the mechanism size to obtain a high sensitivity for a given area

occupied by the mechanism. The area occupied by the entire accelerometer ( Area ) is


fixed to 25 2mm . Form the optimized value of mechl and suspl , the proof-mass dimensions

can be obtained from Eq. 4.16 and Eq. 4.17.

4.5 Design of sense combs and external suspension

The need for having sense combs at the output of the DaCM was explained in Section

4.1.1. It was then argued that the inertia due to the sense combs at the output causes

sufficient cross-axis displacement. Thus the design of sense-combs and the external

suspension requires a sufficient understanding of the cross-axis sensitivity at the output of

the DaCM.

4.5.1 Cross-axis Sensitivities

Cross-axis sensitivity is defined as the ratio of the displacement of the output of the

accelerometer when the acceleration is applied in the cross-axis direction to the

displacement of the accelerometer output when the same acceleration is applied in the

desired direction. It arises due to a moment at the output of the DaCM which tends to

sway the combs in the perpendicular direction. With reference to Fig. 4.9, the cross-axis

sensitivities can be expressed as

100crosscross

des

xXx

= × (4.19)

This definition of the cross-axis sensitivity requires us to minimize the ratio of crossx and

desx and not just crossx . For example, adding a suspension at the sense-combs may

increase the stiffness in both the desired as well as cross-axis directions by equal

proportions thus maintaining the ratio of the displacements constant. Thus, the external

suspensions have to be designed such that the sensitivity in the desired direction is not

affected.


(a)

(b)


(c)

Figure 4.8 The optimized mechanism size and suspension-length of accelerometer with DaCMs (a) M1 (b) M2 and (c) M3

(a) (b) Figure 4.9 Cross-axis sensitivity (a) The displacement of the output of the mechanism M1 with the sense-combs in the desired direction and (b) displacement of the output of the mechanism M1 in the cross-axis direction

4.5.2 Design of the sense combs

The sense-comb’s mass is determined by the number of comb-fingers needed to obtain

sufficient sense capacitance and also the type of the capacitance detection scheme. The

sense capacitance value should be more than the parasitic capacitance of the electronic

circuit, which for most circuits is around 0.5 pF . Therefore, the sense capacitance

desx

crossx


chosen for this application is 1 pF . The number of comb-fingers needed for this is

determined by Eq. 4.20.

0 1

0

Nbhd

ε = 1 pF (4.20)

Taking permittivity of air 120 8.854 10ε −= × /F m , thickness of the electrode 1 25h mµ=

and sense gap between electrodes 0 5d mµ= , we can fix the number of combs N and the

length of each comb b . By noting that 42.2589 10Nb = × , we fix 40N = and

700b mµ= . The sense comb arrangement at the output of the DaCM is shown in Fig.

4.10. It consists of a comb holder of length combl which acts as a stem and all the combs

branch out from it. The holder has a grillage structure with cross beams, which gives it

large stiffness with a small mass.

The capacitance detection method is of the type (b) shown in Section 4.2. In this

arrangement the positively and the negatively charged static electrodes are on either side

of the proof-mass. One pair of differential capacitance consists of a movable and static

electrodes separated by a sense gap of 0d = 5 mµ . The movable electrode of one pair and

the static electrode of the other pair are isolated from each other by a large distance

1 0d dα= , where 1α > as shown in Fig 4.3, Section 4.2. Larger the value of α , the

greater the sensitivity as given by Eq. 4.9. However, a large α also increases the comb-

holder’s length because of the large spacing. This leads to large cross-axis sensitivity as

well as reduction in bandwidth. Figure 4.11a-b show how the cross-axis sensitivity and

the natural frequency of the structure vary with the length of the comb holder combl . The

value of α for this application is taken to be 10. This results in a 1% loss in sensitivity as

shown in Eq. 4.9

It can be seen from Fig. 4.11 (a) that cross-axis sensitivities are above 1% for all

the mechanisms without an external suspension. To bring these cross-axis sensitivities

down, an external suspension is added. The effect of the suspension in terms of its

optimum location is studied in the next section.


Figure 4.10 The sense capacitance at the output of the DaCM.

(a)

combl

Static combs

Moving combs

Comb holder

0d

1d

b


(b)

(c)


(d)

Figure 4.11 Effect of the sense-comb size and mass on the (a) Cross-axis sensitivity without external comb suspension (b) Natural frequency (c) cross-axis sensitivity with external suspension and (d) Sensitivity of the capacitance detection circuit.

4.5.3 External sense-comb suspension

Figure 4.12 shows the sense-combs with an external suspension. The suspension should

be stiff in the cross-axis direction but flexible in the in-plane direction. The external

suspension is thus made of two folded beams on either side of the sense-combs. The

stiffness of the beams should be such that the ratio of the displacement of the cross-axis

direction and the in-plane direction should be as small as possible. The stiffness of the

external suspension, given by Eq. 4.21 is determined by its length 2suspl as its in-plane

width 2suspb and out-of-plane thickness 1h are fixed by the fabrication process to 5 mµ

and 25 mµ respectively. 3

13

2

suspext

susp

Eb hk

l= (4.21)

Figure 4.13 shows the resolution and cross-axis sensitivity for different beam lengths

( 2suspl ). It is seen that as 2suspl increases, the cross-axis sensitivity decreases and the

resolution becomes better except for DaCM M8 shown in Fig. 4.13 (d) where an


optimum length of 2suspl gives minimum cross-axis sensitivity. Figure 4.14 with

reference to Fig. 4.12 shows the optimum position ( posl ) of the external suspension to

reduce cross-axis sensitivity. It is seen that the suspension needs to be placed at the end

of the sense-combs for minimum cross-axis sensitivity. This means that large values of

α are possible leading to small losses in sensitivity as given by Eq. 4.9. The suspension

length 2suspl for all designs is taken to be 1000 mµ .

Figure 4.12 Sense combs with the external suspension. posl indicates the position of the suspension along the comb holder.

(a)

comblposl

2suspl

2suspb


(b)

(c)

Figure 4.13 Effect of the external suspension length 2suspl on the cross-axis sensitivities and resolution of accelerometers with DaCMs (a) M1 (b) M2 and (c) M3


Figure 4.14 A plot of cross-axis sensitivity with respect to the position of the external suspension posl

4.6 Analysis of the accelerometer designs

In the previous sections, various parameters of the accelerometer with the DaCM were

studied and they were optimized to obtain high sensitivity and limit undesirable effects.

The three designs with DaCMs M1, M2 and M8 were optimized based on the proposed

methods. The complete designs are shown in Fig. 4.15 (a), (b) and (c). These designs are

analyzed using linear Euler-Bernoulli finite-element beams programmed in MATLAB

and evaluated for sensitivity along the desired axis, cross-axis sensitivity, natural

frequency, and other quantities important for the accelerometer application. It is seen

from Table 4.8 that the design with DaCM M2 has high sensitivity while the design with

DaCM M8 has lower cross-axis sensitivity, higher pull-in voltage and higher natural

frequency. The value of resolution mentioned in table is the open-loop resolution

assuming an electronic circuit with resolution capability of 10 parts per million.


Table 4.8 Analysis of accelerometers with DaCMs M1, M2 and M8

Quantities M1 M2 M8

Area occupied by the structure in 2mm 25 25 25

Base-capacitance ( 0C ) in pF 1.2 1.2 1.2

Sense-gap in mµ 5 5 5

Open-loop Resolution (in gµ ) 23.5 6.25 30

Cross-axis sensitivity (expressed as %

of the desired axis sensitivity)

0.01 0.05 0.007

Natural frequency in Hz 771 375 940

4.7 Closure

In this chapter, DaCMs has been chosen for the accelerometer application based on the

selection criteria proposed in Chapter 3. These DaCMs are combined with a proofmass,

suspensions and sense combs. The designs are further optimized for high sensitivity in

the desired direction and low cross-axis sensitivity. They are analyzed using linear-beam

elements. Designs with open-loop resolution of 6 gµ and cross-axis sensitivities of less

than 0.01% have been proposed.

The designed accelerometers need to operate in a force-rebalance mode. So, the

system modeling of the accelerometer structure with the electronic capacitance detection

needs to be done to design the feed-back combs and also characterize the accelerometer’s

closed-loop performance. This is done in Chapter 5. The above designs need to be further

evaluated using continuum elements and eventually using prototyping to confirm their

resolution and cross-axis sensitivities.


(a) (b)

(c)

Figure 4.15. Accelerometer designs with DaCMs (a) M1 (b) M2 and (c) M8

5.1

Chapter 5

5 SYSTEM LEVEL SIMULATION OF A MICRO-g

BULK-MICROMACHINED ACCELEROMETER

Summary

This chapter deals with the closed-loop system-level simulation of the accelerometer

along with the capacitance-measurement circuit components. The mechanical

components that consist of the proof-mass, suspension, and the DaCM, are modeled by

mode-summation method. The electronics consists of a modulator, an amplifier, and a

demodulator whose output is the open-loop readout of the accelerometer system. For an

accelerometer that operates in a force re-balance mode, the output signal from the

electronic circuit is fed to a PID controller. This in turn is applied as an electrostatic

feedback force on the proof-mass and the sense-combs. The output of the PID controller

is the measure of the applied acceleration. System simulation done on the accelerometers

designed in Chapter 4 were found to have sensitivities of 0.125 V/mg, 0.445 V/mg, and

0.07 V/mg and a resolution of 40 gµ , 20 gµ , and 70 gµ respectively with a circuit

noise of 2 mV.

5.1 Introduction

An accelerometer with a DaCM has been designed in the previous chapter. The open-

loop sensitivity of the accelerometer was ascertained. However, it was stated in the

literature review that high-sensitivity accelerometers operate in the closed loop (force re-

balance) mode to obtain a high dynamic range and bandwidth. Thus, in this chapter, the

closed-loop response of the accelerometer with a DaCM is studied. We describe the

Chapter 5: System-level simulation of a Micro-g Bulk-micromachined accelerometer 5.2

various components of the accelerometer and the way they are modeled at the system-

level.

The accelerometer-system components can be classified as open-loop and closed-

loop components. Open-loop components consist of the mechanical structure and

electronic capacitance-detection circuitry. The mechanical components consist of the

proof-mass, suspension, and the DaCM. They are analyzed using mode-summation

method by considering the first four natural modes. The electronic circuitry consists of

sense-combs, capacitance-detection circuit, amplifier, filters, and various noise-reduction

components. The detailed modeling of the electronic circuitry is discussed by Kshirasagar

(2006) and is beyond the scope of this thesis. In the system-level simulation done in this

chapter, ideal behavior is assumed ignoring noise and second-order effects for individual

circuit components. However, the overall estimate of the noise is considered at the output

of the accelerometer.

When the accelerometer is operated in the force re-balance mode, the output of

the open-loop electronic circuitry is fed back as an electrostatic force to the proof-mass

and sense-combs. This force counter-balances the inertial force and maintains the proof-

mass and the sense-combs in the stationary position. However, there will be a slight delay

between the applied inertial force and the feedback signal leading to incomplete

stabilization of the proof-mass and sense-combs. A PID controller is thus used to

minimize the delay time, and improve the overall dynamics of the system (Kraft, 1997).

In short, the design of the closed-loop accelerometer requires additional feedback combs

at the proof-mass and sensing port, and a PID controller, in conjunction with the open-

loop components.

The system-level simulation is done in Simulink toolbox of Matlab. Furthermore,

various noise sources in the system are identified and simplified models are used to add

them into the system-level simulation. We now explain each of the components used in

the system-level simulation of the accelerometer.

5.2 Mechanical components: Mode-Summation Method

Mechanical structures have continuous distribution of stiffness and inertia. The dynamics

of each point of such a structure can be captured by representing the continuum as a


system with infinite degrees of freedom. However, discretizing the continuum by using a

finite element mesh approximates it to a finite N degree-of-freedom system. The

equation of motion of such a system is represented by

M x + Dx + K x =f&& & (5.1)

where K , M and D are the N N× non-singular symmetric stiffness, mass, and

damping matrices, respectively, which are assembled from the discretized finite element

model. And, x is the 1N × vector containing the displacements of all the degrees of

freedom and f the 1N × force vector which may vary with time. Integrating the above

system of N coupled equations is computationally expensive if N is large. However,

the number of degrees of freedom for the system can be reduced by the mode-summation

method. Each mode in the mode-summation method corresponds to a single decoupled

degree of freedom when D is appropriately chosen. It is represented by a spring-mass-

damper as shown in Fig. 5.1 (a) and its Simulink representation is given in Fig. 5.1b. The

various modes are obtained by solving the eigenvalue problem arising out of Eq. 5.1

without considering D and f .

K U = ΛMU , (5.2)

where Λ is the diagonal eigenvalue matrix and U is the modal matrix whose columns

correspond to the mode shape of each mode. In the mode-summation method, we convert

the N degree-of-freedom system into a system with fewer degrees-of-freedom. This is

done by transforming the mass and the stiffness matrices by the using the modal vectors

corresponding to the first four modes given by

Td d dU KU = K and T

d d d=U MU M (5.3)

where dU is a matrix of size 4N × that consists of the modal vectors corresponding to

the first four modes, dK and dM are 4 4× diagonal stiffness and mass matrices

corresponding to the four uncoupled modes used for analysis. The force vector f , which

is of size 1N × can also be converted to df of size 4 1× by the following transformation

1T .

1T

d d⇒ =T f U f (5.4)


(a)

(b)

Figure 5.1 A conventional accelerometer represented as a spring-mass-dashpot system and its equivalent in Simulink The same transformation is applied to the damping matrix too. With these four modal

parameters ( dM , dK , df and dD ) independent transient analysis is performed to obtain

the state variable corresponding to each mode given by , 1 4iy i = L . These individual

modal displacements are converted back into the N -dimensional space by the following

transformation.

2 d⇒ =T u U y (5.5)

where u is the 1N × displacement vector and y is the 4 1× modal displacement vector.

The displacement at the desired point of the structure can be extracted form the vector u .

This superposition is summarized in Fig. 5.2. In the representation below, , 1 4if i = L

K

Mx

D


are components of df vector. , 1 4riU i = L is given by ( , )dU r i , as we are interested at

the displacement at the thr degree of freedom. , 1 4iK i = ⋅⋅⋅ , , 1 4iM i = ⋅⋅⋅ and

, 1 4iD i = ⋅⋅⋅ are the diagonal components of the dK , dM and dD respectively.

Figure 5.2 A state space (Simulink) representation of the mechanical structure using modal-summation method

Considering only a few modes is a valid approximation because only these modes

have maximum contribution towards the displacement of the structure for low excitation

frequencies. For example, if all other natural frequencies are very far away from the first

natural frequency, the entire system can be approximated as a single degree-of-freedom

system corresponding to that mode. However, in the case of an accelerometer with a

DaCM, only a few kHz separate the first few natural frequencies from the next. Thus, the

superposition of the first four modes is considered for the mode-summation method to

analyze the dynamics of the accelerometer with a DaCM. The modal parameters, as

explained above, are calculated form the mass, stiffness, and the damping matrices.

Transformation 1T

Transformation2T

Mode 1

Mode 2

Mode 3

Mode 4


While the mass and the stiffness matrices are determined by the geometry and

mechanical properties of the structure, damping depends upon the interaction of the

structure with the surroundings. In the next sub-section we explain how the damping

parameters in the system are evaluated.

5.2.1 Calculating the Damping Coefficients

The accelerometer with a DaCM is a two degree-of-freedom system as shown in Figs.

3.2a-b. Thus, damping is experienced both at the sense-comb end as well as the proof-

mass end. Damping at the sense-comb end mainly arises due to squeezed-film effect of

the gas between the static and the moving combs. On the other hand, the damping at the

proof-mass end is contributed both by the squeezed-film effect and the Couette flow of

air between the mass and the substrate. The squeezed-film damping coefficient is given

by (Senturia, 2001) 3 3/sqd n b l hµ= (5.6)

and the Couette flow damping coefficient is given by

proofCf

Ad

hµ

= (5.7)

where µ = viscosity of air = -5 21.735 10 - /N s m×

n = no. of comb fingers = 40.

b = width of the comb fingers = 25 mµ

l = length of the fingers = 700 mµ

h = gap between the fingers = 5 mµ

proofA = Area of the proof mass =3000×2000 2mµ

The squeezed-film damping coefficient sqd is calculated to be -56.0725 10 /N s m× −

while the coefficient arising out of the Couette flow has a value of -52.1 10 /N s m× − .

Thus the damping coefficient at the sense-comb end is

-5 6.0725 10 /sc sqd d N s m= = × −

while at the proof-mass end is

-58.2 10 /pm sq cfd d d N s m= + = × − .


5.2.2 Mechanical noise in the system

The source of mechanical noise was discussed in Section 2.3.4. It is further seen from Eq.

2.12 that the spectral density of the force arising out of Brownian motion is proportional

to the square root of the damping coefficient. Owing to the different damping coefficients

at the proof-mass and the sense-comb ends, the spectral density of noise acting at these

ends are different and are given as follows.

The spectral density of the noise sources that act on M and m are given by

4n pm B pmF K Td− = and 4n sc B scF K Td− = (5.8)

where, BK = Boltzman’s constant = 1.238e-23 1JK −

T = Temperature in Kelvin = 300 K

pmd = proof-mass damping co-efficient = -56.0725 10 /N s m× −

scd = Damping co-efficient at the sense-comb end = -56.0725 10 /N s m× −

n pmF − = -121.15 10 /N Hz× .

n scF − = -121 10 /N Hz×

For a given acceleration sA acting on the system, Eq. 3.1 of Chapter 3 gives the

displacement of the output. For a noise signal of spectral density given by Eq. 5.8 acting

on the proof mass M and the sense-comb mass m , the displacement is given by

2

noise 2

4 ( )( ) ( )

B sc s co ci co

co s ci ext co ci s

K Td k k n k nkx

k k k k n k k kα+ + ±

=+ + + +

(5.9)

where /pm scd dα = . Thus, the signal to noise ratio ( SNR ) is given by

( ) 22 222

2

( )4 ( )

s co ci cos

noise B sc s co ci co

M k k n k m nkAxSNRx K Td k k n k nkα

⎛ ⎞+ +⎛ ⎞ ⎛ ⎞= = ⎜ ⎟⎜ ⎟ ⎜ ⎟ + +⎝ ⎠ ⎝ ⎠⎝ ⎠

m

m (5.10)

The mechanical components and its system-level representation have been

discussed above. The displacement of the sense-combs produces a change in capacitance

as shown in Section 4.2. In the next section we explain how the change in capacitance

due to the displacement of the sense-combs is detected and converted to a measurable

output voltage.


5.3 Capacitance detection circuit

Figures 5.3 and 5.4 shows the differential capacitance in a bridge arrangement with

capacitance sC . The output voltage for such an arrangement is proportional to / sC C∆ as

shown in Section 4.2. This capacitance change is modulated with a high frequency pulse

of opposite phase, shown as mnV and mpV respectively. This modulation voltage mV

applied to the sense-combs is around 5 V at 1 MHz frequency. This modulation

attenuates the low-frequency 1/ f noise of the amplifier. The modulating voltage on the

sense-combs causes an electrostatic force at the sensing end of the accelerometer.

However, the mechanical structure acts as a second-order filter, filtering out any force at

frequencies well beyond its resonance frequency, which is around 1 kHz. Thus, it is safe

to neglect the effect of the mechanical force exerted by the high-frequency sensing

voltage on the sense-combs. The output of the differential-capacitance arrangement is

given by a pulse of 1 MHz and voltage diffV shown below.

2m

diffs

VCVC∆

= (5.11)

The amplifier then amplifies this modulated signal with special care taken to limit the ac

and the dc offsets. The gain of the amplifier A is around 100. The amplified signal is

then demodulated with a pulse of the same frequency. During this demodulation, all the

dc offsets are converted to high frequency noise. The demodulated signal is then passed

though a low-pass filter to eliminate the high-frequency noise. This type of noise-

reduction technique is called as chopper stabilization (Kshirasagar, 2006). The net output

of the circuit is given by

4m dm

outs

V VCV AC∆

= (5.7)

where dmV is the demodulating pulse whose value is 2 V. The details of the electronic

circuits along with the system-level simulation of the electronic components in T-Spice

are presented by Kshirasagar (2006).


Figure 5.3 Block diagram of the capacitance detection circuit (Boser et al., 1994)

Figure 5.4 Simulink representation of the electronic capacitive sensing circuit

The resolution of the electronic circuitry depends upon the noise in the circuit.

The various noise sources in the circuit include the Johnson’s noise in the resistors, 1/ f

noise in the amplifiers, and dc offsets. The detailed modeling of noise is carried out in T-

Spice in (Kshirasagar, 2006). The electronics circuit was found to resolve better than 10

parts per million. Since the base capacitance is close to 1 pF , a minimum change in

Differential Capacitance

Capacitance Detection Circuit


capacitance of up to 10 aF can be detected. In the above circuit, this translates to an

output voltage noise of 2.5 mV . This means that any acceleration signal, which produces

a voltage change of more than 2.5 mV can be detected. The value of this acceleration is

the open-loop resolution of the accelerometer. The open-loop resolutions of the three

accelerometer designs are given in Table 4.8. Their open-loop sensitivities are 0.125

V/mg, 0.445 V/mg, and 0.075 V/mg respectively. In the next section, we study the

closed-loop response of the accelerometer.

5.4 Closed-loop response

In force re-balance mode of operation, the output of the open-loop system is fed in as a

force to the proof-mass as well as the sense combs. This is accomplished by applying the

output voltage on a set of feedback combs at both the proof-mass end and the sensing

end. This force counter-balances the inertial force, thus limiting the deflection of the

proof-mass and sense-combs to a very small value. Operating in the force re-balance

mode increases the bandwidth, linearity and the dynamic range of the entire system as

shown in Section 2.4.3. Since the output of the open loop circuit is directly fed as a force

to the proof mass, it would naturally lag with the applied input acceleration signal, which

is to be measured. This would lead to incomplete stabilization of the proof-mass and the

sense-combs. Thus, to reduce the time delay between the applied feedback and the input

acceleration, a PID controller needs to be incorporated. The output of the PID is the

measure of the input acceleration.

The design of the feedback combs and the PID controller is explained in the next

section.

5.4.1 Feedback combs

The DaCM with a proof-mass is a two degree-of-freedom system and thus requires

stabilization of both the proof-mass as well as the sense combs. The proof-mass and the

sense-combs are subjected to feedback force by applying the output voltage of the

electronic circuit to a set of feedback combs. However, it is to be noted that forces

produced by the voltage applied between two electrodes is always attractive in nature.


Assuming an area of overlap of A and a sense gap of d , the electrostatic force between

two electrodes is given by 2

02el

AVFd

ε= (5.12)

It can be seen from the above equation that the electrostatic force varies nonlinearly with

the applied voltage. To linearize the force vs. voltage relationship, the arrangement

shown in Fig. 5.5 is used. Here, a dc bias is applied to two static combs so that the net

force on the moving comb is zero. The feedback voltage is then added to the dc bias on

one comb and subtracted from it in the other comb.

Figure 5.5 The feedback combs with dc bias

The feedback force is given by

2 2

0 02 2

( ) ( )2( ) 2( )

fb b fb fb b fbfb

An V V An V VF

d x d xε ε+ −

= −− +

(5.13a)

Assuming d x>> , the above equation simplifies to

02

2 fb b fbfb

n AV VF

dε

= (5.13b)

where fbn is the number of feedback combs. It can be seen from the above equation that

the feedback force is proportional to the voltage fbV . It is further seen from Eq. 5.13a that

the maximum value of the feedback voltage fbV is the bias voltage bV . This is because

the second term of the right-hand side of the equation becomes zero at this voltage. This

determines the operating range of the accelerometer. Care should be taken to see that the

dc bias bV is far from the pull-in voltage of the structure. Since, the maximum voltage

applied on combs (as seen from the left set of feedback combs in Fig. 5.5, when fb bV V= )

b fV V+

b fV V− d x−d x+


is 2 bV , the bias voltage bV should be less than half the pull-in voltage. The accelerometer

with a DaCM has a stiff proof-mass end with very flexible sense-comb end. Thus, the

pull-in voltages for the proof-mass p pmV − and the sense-comb end p scV − are different.

They are given by the formula: 3

0

0

827

inp pm

pm

k dVn Aε− = and

30

0

827

outp sc

sc

k dVn Aε− = (5.14)

where pmn and scn are the number of feedback comb fingers at the proof-mass end and

the sense-comb end, respectively. Furthermore, A is the area of overlap between the

comb fingers, 0d is the gap between the two combs, and ink and outk are the stiffnesses

at the input and the output ends of the accelerometer, which are given by (see Fig. 3.2 in

Chapter 3) 2

( ) co extin s ci

co ext

n k kk k kk k

= + ++

and 2

2

( )( )s ci ext co co extout

s ci co

k k k k n k kkk k n k

+ + +=

+ + (5.15)

The number of comb-fingers in the proof-mass and the sense-comb end along with the

bias voltage and pull-in voltage are given for the three designs in Table 5.1. The length of

overlap for each comb in the proof-mass end is taken to be 500 mµ and that on the

sense-comb side is taken to be 250 mµ .

Table 5.1 Details of the feedback combs for the three designs discussed in Chapter 4

Accelerometer design

Feedback combs in the proof-mass

pmn

Feedback combs in the sense-mass

scn

Proof-mass bias voltage

b pmV − in V

Proof-mass bias voltage

b scV − in V

Pull-in voltage at the sense-end p scV − in V

M1 (Fig.

4.16(a))

10 1 5 5 20

M2 (Fig.

4.16(b))

16 6 5 3 9

M8 (Fig.

4.16(c))

27 2 5 5 40


The bias voltages shown in Table 5.1 are well below half the pull-in voltage. The range

of the accelerometer is given by the acceleration which gives an output voltage equal to

the bias voltage. This is given by

b scrange

Vasensitivity

−= (5.16)

For the above three designs and the bias voltage given by Table 5.1, the maximum

detectable acceleration is 40 mg , 6 mg and 71 mg respectively. The small working

range of the accelerometer is due to the high circuit sensitivity given in Table 5.2. This

implies that output voltage of the accelerometer becomes equal to the bias voltage for a

small value of acceleration. The working range of the accelerometer can be improved by

decreasing the circuit gain so as to permit large working range as given by Eq. 5.16.

However, this leads to higher noise levels in the circuit thus limiting the resolution. The

working range can also be increased by increasing the dc bias voltage b scV − . But, this

voltage is limited by the pull-in at the sense-comb end. It can be seen that the most

sensitive design has the least range. Thus, there is a clear trade-off between the sensitivity

and the working range of the accelerometer.

The next section deals with the design of the PID controller for the closed-loop

accelerometer.

5.4.2 Design of the PID controller (Kraft, 1997)

Before designing a PID controller, various components of the accelerometer are

represented as a transfer function in the s-domain. Assuming small deflections, as is done

throughout the chapter, the simplified mathematical model of the closed-loop analog

accelerometer is shown in Fig. 5.6. Using a single normal mode approximation for the

mechanical structure, it can be represented by a simple second order system of the form

shown below. Transfer function of the mechanical components

2( ) fGTm s

ms ds k=

+ + (5.17)

where fG signifies the gain of the mechanical amplification. The entire electronic

circuitry can be considered as a gain cG if the dynamics of the op-amp are ignored. The

PID controller can be represented by


IPID p d

kT k k ss

= + + (5.18)

For small deflections, the feedback force can be represented by a gain fbG . The transfer

function of the system is represented symbolically in Fig. 5.7

Figure 5.6 Simplified system design of the closed loop accelerometer

Figure 5.7 Mathematical model of the analog closed loop accelerometer

In Fig. 5.7, a is the applied acceleration, f is the conversion factor of acceleration to

force. The open loop transfer function is given by 2

3 2

( )m c p i dOP

fG G k s k k sF

ms ds ks+ +

=+ +

(5.19a)

The closed loop transfer function is given by

f 2( ) mm

GT sms ds k

=+ + cG

IPID p d

kT k k ss

= + +

fbG

+ - a


2

3 2

( )( ) ( )

m c p i dCL

fb m c d fb m c p fb m c i

fG G k s k k sF

ms d G fG G k s k G fG G k s G fG G k+ +

=+ + + + +

(5.19b)

From the denominator of the expression in Eq. 5.19b, it can be seen that coefficients of s

and 2s which signify stiffness and damping respectively have been increased due to

feedback. From Fig. 5.8, it can be seen that the increasing ik increases the sensitivity but

does not affect the bandwidth. From Fig. 5.9, it can be seen that increasing pk increases

the system gain thereby increasing the bandwidth. However, this makes the system

unstable because the phase margin decreases. A small pk and a high ik would increase

the sensitivity as well as the stability of the system. For this reason 2pk = and 1500ik =

was chosen for the simulations. Figure 5.10 shows the combined accelerometer system

with the electronics. Figures 5.11- 5.13 show the closed-loop resolution of the three

designs of accelerometers, M1, M2 and M8, for a step pulse applied. It can be seen in

Figs. 5.11b, 5.12b and 5.13b, the proof-mass displacement comes to undeformed position

soon after the acceleration is applied. Figs. 5.11c, 5.12c and 5.13c show the output of the

PID controller

Figure 5.8 Bode plot of the open loop transfer function for design M1 for increasing integral gain ik .

Mag

nitu

de in

dB

Ph

ase

in d

eg

Frequency in rad/s


Figure 5.9 Bode plot of the closed loop transfer function for increasing proportional gain pk .

Table 5.2 Specifications of the three designs of the accelerometer.

Quantities M1 M2 M8

Base-capacitance ( 0C ) in pF 1.2 1.2 1.2

Sense-gap in mµ 5 5 5

Cross-axis sensitivity (expressed as % of the

desired axis sensitivity)

0.01 0.05 0.007

Natural frequency in Hz 771 375 940

Open-loop sensitivity in /V mg 0.125 0.445 0.075

Open-loop Resolution (in gµ ) 23.5 6.25 30

Closed-loop Resolution (in gµ ) 40 20 60

Pull-in voltage at the sense-comb end in V 20 9 40

Maximum working range in mg 40 6 71

Log10( SNR ) for 1 gµ acceleration 8.1 15.2 15.3

Increasing pk

Phas

e in

deg

M

agni

tude

in d

B

Frequency in rad/s


Figure 5.10 Complete system representation of the accelerometer and the electronics


(a)

(b)


(c)

Figure 5.11 Closed loop response of accelerometer M1 .(a) A step signal of 40 gµ at 0.1 s (b) Output at the PID controller (c) Displacement of the sense-combs

(a)


Figure 5.12 Closed loop response of accelerometer M2 .(a) A step signal of 20 gµ at 0.1 s (b) Output at the PID controller (c) Displacement of the sense-combs


(a)

(b)


(c)

Figure 5.13 Closed loop response of accelerometer M8. (a) A step signal of 60 gµ at 0.1s (b) Output at the PID controller (c) Displacement of the sense-combs

5.5 Closure

In this chapter, the closed loop system-level simulation of the accelerometers designed in

Chapter 4 was performed. The three designs were found to have a sensitivity of 0.125

V/mg, 0.445 V/mg, and 0.07 V/mg; a resolution of 40 gµ , 20 gµ , and 70 gµ ; and a

working range of 40 mg , 6 mg , and 70 mg , respectively. Modal summation method

was used to model the mechanical components. Mechanical noise arising out of

squeezed-film damping was modeled. The electronic components which included the

capacitance-detection circuit, an amplifier gain, and a PID controller were modeled in

Simulink assuming ideal behavior. The actual electronic circuit incorporates various

techniques such as chopper stabilization and high-frequency modulation of the sense

signal was used to reduce electronic noise.

The working range of the accelerometer was found to be low. This calls for

reducing the sensitivity of the electronic circuit, and at the same time keeping the noise

level under control.

6.1

Chapter 6

6 TOPOLOGY OPTIMIZATION OF DaCMs FOR

SENSORS

Summary

This chapter deals with using topology optimization to synthesize new DaCMs for sensor

applications. Topology optimization of DaCMs requires various nonlinear constraints

such as those on the cross-axis sensitivity and the natural frequency. These constraints

are taken care of by sequential linearization in the optimality criteria method.

Furthermore, topology optimization for the accelerometer problem is attempted by using

objective functions and constraints derived from the spring-mass-lever model proposed in

Chapter 3. The topologies thus obtained, were modified to comply with the fabrication

constraints.

6.1 Introduction

It was pointed out in Chapter 3 that most of the DaCMs available in the literature were

designed for actuator applications. It was concluded that sensors required certain criteria

such as net amplification ( NA ), natural frequency, and cross-axis stiffness which were

not important for actuators. Thus, when the existing mechanisms were compared for

various criteria important for sensor applications, it was observed that no mechanism

gave a favorable value for all the criteria. Furthermore, it was noted that mechanisms

from the literature lacked sufficient cross-axis stiffness. Thus topology optimization is

used to synthesize new DaCMs for sensor applications with constraints on cross-axis

sensitivity and natural frequency.

Chapter 6: Topology Optimization of DaCMs for sensors 6.2

In this chapter, we aim at achieving two objectives. The first objective is to

generate DaCMs with constraints on cross-axis sensitivity and natural frequency. This

aims to add new mechanisms with high cross-axis sensitivities to our catalog of DaCMs,

making the catalog more useful in the future. The second objective is to generate DaCMs

specifically for the accelerometer and the force sensor applications. This aims at

obtaining DaCMs for specific constraints required by the applications, thus generating

topologies that are optimum for the given design domain and loading conditions. These

are discussed in subsequent sections. The next section gives a brief review of topology

optimization using the optimality criteria method.

6.2 Topology optimization for sensor applications

Topology optimization is recognized as one of the systematic methods to design

complaint mechanisms (Ananthasuresh, 1994). This method operates on a fixed mesh of

finite elements and defines a design variable, which is associated with each element in

the mesh. The optimization algorithm determines the value of the design variables. The

values of the design variables define the optimal topology of the mechanism. The design

obtained from topology optimization is specific to the design domain, loading, and

boundary conditions. The design variables are driven towards the optimal topology by the

objective function and the constraints, which are specific to the problem. Objective

function is a scalar quantity, which is a function of the design variables. For example,

stiffness of the structure is indicated by the sum of the strain energy associated with each

element in the continuum. The strain energy at each element is a function of the design

variable.

Compliant mechanisms involve relative motion of two points in a continuum. The

two points of intent are a point where the force is applied called the ‘input’, and another

which displaces in a specified direction called the ‘output’. Figure 6.1 shows a design

domain made of an assembly of inter-connecting frame elements, also known as the

‘ground structure’. For a DaCM topology to be obtained from this ground structure, it is

required that the ratio of the output displacement to the input displacement is maximized.

This demands some stiffness at the input side and flexibility at the output side. These two

quantities can be represented as a sum of some energy measure for all the elements in the


domain. The measure of stiffness at the input side where the force is applied can be

expressed as the sum of the strain energies stored in the each element. The flexibility or

the output displacement can be expressed as the sum of mutual strain energy ( MSE ) of

each element (Saxena and Ananthasuresh, 2000). The mathematical formulation of this

energy is shown in the next section. The mutual strain energy ( MSE ) of each element

signifies the contribution of that element towards the displacement of the desired output

point, for a force applied at the input point. At the same time the strain energy of each

element signifies the contribution of that element towards the stiffness at the input side.

These two energy measures are conflicting as one demands stiffness and the other

flexibility. Ananthasuresh (1994) proposed taking a weighted average of the stiffness

and flexibility measures, while Frecker et al. (1997) proposed minimizing a ratio of

flexibility and stiffness measures. Saxena and Ananthasuresh (1998) proposed a

formulation which attempted to maximize the mechanical advantage of the mechanism,

by letting it store as little strain energy as possible. A summary of various other

combinations of objective functions was made by Saxena and Ananthasuresh (2000).

In this work, we have identified various criteria apart from inherent geometric

amplification (see Chapter 3) that are important for sensor applications. From the

comparison of various DaCMs from the catalog, it was noticed that cross-axis stiffness of

mechanisms from literature was not up to the mark. Thus in this chapter, apart from the

normal objective functions for compliant mechanism synthesis, constraints on cross-axis

stiffness and natural frequency are incorporated. These constraints, unlike the volume

constraints have a nonlinear dependence on the design variable of each element. In the

next section, we explain the various objective functions and constraints used for topology

optimization.

6.2.1 Objective functions and constraints used for topology optimization

Various objective functions and constraints, inspired by the sensor applications are

considered in this section. The first two objective functions aim at generating compliant

mechanisms for high cross-axis sensitivities as well as natural frequencies. It was shown

in Chapter 3 that none of the mechanisms had sufficient cross-axis stiffness. Addition of

a DaCM with high cross-axis stiffness would then make the catalog more comprehensive


for further use. The third and the fourth objective functions are aimed at obtaining

topologies specific to the accelerometer and the force sensor applications.

Figure 6.1 Ground structure made of frame elements used for topology optimization. The black rectangular boxes stand for fixed supports while the arrows show the input and the output points. The nodes are numbered as shown. Case 1) Generating DaCMs with specified cross axis stiffness

Maximizex

/MSE SE

subject to *crossSE SE≤ and equilibrium equations.

ilb x ub≤ < , where , 1ix i p∈ = ⋅⋅⋅x

In the above formulation, x is a vector containing all the design variables. For a ground

structure made of beams, the in-plane widths are taken as the design variables. The

mutual strain energy ( MSE ) is given by

MSE = outu = Tinv Ku (6.1)

where v denotes the displacement field due to a unit load applied at the point where the

output displacement is desired. Displacement field due to the load applied at the point of

input is given by inu . The design domain for this formulation is shown in Fig. 6.2. K is

the global stiffness matrix. Strain energy ( SE ) is given by

SE = 2inu = 1

2T

in inu Ku (6.2)


where inu is the displacement field due to the unit applied at the input point.

Strain energy in the cross-axis direction ( crossSE ), which is indicative of the cross-axis

displacement is given by

crossSE = 2

crossu = 12

Tcross crossu Ku (6.3)

where crossu is the displacement field due to the unit load applied in the cross-axis

direction. The cross-axis strain energy is constrained to have a value of *SE .

Figure 6.2 Design domain for generating compliant mechanisms with cross-axis displacement constraint

Case 2) Generating DaCMs with specified frequency constraints

Maximizex

MSESE

subject to the ω ≥ 0ω and equilibrium equations.


This formulation generates a DaCM whose first natural frequency is greater than a

specified value. This is done if the bandwidth of the application, which is dependent on

the natural frequency, needs to be increased. The natural frequency 0ω is the square root

of the eigen-values of the generalized eigen value problem given by

0ω = 1det( )−K M (6.4)

where K is the stiffness matrix of the design domain and M is the inertia matrix. There

are certain problems associated with dealing with the frequency constraint such as mode-

shifts occurring during optimization. These are explained with later in this chapter.

in inF and u

outu

cross crossF and u


Case 3) Generating DaCMs for accelerometer applications

Maximizex

NA (see Eq. 3.2, Chapter 3)

Subject to ω ≥ 0ω and *crossSE SE<

and equilibrium equations.


In this problem, topology optimization is used to obtain topologies specific to the

accelerometer application. The net amplification for an accelerometer with a DaCM is

given by Eq. 3.2. It is important to note that this quantity depends upon proof-mass ( M )

dimensions, suspension-stiffness ( sk ), the comb-mass ( m ) and the external-suspension

stiffness ( extk ). So, given a fabrication process, which determines the values for these

quantities, one can perform optimization to obtain a mechanism that gives a maximum

NA . The formula for the natural frequency ω is given by Eq. 3.4. This approach

simplifies the optimization procedure considerably as it is not necessary to include the

proof-mass, suspension, comb-mass and its suspension explicitly in the design domain.

The spring-mass-lever model, which defines the objective function and the constraint,

takes them into account. This quickens the optimization process and also prevents

problems such as ill-conditioning of the stiffness and the inertia matrices. However, it is

hard to fabricatable topologies as some of the beams of the design domain might have

dimensions that cannot be fabricated.

Case 4) Generating DaCMs for force sensor applications

Maximizex

MSESE

Subject to *SE SE< and *cross crossSE SE<

along with equilibrium equations


The sensitivity of the force sensors depend upon the unloaded output-displacement which

is the output displacement per unit input-load. This is a measure of the flexibility of the

device alone, and thus cannot define a DaCM. To obtain a DaCM, we need to use the


usual objective function, which is a function of stiffness and the flexibility of the device.

To make the mechanism more compliant, we incorporate a constraint limiting the

stiffness of the mechanism. Additional cross-axis and frequency constraints could also be

specified. The topology optimization of a DaCM for a force sensor problem is dealt with

in Chapter 7.

6.3 Optimality Criterion with nonlinear constraints

Optimality criteria methods have been used widely to obtain optimum topologies in

structural and topology optimization with various constraints. These methods use the

optimal conditions of the problem to update the values of design variables at each

iteration (Venkayya, 1989). In contrast, some mathematical programming methods use

sequential linearization or sequential quadratic programming to approximate the

objective functions and constraints. For simpler structural optimization problems with

linear or no constraints, optimality criterion method is recommended. Nonlinear

constraints are better dealt with in mathematical programming than in optimality criteria

methods. However, nonlinear constraints have been previously incorporated in the

optimality criteria method by Yin and Yang (2001), and Yin and Ananthasuresh (2001).

In this chapter an effort is made to incorporate nonlinear constraints into optimality

criteria method to generate compliant mechanisms for sensor applications. The

constrained optimization problem can be stated as follows

Minimizex

( ) 1if x i p= = ⋅⋅⋅x x

Subject to equilibrium equations and

( ) *g g<x and ilb x ub< < (6.5)

where ix , 1i p= ⋅⋅⋅ represent the design variables that define the topology. The objective

function is given by ( )f x , while ( )g x is the constraint. The necessary condition for

optimality can be written as

( ) ( ) 0 , 1 1 , 0ij j

f g x i p and j px x

∂ ∂+Λ = = = ⋅⋅⋅ = ⋅⋅⋅ Λ ≥

∂ ∂x x x (6.6)


where Λ is the Lagrange multiplierIn optimality criterion, we use this necessary

condition (also called as the Karush Kuhn-Tucker conditions when expressed in the

discretized form) to update the design variables ix , 1i p= ⋅⋅⋅ in the iteration as

1 ( ) ( ) 1k ki i

i i

f gx x sc i px x

+ ⎛ ⎞∂ ∂= + + Λ = ⋅⋅⋅⎜ ⎟∂ ∂⎝ ⎠

x x (6.7)

The scale-factor sc determines to a large extent the dynamics of convergence of the

algorithm. A large scale-factor might lead to oscillations, while a small factor a large

number of iterations to reach the converged solution. The upper and lower bounds on the

design variables are taken care of as follows. If, by using the update-scheme of Eq. 6.7

some elements of 1 kix + exceed the upper bound or become lesser than the lower bound,

then the value of the design variable for that element is made equal to the bound which it

has surpassed. Sometimes to prevent rapid changes of the design variables during

optimization, a move-limit is introduced to restrict the maximum change in the design

variables. The value of Λ is iteratively computed until the constraints are satisfied.

In the optimization problems posed in the previous section, the constraints are

nonlinear with respect to the design variables. To satisfy the constraints at each iteration,

we linearize them with respect to the value of the design variable at that iteration. This

simplifies the constraint handling technique in the algorithm. The linearization

approximation of the constraint is justified if the change in the design variables at each

iteration is sufficiently small.

0

0 0

0

0

0

0 0

0 0

( ) *

( ) ( ) *

( ) ( ) ( ) *

( ) , , 1

( ) ( ) *

i

i i

ii

i

g g

g g g wherex

g g g g orx x

gwhere x x i px

g g gx

δ δ=

= =

=

=

≤

⎛ ⎞∂+ ≤ = −⎜ ⎟∂⎝ ⎠

⎛ ⎞ ⎛ ⎞∂ ∂≤ − +⎜ ⎟ ⎜ ⎟∂ ∂⎝ ⎠ ⎝ ⎠

⎛ ⎞∂≤ Α = = = = ⋅⋅⋅⎜ ⎟∂⎝ ⎠

⎛ ⎞∂= − +⎜ ⎟∂⎝ ⎠

0x x

x x x x

x x

x x

x

x x x x x x

x xx x x

xAx b x

xb x x

(6.8)

By substituting the value of Λ in the above equation, we get


*

*

Tm

mT

m m

fsc Ax

sc A A

⎛ ⎞∂⎛ ⎞− − +⎜ ⎟⎜ ⎟∂⎝ ⎠⎝ ⎠Λ =b Ax

(6.9)

where mA and ( )m

fx

δδ are the components of A and ( )f

x∂

∂ whose corresponding

elements have neither reached the upper nor the lower bound. The flowchart for the

algorithm is shown in Fig. 6.3.

One of the most important steps in any optimization algorithm is the

determination of sensitivities. The sensitivities of the objective functions and constraints

introduced in Section 6.2 are derived in the next section.

6.3.1 Sensitivity analysis for the Objective functions and constraints

Sensitivities are the derivatives of the objective function and the constraints with respect

to the design variables. They can be calculated analytically, i.e., from the physics of the

problem, or by finite-difference techniques. Sensitivity analysis is an important part of

optimization by both mathematical programming as well as the optimality criteria

methods. Analytical sensitivities speed up the optimization process owing to fewer

function evaluations at each iteration. In this section, derivatives of the objective

functions and constraints proposed for all the four cases shown in Section 6.2.1 are

presented.

i) Ratio of mutual strain energy to the strain energy MSEnSE

⎛ ⎞=⎜ ⎟⎝ ⎠

TinMSE = v Ku and T

in inSE =u Ku

where the individual terms are explained in the Section 6.1 T

T T inin in

i i i i

MSEx x x x

∂∂ ∂ ∂= + +

∂ ∂ ∂ ∂uv KKu v u v K (6.10)

But we have

in in=Ku F and

v=Kv F

where inF is the force applied at the input of the structure and vF is the virtual load

applied at a point in the structure where the output displacement is desired.


Figure 6.3 Schematic of the optimality criteria with non-linear constraints.

Update Λ using Eq.6.9

YesNo

YesNo

STOP

Initial guess 0x

Update design variables using Eq. 6.7

Guess value for 0Λ = Λ

Is 1max( )k kx x tol+ − <

Is previous tolΛ −Λ ≤

Problem Specifications Objective constraintf and g

Evaluate ( ) ( )f x and g x

Calculate sensitivities

i i

f gandx x∂ ∂∂ ∂


Differentiating these two equations and assuming that the applied and the virtual loads

are independent of the design variables, we get

in in inin in

i i i i i

orx x x x x

∂ ∂ ∂∂ ∂+ = = = −

∂ ∂ ∂ ∂ ∂u F uK KK u 0 K u (6.11)

Eq. 6.7 makes use of the symmetry of the stiffness matrix. Similarly, we have

andT

T

i i i ix x x x∂ ∂ ∂ ∂

= − = −∂ ∂ ∂ ∂

v K v KK v K v (6.12)

Substituting the above two equations in Eq. 6.10, we get

Tin

i i

MSEx x

∂ ∂=−

∂ ∂Kv u (6.13)

Similarly we have

Tin in

i i

SEx x

∂ ∂=−

∂ ∂Ku u (6.14)

So, 2

T Tin in in

i i

i

MSE SE MSEx xSE

x SE

∂ ∂⎛ ⎞ − × + ×∂⎜ ⎟ ∂ ∂⎝ ⎠ =∂

K Kv u u u (6.15)

ii) Cross-axis displacement ( crossSE )

Tcross cross crossSE =u Ku

The derivation of the sensitivities of cross-axis displacement with respect to design

variables is similar to the sensitivities of strain energy ( SE ) given in Eq. 6.14.

Tcrosscross cross

i i

SEx x

∂ ∂= −

∂ ∂Ku u (6.16)

iii) Natural Frequency ω

The generalized eigen-value equation in dynamics is given as 2

i i iω=Ku Mu (6.17)

where iω is the thi eigen value and the corresponding modal vector is iu . Pre-

multiplying the above equation by Tiu , we get the following equation

0Ti i iand= =u Su Su 0 (6.18)


where 2( )iω= −S M K is the dynamic compliance.

Since most eigen-value problems have normalized mass matrices, we have

1Ti i =u Mu (6.19)

Differentiating Eq. 6.14 with respect to the jth design variable, we get

0T T

T Ti ii i i i

j j j

Sx x x

∂ ∂∂+ + =

∂ ∂ ∂u uSu u u u S .

Substituting the values for S , in the above equation, we get

0Ti i

jx∂

=∂

Su u (6.20)

Expanding S in terms of M and K in the above equation, we get

2[2 ] 0T ii i i i

j j jx x xωω ω∂ ∂ ∂

+ − =∂ ∂ ∂

M Ku M u

Expanding the above equation, we get

22 0T T Tii i i i i i i i

j j jx x xωω ω∂ ∂ ∂

+ − =∂ ∂ ∂

M Ku Mu u u u u (6.21)

Further, using Eq. 6.15 and re-arranging the terms, we get

1 02 2

T Ti ii i i i

j j i jx x xω ω

ω∂ ∂ ∂

= − =∂ ∂ ∂

M Ku u u u (6.22)

iv) Net Amplification (NA)

The formula for the net amplification is given by Eq. 3.4 in Chapter 3. Of all the

quantities in this expression, only cik , cok and n are dependent on the design variables.

The input stiffness cik is given by

1/( )Tci in ink = u K u (6.23)

where inu is the displacement field for a unit input load applied . The derivative of

Tin inu K u is as given by Eq. 6.10. The output stiffness cok is given by (see Fig. 3.3,

Chapter 3)

2 2/( ) /( )Tco ci in cik k n k MSE n= − = −v K u (6.24)


where v is the displacement field due to a virtual load applied at the output point.

Knowing the derivatives of the stiffness cik , inherent magnification n (Eq. 6.15), and

mutual strain energy ( MSE ) ( Eq. 6.13), the derivative of the net amplification (NA) can

be found. Same is in the case with natural frequency given by Eq. 3.4.

We have explained the basic technique of topology optimization using optimality

criteria method. We have also derived the sensitivities for the various objective functions

proposed for optimization. In the next section, we present some numerical examples and

the results of topology optimization.

6.4 Numerical Examples

Incorporating frequency constraints is usually troubled with the problem of mode- shifts.

It is mainly the first natural frequency which is optimized for in most applications. Every

natural frequency has its corresponding mode shape. The derivative of the natural

frequency with respect to the design variables has terms involving the modal vector iu as

shown in Eq. 6.22. The modal vectors used to calculate the derivatives should be of the

corresponding natural frequency that needs to be optimized. However, there is a

possibility that the mode corresponding to the first natural frequency shifts its position.

This happens because some other mode might have a smaller frequency than the mode

under consideration. When this occurs during the course of optimization, there will be a

discontinuity from the previous iteration leading to a drastic change in the sensitivity of

the objective function and the constraints.

To overcome the problem of mode-shifts, modal assurance criteria (MAC) is used

(Allemang, 2003; Maske et al., 2006). This criteria at each iteration tracks the mode

which was being optimized for in the previous iteration, thus avoiding discontinuity. It

works on the principle that the mode which is being optimized for is orthogonal to all

modes in the previous iteration, except with itself. This holds good, for small changes in

the design variable between two iterations. In this chapter, a modal assurance criterion is

used to circumvent the problem of mode-shifts. However, this method was not found to

prevent shifts completely because of large changes in the design variables during the

course of optimization. But the number of mode shifts was found to reduce during the

course of the optimization process by incorporating this criterion.


Below, we present the results obtained from topology optimization for DaCMs

with constraints on the cross-axis displacement and natural frequency. A ground structure

made of a grillage of beams was used as the design domain for optimization. The

specifications of the optimization problem and the optimized topology are shown.

6.4.1 Topology optimization of DaCMs with constraints on cross-axis displacement

and natural frequency

Case 1. Generating DaCMs with specified cross axis displacement.

Maximizex

MSESE

subject to *crossSE SE< and equilibrium equations

ilb x ub≤ < , where , 1ix i p∈ = ⋅⋅⋅x (6.25)

For the example solved here,

1. Ground structure is shown in Fig. 6.4a with the input, output and fixed points.

2. Size of the grid = 500 mµ × 500 mµ

3. Thickness of each element = 25 mµ

4. Upper bound for the element width = 10 mµ

5. Lower bound for the element width = 1e-5 mµ

6. Force at the input = 1 Nµ

7. Value of *SE = 5e-4 mµ

8. Maximized Objective function value = 5.89

The optimized topology is shown in Fig. 6.4b. It has an inherent amplification ( n ) of

around six. This mechanism was found to have the highest cross-axis stiffness when

compared with DaCMs from literature in Chapter 3. The deformed profile is shown in

Fig. 6.4c. Figs. 6.5b-c show the convergence history of the objective function and the

constraint.


(a) (b)

(c)

Figure 6.4 (a) Ground structure used for optimization for the formulation given by Eq. 6.25 (b) Symmetric half of the optimized topology (c) Deformed plot of the mechanism

Case 2. Generating DaCMs with specified frequency constraints

Maximizex

MSESE

Subject to 0ω ω≥ and equilibrium equations

ilb x ub≤ < , where , 1ix i p∈ = ⋅⋅⋅x (6.26)

For the example solved here,

1. Ground structure is shown in Fig. 6.6 (a) with the input, output and fixed points.







7. Value of *ω = 10e5 Hz

8. Maximized Objective function value = 1.3

(a) (b)

Figure 6.5 A plot of the (a) objective function and (b) constraint history during optimization for the Case 1.

Figures 6.6b-c show the topology and the deformed plot. The problem of mode shifts was

not eliminated with the use of modal assurance criterion (MAC). This is because large

changes in the design variables between two successive iterations disrupt the

orthogonality between two dissimilar modes.

(a) (b)


(c)

Figure 6.6 (a) Ground structure used for optimization for case (2) formulation (b) Optimized topology (c) Deformed plot of the optimized topology

6.4.2 Topology optimization of DaCMs for accelerometer applications

In this section, we use topology optimization for the accelerometer application. Here, we

aim at getting an optimum topology of a DaCM to be used in conjunction with the

accelerometer. The objective function derived from the spring-mass-lever model is used

to generate topologies for the accelerometer application. Using the spring-mass-lever

model is equivalent to modeling the design domain with the proof-mass, suspension,

sense-comb mass, and external suspension. Additionally, modeling these would result in

ill-conditioning of the stiffness and the inertia matrices as the dimensions of the proof-

mass is around 200 times more than the minimum dimension of the DaCM. The mode-

shift problem can also be avoided by using an estimate of the natural frequency from the

spring-mass-lever model than performing eigen-analysis on the entire domain.

There are, however, severe drawbacks in using the ground structure made of

beams for topology optimization. Here, the width of each beam is considered as the

design variable. Each element is assigned an upper and lower bound within which the

width of the beam can lie. If the design variable reaches the lower bound, its contribution

to the objective function can be considered minimal and thus the element is neglected.

Though the design variables are driven towards the upper or lower bounds, there may be

elements with intermediate widths in the optimal topology. These intermediate widths

may be smaller than the fabrication tolerance of the process. Thus, these topologies need


to be modified by a separate size-optimization process to ensure that the fabrication

constraints are met. Though useful, this makes the topology sub-optimal. This problem

occurs even in continuum elements if fabrication constraints are not included explicitly in

the algorithm.

Below, we present some examples of topology optimization for the accelerometer

application.

Case 1(a).

Maximizex

( )

2

2

( )( )



F k k n k F nk kNA

k k k k n k k k F

+ + +=

+ + + +x (6.27)

subject to 0ω < *ω and equilibrium constraints

Where ω is given by Eq. 3.4. All the terms are explained in Chapter 3. As mentioned in

Section 6.2, only cik , cok and n are dependent on the topology of the DaCM to be

optimized. All the other quantities are dependent on the suspension stiffness, proof-mass

dimensions and external comb-drive mass. For this example we try to obtain a DaCM

which adheres to the process specification of bulk-micromachining using deep-reactive

ion etching (DRIE) with silicon on insulator (SOI) wafers as explained in Chapter 4. The

corresponding values of the fixed quantities are






6. Value of the Sensor stiffness ( sk ) = 197 /N m

7. Value of the proof-mass inertia ( M ) = 7.5e-6 kg

8. Value of the comb suspension ( extk ) = 1.25 /N m

9. Value of *ω = 2500 Hz

10. Value of the maximized Objective = 2.75

11. Value of inherent amplification ( n ) = 3.1


It can be seen from Fig. 6.7b that not all the elements that define the topology have

reached the upper bound. The beams which are grey in color have intermediate widths

less than 5 mµ , which is the minimum fabricatable dimension using the DRIE with SOI

process. The topology obtained above is first refined by removing dangling elements, and

is then optimized for its in-plane width so as to meet the fabrication constraints.

(a) (b)

(c)

Figure 6.7 (a) Ground structure used for optimization for case (3b) formulation (b) Optimized topology (c) Deformed plot of the optimized topology.


The formulation for the size optimization problem is shown below

Maximizex

NA given by Eq. 3.2

Subject to equilibrium equations

and bounds on x , such that max minix x x< < (6.28)

In the above formulation all quantities used for the evaluation of NA are shown above.

The minimum value for the in-plane width minx is taken to be 5 mµ which is the

minimum fabricatable dimension in the DRIE with SOI process. Figure 6.8a represents

the topology as obtained from the optimization algorithm with the dangling elements

removed. The DaCM after size-optimization shown in Fig. 6.8b is coupled with an

accelerometer proof-mass and suspension. The procedure followed is as shown in

Chapter 4. The resulting accelerometer was found to resolve better than 10 gµ with an

open-loop circuit resolution of 10 parts per million. However, it was found to have a high

cross-axis sensitivity of 0.2%. This can be attributed to just two supports on which the

mechanism is fixed. In the next example, we generate a DaCM for the accelerometer

application with a constraint on cross-axis stiffness

(a) (b)

ix


(c)

Figure 6.8 (a) Symmetric half of the DaCM obtained from the topology optimization (b) Modified topology after optimization of the in-plane width to comply with the fabrication process (c) Complete topology Case 1(b)

Maximizex

NA

( )

2

2

( )



F k k n k F nk kNA

k k k k n k k k F

+ + +=

+ + + + (6.29)

subject to the equilibrium equations and constraints

*crossSE SE<

Where crossSE is the strain energy for a load applied in the cross-axis direction. The cross-

axis stiffness is given by Eq. 6.3 and is represented in Fig. 6.2.

For the following problem,






6. Value of the Sensor stiffness ( sk ) = 300 /N m

7. Value of the proof-mass inertia ( M ) = 7.5e-6 kg

8. Value of the comb suspension ( extk ) = 1.25 /N m


9. Value of crossSE = 0.01 mµ

10. Value of the maximized objective function = 2.58

Figure 6.9a shows the optimized topology of the mechanism. In this DaCM, the in-plane

widths of all the beams were greater than 5 mµ . It has an inherent amplification of 3.2.

(a) (b)

(c)

Figure 6.9 (a) Ground structure used for optimization for formulation given by Eq. 6.29 (b) Optimized topology (c) Deformed plot of the optimized topology The in-plane widths of this mechanism were further optimized to obtain a high net

amplification. The procedure followed is as shown in Chapter 4. The resulting

accelerometer was found to resolve better than 12 gµ with an open-loop circuit


resolution of 10 parts per million. Its cross-axis sensitivity was found to be around

0.02%. Figure 6.9 shows the optimized topology coupled with an accelerometer proof-

mass and suspension.

6.5 Closure

In this chapter topology optimization is explored to generate DaCMs for sensor

applications. The objective function formulation proposed by Frecker et al. (1997) was

extended to incorporate non-linear constraints such as that of cross-axis sensitivity and

natural frequency. The sensitivity analysis for the proposed objective function and the

constraints along with the flow-chart of the optimization algorithm has been presented.

For the accelerometer application, the net amplification (Eq. 3.2) has been used as the

objective function along with the natural frequency (Eq. 3.4) and cross-axis sensitivities

using the insight obtained in Chapters 3 and 4. Optimization for cross-axis sensitivities

yielded a DaCM whose cross-axis stiffness was better than any other mechanism in the

catalog. Optimization for the accelerometer applications yielded two designs. The best

design (Fig. 6.9) was found to have an open loop resolution of 12 gµ with cross-axis

sensitivities of less than 0.02%.

Topologies obtained from optimization were observed to have dangling elements

and also elements having with very narrow features. Thus, they cannot be fabricated

directly. These topologies were modified for its shape and size and made to comply with

the fabrication limitations. The next step towards optimization for sensor applications is

to incorporate fabrication constraints to define topologies that can be realized by standard

bulk micro-machining techniques. Also, use of continuum elements could be investigated

to generate DaCMs for the aforementioned sensor applications.


Figure 6.10 Optimized mechanism in conjunction with a proof-mass and suspensions

7.1

Chapter 7

7.A DISPLACEMENT-AMPLIFYING COMPLIANT

MECHANISM AS A MECHANICAL FORCE

SENSOR

Summary In this chapter, we introduce a force sensor incorporating a DaCM for vision-based

measurement of force for possible application in cell manipulation and laparoscopic

surgery. The criteria required for DaCMs to be suitable for this application are

formulated. These criteria are used for comparison of mechanisms and selecting the best

mechanism based on the figure of merit criteria proposed in Chapter 3. Furthermore, an

attempt is made to design a new mechanism by topology optimization. A scaled up

version of this mechanism is fabricated using a Hall-effect proximity sensor, tested, and

calibrated as a force sensor. It was found to have a sensitivity of 324 mV/N with a

resolution of 30 mN, and a maximum working range of 3 N. This prototype force sensor

is used to measure the force required to rupture an inflated balloon. This demonstrates

the use of the sensor, when fabricated at the micron scale, to measure the forces in intra-

plasmic live-cell injection.

7.1 Introduction Force sensors are integral parts of integrated systems and testing and characterization

setups. Traditional force sensors use the principle of resistance change, e.g., a load cell,

due to the strain caused by the applied load. But at small length scales embedding such

load cells my not be effective. These sensors occupy a lot of space and might suffer from

errors due to misalignment. There have been attempts to use compliant mechanisms to

Chapter 7: A DaCM as a mechanical force sensor 7.2

apply forces as well as to use its deformation at either a single point or a set of points to

measure the applied force (Wang et al., 2001; Greminger and Nelson, 2004).

The mechanisms used for force sensor applications should be flexible enough so

as to detect minute forces. However, they have to be stiffer than the object being grasped

so that large forces can be exerted. In this chapter, we investigate the use of DaCMs as

force sensors. We first start out by selecting the most suitable mechanism from the

catalog of DaCMs obtained from literature. The specification for selection is obtained

from vision-based force sensing in inter-plasmic cell injection and in laparoscopic

surgery. We then obtain new DaCMs using topology optimization. The criteria required

by the DaCMs for the force sensor application are derived from the spring-mass-lever

model as shown in the next section.

7.2 Use of DaCMs as force sensors Displacement-amplifying compliant mechanisms (DaCMs) and their spring-mass-lever

models were discussed in Chapter 3. Referring to Figs. 3.3a-b, we note that the input side

of these mechanisms should be stiff so that large forces can be applied on them. On the

other hand, the output side should be flexible so as to allow large output displacements.

The sensitivity of the DaCM-based force sensor is given as the ratio of the output

displacement to the force applied at the input. This quantity is termed as the unloaded

output displacement (Uod). This is directly obtained from the expression for output

sensitivity given by Eq. 3.1 by equating 2 0F = , 0sk = , and 0extk = . This gives

odU = 2

1 ci

x nF k

±= (7.1)

The terms of the above equation and the procedure to evaluate them are explained in

Chapter 3. The positive sign is for the non-inverter while the negative sign is for the

inverter (see Figs. 3.3a-b).

Apart from sensitivity, various other criteria such as the maximum working range

( maxF ), cross-axis sensitivity ( crossK ), and natural frequency ( f ) are important for sensor

applications. The working range is determined by the maximum force at which the stress

in the mechanism exceeds the yield stress with a specified factor of safety. For some

mechanisms, such as mechanism M2, impending contact limits the maximum force.


Cross-axis sensitivity is important for the sensor applications. A two-axis force sensor

should be equally sensitive to forces acting in any direction within the plane of the

mechanism, while a single-axis force sensor should be sensitive only to the forces in the

direction of amplification. In this chapter, we consider only single-axis force sensors. It

should be noted that force sensors are operated in open-loop mode and thus undergo large

displacement for large applied forces. Thus, geometric nonlinearity is used in computing

all the quantities involved. In the next section, we select a mechanism from the catalog of

the DaCMs based on the criteria mentioned above for two case studies, one at the mµ

scale and the other at the cm scale. The specifications for these are guided by two

potential applications, viz., force sensor measurement in inter-plasmic cell injection and

in laparoscopic surgery. We do not, however, consider all aspects of the two applications

in this work; rather we obtain plausible topologies for these applications.

7.3 Force sensor at the micron scale In this section, we select a mechanism for the vision-based force measurement of intra-

plasmic live-cell injection. Typical forces that are exerted on the cell during manipulation

are in the order of 1 Nµ (Cappelleri et al., 2006). This usually generates a displacement

of 5 mµ , which corresponds to one pixel displacement in most standard microscopes

used for these purposes. Assuming that the minimum displacement that could be resolved

is 1 pixel, it is required to resolve a force of 0.25 Nµ . This corresponds to an unloaded

output displacement odU of 20 /m N . In addition to high unloaded-output displacement,

it is necessary for the mechanisms to take large forces before failure and have little cross-

axis sensitivity. To find the most suitable mechanism for this application all the

mechanisms introduced in Chapter 3 are made fit into an area of 3.368 mm × 2.526 mm,

which corresponds to the filed of view in a microscope (Cappelleri et al., 2006).

It was explained earlier in Chapters 3 and 4 that the in-plane widths of each

mechanism have to be optimized to ensure the comparison of topologies alone. Thus, size

optimization of in-plane beam widths to obtain a high sensitivity ( odU ) is performed. The

formulation of optimization problem is given by Eq. 7.2 with reference to Fig. 7.1.

Maximize Ux

given by Eq. 7.1


odci

nUk

=

Subject to the equilibrium and bounds on the design variables x

Such that max minix x x< < (7.2)

where x is the design-variable vector containing the in-plane widths of all the elements.

The minimum width of the beam minx was limited by the fabrication requirement.

Assuming a deep-reactive ion etching (DRIE) process on silicon-on-insulator (SOI)

wafer, the minimum width of the beam minx was fixed to 5 mµ . The out-of-plane

thickness of the mechanisms was fixed to 25 mµ . Table 7.1 shows the comparison of

eight mechanisms M1-M8. The weights assigned to the different quantities to obtain the

figure of merit for this application is given in the penultimate row of Table 7.1. If a high

sensitivity is desired, then mechanism M2 is most suitable. If maximum working range is

desired, then applying the weights shown in Table 7.1 (last row) mechanism M5 was

obtained to be the best.

The mechanism M2 obtained from this comparison as the most suitable

mechanism still did not meet the specification of having an unloaded output displacement

of 20 m/N as demanded by the vision based force sensor application. To achieve such an

unloaded output displacement, the mechanism had to be modified by changing the out of

plane thickness of 7.5 mµ . Figure 7.2 shows mechanism M2 as a force sensor. The

supports of the mechanism are fixed to the laparoscopic tube or the manipulator.

Figure 7.1 Size optimization of the mechanisms for the force sensor application.

ixArea occupied by the Mechanism is 3.5 mm × 2.5 mm


Table 7.1 Weights associated with a quantity for the computation of the overall figure of merit for a forced sensor application

M1 M2 M3 M4 M5 M6 M7 M8

nGA 0.91 0.11 0.25 1.00 0.91 0.40 0.02

0.10

odU 0.04 1.00 0.01 0.01 0.00 0.02 0.04

0.00

crossnK 0.00 0.00 0.01 0.01 0.01 0.00 0.00 1.00

nf 0.21 0.12 0.41 0.22 0.32 0.37 0.43

1.00

maxF 0.08 0.00 0.19 0.56 1.00 0.16 0.04

0.29

Case1 0.15 0.62 0.09 0.18 0.23 0.11 0.07

0.24

Case 2 0.37 0.15 0.20 0.57 0.70 0.24 0.07

0.34

Weights nGA odU crossnK nf maxF

Case 1 0.10 0.6 0.1 0.1 0.1

Case 2 0.35 0.1 0.1 0.1 0.35

Figure 7.2 Vision based force sensing of cells 7.4 A force sensor fabricated at the meso-scale In this section, we aim at designing a DaCM for a force sensor, which can resolve a force

of around 30 mN . The force sensor is to be made of metal (spring-steel) using wire-cut

electronic discharge machining (EDM). The minimum in-plane width that can be

fabricatable by the process is fixed to 0.25 mm .

Cell

Point of contact

Sensing port

Input motion

Input motion


It was seen in Table 7.1 that mechanism M2 had greater sensitivity than any other

mechanism. Since the sensitivity ( odU ) is found to be independent of the applied load, we

expect to obtain the same mechanism for this application. When mechanism M2 is scaled

to fit the specifications for this application, we find that its stiffness is 0.5 /MN m

making its sensitivity equal to 10.7 /m Nµ . This means that output displaces by 0.3 mµ

for a force of 30 mN . Such small displacements might be hard to detect. The mechanism

size, for the sake of prototyping, is increased to fit an area of 4 cm × 4 cm . For this size,

the mechanism gives a sensitivity of 400 /m Nµ with an output displacement of 12 mµ

for a force of 30 mN . In the next section, we use topology optimization with shape and

size refinements to obtain a more sensitive force sensor.

7.4.1 Topology optimization of DaCMs for force sensor application

Topology optimization for DaCMs was introduced in Chapter 6. The need for an

objective function with a tradeoff between stiffness at the input point and flexibility at the

output point to obtain a DaCM was discussed. However, the force sensor application

requires a high ratio of inherent amplification ( n ) and the stiffness ( cik ). This can be

expressed as

odU = ci

nk

= out in

in in

x xx F

= out

in

xF

(7.3)

where outx is the output displacement for a input load inF . Since the input load inF is

constant, odU seeks to maximize the output displacement only. This objective function

has no tradeoff on stiffness and seeks to maximize only the flexibility. Hence, a DaCM

cannot be obtained by this formulation. Thus, the approach adopted is to obtain a DaCM

by using the usual formulation of maximizing the ratio of the displacements at the input

and the output points using beam grillages as the ground structure. The formulation of the

optimization problem is shown below

Maximizex

out

in

u MSEu SE

= with respect to the design variables x

Subject to equilibrium constraints and


volume constraint b≤Ax

and bounds min maxix x x< < (7.4)

In the above equation, x represent design variables, which are the in-plane widths of the

beam ground structure shown in Fig. 7.3a. Figure 7.3b shows the optimal topology.

(a) (b)

Figure 7.3. Topology of a DaCM. (a) Ground structure made of grillages (b) Optimized topology

It can be seen from Fig. 7.3b that the topology obtained from optimization has

deformable regions in its top part while most of the bottom part is significantly rigid. The

basic topology from Figure 7.3b was then optimized for its width (keeping fabrication

constraints in mind) and the angle between the beams in order to obtain a high unloaded

output displacement ( odU ). The flexible top part of Fig. 7.3b is shown in Fig. 7.4a. The

formulation for this shape and size optimization problem is shown in Eq. 7.5.

Maximizey

odci

nUk

=

Subject to equilibrium constraints

And bounds min maxiy y y< < (7.4)

In the above equation y is the vector of design variables. It consists of a set of element

lengths ( il ), orientation of the element ( iθ ), and width of the element ( ix ). These are

shown in Fig. 7.4a.


(a) (b)

Figure 7.4. Shape and size optimization of the DaCM. (a) Skeletal topology from topology optimization (b) Final mechanism from shape and size optimization Figure 7.4b shows the optimized mechanism. The area occupied by the mechanism is 4

cm × 4 cm . It has an inherent amplification ( n ) for this mechanism is 2.53. Its

sensitivity is 540 /m Nµ compared to 400 /m Nµ of mechanism M2. Thus we have

achieved a more sensitive mechanism than that in the literature. Figure 7.5 shows the

SOLIDWORKS model of the mechanism with all the dimensions. The minimum feature

size is fixed to 0.25 mm , as it minimum possible in-plane dimension manufacturable in

the wire-cut EDM. The choice of the fabrication process and material for manufacturing

are explained in the next section.

7.4.2 Fabrication Process of the mechanism

Fabrication of the mechanism is very important since it has a bearing on the design

process. For example, the shape and size optimization of the mechanism had to take into

account the minimum feature size that was permissible by the fabrication process. The

fabrication process in turn depends upon the material in which the mechanism is to be

made and the overall size of the mechanism. Since the mechanism needs to be

sufficiently elastic, the suitable choices are hardened spring-steel, Copper- beryllium and

titanium. Hardened spring steel was chosen because it is readily available and is

inexpensive. Figure 7.6 shows the spring-steel sheet being bent almost at 090 . The sheet

comes back to its original shape once the force is released, indicating its elastic nature.

Some other choices include plastics such as polypropylene, but with conventional

fabrication techniques (eg., CNC milling) it becomes difficult to scale down the size of

the plastic prototype.

il

iθ

ix


Figure 7.5 SOLID WORKS model of the mechanism showing the dimensions in mm

The size of the mechanism is around is 4 cm × 4 cm . Intricate shapes need to be

cut within this size accurately. Wire-cut EDM is an excellent tool for prototyping with an

accuracy of close to a micron. However, there may be misalignment errors for complex

parts with a number of islands. Photochemical etching can also be used to fabricate the

mechanisms in the cm -scale. This process suffers from wavy and sloping edges due to

improper penetration of the etchant, but is suitable for batch fabrication. Another process

which could be considered is laser-cut machining. This also suffers from edge-

undulations if the focus-area of the laser is not concentrated. Since prototyping is the aim,

wire-cut EDM is used. Figure 7.7 shows the wire-cut EDM that was used to fabricate the

mechanism in the Dept. of Mechanical engineering. Figure 7.6 shows the spring steel

sheet and the fabricated mechanism. The above model was then analyzed in an FEA

package to determine the maximum stress experienced.

Input port

Output port

Fixed end Fixed end


Figure 7.6 The Spring steel sheet shown along with the fabricated mechanism

Figure 7.7 Wire cut EDM machining the mechanism

7.4.3 Finite Element Analysis of the mechanism using COMSOL

The mechanism was analyzed using plane stress elements with geometric non-linearity in

Comsol Multiphysics. The force vs. displacement curve is shown in Fig. 7.8. For a small

range of forces (0-0.5 N), the displacement is linear with respect to the force and is also

symmetric about the origin. Larger displacements can also be calibrated in terms of the

force by either having a look-up table or by using polynomial fits. The amplification

EDM wire


factor for the mechanism is 2.53. The maximum stress that the mechanism can handle is

700 MPa (the failure stress of Spring Steel). The maximum load before failure was found

by FE analysis in COMSOL to be 3.58 N (Fig. 7.9).

In the next section, we explore the various displacement detection techniques

which can then be calibrated for force.

Figure 7.8 Force vs. Displacement relation for the mechanism by FE analysis using COMSOL software


Figure 7.11 Analysis in COMSOL indicating the maximum stress of 700 Mpa

7.4.4 Displacement Sensing Technique (Hall-effect Sensor)

Mechanical force sensors act as springs to convert force to displacement. Thus, accurate

detection of displacement is useful for force measurement. For cell manipulation in the

micron scale, vision based force sensing is commonly used (Greminger and Nelson,

2004; Wang et al., 2001). Placing any other sensor is quite cumbersome due to space

constraints. However, at the cm -scale at which the device is fabricated, we can use a

number of non-contact sensors to detect the displacement. Some of the techniques worth

considering are linear-varying differential transducer (LVDT) and the Hall-effect

principle. LVDTs are bulky and are quite hard to miniaturize. Thus Hall-effect sensor is

used in this project for the purpose of distance measurement. The main advantage of this

sensor is its high sensitivity and small size (Hall-effect sensors, Honeywell). The

principle of operation of this kind of a sensor is that when a current-carrying conductor is

placed in a magnetic field, a voltage will be generated perpendicular to both the current

and the field. The voltage depends on the strength of the magnetic field and this property


is used to make it a proximity sensor. The Hall-effect sensor used here is of Allegro make

and has the model no. A1321. It is mounted on a printed circuit board (PCB) as shown in

Fig. 7.9. The board is made with metal pads and can be mounted close to the point whose

displacement is to be detected. It is recommended that the magnet is placed at the moving

point while the sensor is fixed because the wires taken from the PCB might disrupt the

displacement. The experimental setup with the characterization of the device is described

in the next section.

Figure 7.9 Hall Effect Sensor surface mounted on a PCB

7.5 Experimental set-up to calibrate the force sensor and the DaCM The Hall–effect sensor has three ports, an input port where the input voltage (constant

DC voltage of 6V) is supplied, a ground port and output port where the output voltage

dependent on the output displacement is read. The three ports on the PCB are shown in

Fig. 7.9. The magnet needed for effecting the change in the output voltage is fixed to the

output port of the mechanism while the Hall-effect sensor is fixed on the support. The

experimental set-up is shown in the Fig. 7.10. Care is to be taken to make sure that the

center of the magnet and the sensor coincide. The sensitivity of the sensor depends upon

the distance between the magnet and the sensor, input voltage, and the strength of the

magnet. Thus, the sensor is calibrated for one such setting. The sensor and magnet

positions are shown in Fig. 7.11.

Hall Effect Sensor

Ground port Output port Input port

1cm


Figure 7.10 Experimental setup for calibrating the sensor with a DaCM

Figure 7.11 Magnified view of the sensor and the magnet

Standard weights were attached to the input port of the mechanism and the output

voltage could be calibrated for the known input load. The voltage was found to vary

linearly with the force for a range of 0 - 0.25 N . Fig. 7.12 shows the voltage vs force

plot with a linear fit. It was found that for the given arrangement a sensitivity of 324

/mV N is obtained. Since the minimum stable detectable voltage is 1 mV , 3.12 mN is

the minimum possible force that can be detected.

Mechanism with Hall-effect sensor

Voltage supply with a regulator

Output reading

Magnet

Hall Effect sensor

Input Output

Ground


7.5.1 Force required to rupture an inflated balloon

The calibrated force sensor is now used to estimate the force required to rupture an

inflated balloon. This has a direct implication in micro-manipulation of the cell where it

is necessary to measure the force needed to rupture the cell wall. The force required to

rupture the balloon will be dependent on the sharpness of the edge that is used to contact

the balloon.

Shown in the Fig. 7.13 is the output voltage reading before the contact, which is

1.525 V . When the contact of the mechanism occurs with the balloon, the voltage

changes and just when the balloon ruptures, the output voltage reads 1.413 V . The

change in voltage is 0.112 V . Assuming a sensitivity of 0.324 /V N , the force was

measured to be 0.346 N .

Figure 7.12 Experimental Calibration of the Sensor


Figure 7.13 Figure showing the initial and final output reading before and after the rupture of the balloon

7.6 Closure In this chapter, we have demonstrated the use of a DaCM as a force sensor. First, a

mechanism which is most suitable for the force sensor application was selected from the

catalog of DaCMs. We then used topology optimization along with refinement in its size

and shape to obtain a mechanism with a higher sensitivity than the best mechanism in

literature. This DaCM has been fabricated and tested as a single-axis force sensor. The

usefulness of this type of measurement technique has been demonstrated by the balloon

experiment, which has relevance in the micromanipulation of biological cells. With

appropriate fabrication techniques, the size of the mechanism can be scaled down to

occupy a 1 cm × 1 cm area without any reduction in flexibility and thus could be

inserted into a laparoscopic tube to estimate the force exerted by the surgical tool on the

organ.

The force sensor proposed in this chapter is a single-axis force sensor. The next

effort in the direction would be to use topology optimization to synthesize mechanisms


which can be used to measure in any direction within a plane. Furthermore, cross-axis

stiffness constraints need to be included in topology optimization.

8.1

Chapter 8

8.CONCLUSIONS AND FUTURE WORK

8.1 Summary

The principal objective of the thesis is understanding displacement-amplifying compliant

mechanisms (DaCMs) and investigating their use for the sensor applications. Towards

this end, a lumped spring-mass-lever model for the DaCM that captures its static behavior

and the dominant-dynamic mode has been proposed. These models have been used to

identify and evaluate various criteria which are important for sensor applications. Several

insights, specifically the importance of net amplification rather than the inherent

amplification for sensor applications was realized. A number of mechanisms from

literature were considered to develop a catalog of DaCM topologies. These mechanism

topologies were compared for various criteria relevant to the application.

A figure of merit based on a weighted average of all criteria was proposed to

select a DaCM topology that is most suited for a given application. This method is

proposed for the selection of DaCMs in contrast to solving an optimization problem for

each application. In case all the mechanisms in the catalog are found wanting with

respect to a criterion, then it is worthwhile to formulate an optimization problem to obtain

a new topology which suits the application. As an example, it was observed that most

mechanisms from literature had poor cross-axis stiffness. This was first overcome by a

structural modification of adding an external suspension, which considerably improved

its cross-axis stiffness. For further improvement, an optimization problem was posed and

solved to obtain a new mechanism with high cross-axis stiffness.

Based on the figure-of-merit technique, DaCM for an accelerometer application

has been selected and optimized. System-simulation of three accelerometer designs with

electronic components in the forced-feedback mode was carried out in SIMULINK.

modal superposition was used to model the DaCM and the accelerometer in the system

Chapter 8: Conclusions and Future Work 8.2

level. The designed accelerometers were found to have sensitivities of 0.125 /V mg ,

0.445 /V mg , and 0.07 /V mg and a resolution of 40 gµ , 20 gµ , and 70 gµ

respectively. Topology optimization incorporating constraints on natural frequency and

cross-axis stiffness was used to synthesize new designs for this application.

Furthermore, DaCMs have been investigated as a single-axis mechanical force

sensor in laparoscopy and vision based force sensing in cell manipulation. Topology

optimization along with shape and size refinements was used to obtain a DaCM with

sensitivity higher than any mechanism from literature. This mechanism was fabricated

and tested as a force sensor. It was found to have a sensitivity of close to 0.32 /V N and

could resolve better than 3.2 mN . The use of the mechanism in micron-scale cell

manipulation was demonstrated in the macro scale by estimating a force required to

rupture a balloon.

8.2 Contributions

The main contributions of the thesis can be summed up in the following salient points.

• A lumped spring-mass-lever model was proposed which led to a better

understanding of DaCMs for both actuator and sensor applications.

o This model was used as a framework to compare various DaCMs for a

particular application.

o A figure of merit was proposed to select the best mechanism from a

catalog of DaCM topologies from literature for an application.

• It was shown that adding a DaCM could increase the sensitivity of a micro-g

accelerometer from literature (Chae at al., 2004) by at least three times.

o Three accelerometer designs with DaCMs were proposed for a simple

bulk-micromachining process with deep-reactive ion etching (DRIE) on

silicon on insulator (SOI) wafers.

o Additional structural modifications at the output end of the DaCM were

incorporated to reduce the cross-axis sensitivity.

o The system-simulation of the combined mechanical and electronic

components was performed. The accelerometer with a DaCM in the force

re-balance mode was found to resolve better than 20 gµ .


o A bulk-micromachining process with deep-reactive ion etching (DRIE)

with silicon on insulator (SOI) wafers was proposed to fabricate the

accelerometer with a DaCM.

• Topology optimization was used to design new DaCMs for the accelerometer

applications.

o Constraints on cross-axis stiffness were imposed in the topology

optimization problem. These nonlinear constraints were incorporated into

optimality criterion by sequential linearization.

o Topologies obtained from optimization were further refined for their shape

and size to meet the fabrication constraints of the proposed process.

• Use of DaCM as a single-axis force sensor is proposed.

o Unloaded output displacement was identified as the measure of the

sensitivity of the force sensor. These and other attributes were used to

select the best DaCM from the catalog for the force sensor application.

o Topology optimization and shape-size refinement was used to design a

mechanism which is more sensitive than mechanisms from literature.

o The above mechanism was fabricated using Wire-cut EDM and fit with a

Hall-effect sensor for detecting the output displacement. The sensor has a

sensitivity of 0.32 /V N and could resolve better than 3.2 mN .

o This force-sensor was successfully used to estimate the amount force

required to rupture an inflated balloon.

8.3 Future Work

The catalog of DaCMs can never be full. There is a need to identify more mechanisms

and further optimize them for various applications, be it sensors or actuators and add it to

the existing catalog. This would facilitate easier selection of the mechanisms for the

required application, and would save computational effort required to synthesize

mechanisms using topology optimization for every new application. The generality of

spring-mass-lever model makes it applicable for both sensor and actuator applications. In

this thesis, we have dealt with only sensor applications. But all the methods proposed

could be implemented for actuator application with the same ease.


The accelerometer with a DaCM that was designed in Chapters 3 and 4 needs to

be fabricated and tested with a capacitance-measurement circuit to evaluate its

performance. Better designs with higher sensitivities and lower cross-axis sensitivities

need to be obtained. A more comprehensive system-simulation including the electronic

components and realistic-noise models need to be realized to ascertain its sensitivity,

resolution and dynamic-range. The force sensor application illustrated in chapter 6 needs

to be further optimized for lower cross-axis sensitivities. Also, the DaCM designed for

laparoscopic applications in Chapter 7 needs to be fitted into a 1 cm × 1 cm area to be

inserted into the tube. For this there is a need to improve the fabrication in terms of the

minimum fabricatable feature size.

Synthesis of mechanisms for sensor applications using topology optimization is

proposed for accelerometer applications. These examples illustrated use the lumped

spring-mass-lever model to derive the objective functions and constraints required for

optimization. Further optimization of these mechanisms keeping in mind the fabrication

constraints needs to be accomplished. Furthermore, continuum optimization

incorporating multiple thickness layers defining the proof-mass, suspension layers and

the mechanisms could define optimum location of the proof mass and suspensions within

the given area. Overall, this work is to initiate the development of complaint mechanisms

for sensor applications, and future work could continue on similar lines.

A.1

APPENDIX A

A.EFFECT OF FABRICATION LIMITATIONS ON THE RESOLUTION OF AN ACCELEROMETER

In this appendix it is shown how the resolution of an accelerometer is influenced by the

fabrication limitations. To illustrate this, an accelerometer such as the one designed in

Chapter 4 is taken. The dimensions of the accelerometer together with a DaCM are

determined to a large extent by the fabrication process in terms of its minimum feature

size and thickness of the proof-mass and the mechanism. It is shown that slender in-plane

beam widths, large proof-mass thickness, and a smaller suspension thickness can

effectively enhance the sensitivity of the accelerometer.

Figure A.1 An accelerometer proof-mass with suspension, DaCM and the comb-drives

minb

1h 2h

g

Appendix A: Fabrication Limitations on the resolution of an Accelerometer A.2

The minimum feature size denoted by minb determines the stiffness of the

suspension and the DaCM. The relative thicknesses of the proof mass and the

mechanism, i.e. 2 1/h h determines the amount of inertial force created by the proof mass

while the gap g determines the rate of change of capacitance thus contributing to the

overall sensitivity of the accelerometer. Table A.1 shows how these dimensions affect the

sensitivity and thus the resolution of the accelerometer.

The first accelerometer indicates an accelerometer with a DaCM with minimum

feature size ( minb ) and gap ( g ) to be 5 mµ . The suspension, mechanism and the proof-

mass have a uniform thickness ( 1 2h h= ) of 25 mµ . In the subsequent designs, the

accelerometer proof-mass is 250 mµ thick which gives a greater force and the sensitivity

of the accelerometers can be remarkably increased. Such accelerometers can be

fabricated with deep-reactive ion etching (DRIE) on silicon on insulator (SOI) wafers on

the lines of the process proposed in Appendix B. With the gap between the capacitance

fingers reduced to 1.1 mµ , the sensitivity and the resolution can be remarkably improved.

But it should be noted that with the combined surface and bulk micro-machining process

proposed in Chae et al. (2004) it can be shown that 0.3 gµ can be detected.

Appendix A: Fabrication Limitations on the resolution of an Accelerometer A.3

Table A.1 A comparison of the sensitivity and resolution of various accelerometers with different fabrication limitations

µg th

at c

ould

be re

solv

ed

with

10

parts

per m

illio

n

capa

cita

nce

263µ

g

38µg

5.26

µg

0.3µ

g

Impr

ovem

ent

beca

use

of

the

(DA

CM

)

35

19

25.3

7.8

Sens

itivi

ty

(∆C

/C) w

ith

ampl

ifica

tion

0.03

8/g

g=5µ

m

0.30

/g fo

r

g=5µ

m

1.9/

g fo

r a

1.1µ

m g

ap

33.8

/g

for g

= 1

.1µm

Feat

ure

size

of th

e

DaC

M

b min

= 5µ

m

h 1=

25µm

h 2=2

5µm

b min

= 5µ

m

h 1=

25µm

h 2=2

50µm

b min

= 5µ

m

h 1=

25µm

h 2=2

50µm

b min

= h 1

=

3µm

h 2=4

50µm

Sens

itivi

ty

(∆C

/C)

with

out

ampl

ifica

tio

n

0.00

1 /g

for g

= 5µm

0.01

5 /g

for g

= 5µm

0.07

5 /g

for g

=

1.1µ

m

4.3

/g

for g

= 1

.1µm

Feat

ure

Size

25µm

thic

k

susp

ensi

on

and

25µm

thic

k

proo

f mas

s

25µm

thic

k

susp

ensi

on

and

250µ

m th

ick

proo

f mas

s

25µm

thic

k

susp

ensi

on

and

250µ

m th

ick

proo

f mas

s

3µm

thic

k

susp

ensi

on

and

450µ

m th

ick

proo

f mas

s

(diff

til

Acc

eler

ome-

ter

with

its P

roce

ss

Acc

eler

omet

er

(Fig

. A

.1)

with

DR

IE

on

SOI

(5µm

fea

ture

siz

e

and

1.1µ

m g

ap)

Acc

eler

omet

er

(Fig

. A

.1)

with

DR

IE

on

SOI

(5µm

fea

ture

siz

e

thic

knes

s)

Acc

eler

omet

er

(Fig

. A

.1)

with

DR

IE

on

SOI

(5µm

fea

ture

siz

e

and

1.1µ

m g

ap)

Acc

eler

omet

er (F

ig.

A.1

) w

ith c

ombi

ned

surf

ace

and

bulk

mic

ro-m

achi

ning

Si

No . 1 2 3 4

B.1

APPENDIX B

BDRIE WITH SOI PROCESS FOR FABRICATING THE ACCELEROMETER WITH A DaCM

It was shown in Appendix A that accelerometers with thicker proof-mass with thinner

suspension sections yield high sensitivities. To provide for such features, a bulk

micromachining process using deep-reactive ion etching (DRIE) with silicon on insulator

(SOI) wafer was proposed. An SOI wafer has three layers, a structural layer 25 mµ thick,

an intermediate oxide layer, and a base layer which is around 250-500 mµ thick as

shown in Fig. B.1a. The structural layer is used to define the suspension and the DaCM,

while the base layer together with the structural layer defines the proof-mass. The

fabrication steps are shown below.

An SOI (Silicon on insulator) wafer is taken, cleaned and a surface metal layer is

deposited on top of the structural silicon layer which is 25 mµ thick as shown in Fig.

B.1b. The metallization is to provide electrical connectivity between the silicon and the

base of the glass layer on which they will be bonded. The structural layer is then etched

to a depth of 5 mµ to form trenches as shown in Fig. B.1c. The trenches are the regions

where the suspension and the mechanism will be defined. The unetched portions form

protrusions. These protrusions act as support regions, which will be bonded to the glass

wafer. The shallow trenches are further etched to carve out the suspension and the

DaCM. This is shown in Fig. B.1d.

(a) SOI Wafer

Structural Silicon layer 25 mµ thick

Silicon oxide layer, 2 mµ thick

Base Silicon oxide layer, 250 mµ thick

Appendix B: DRIE with SOI process for fabricating the accelerometer with a DaCM B.2

(b) Metallization

(c) Etching a trench on the substrate

(d) DRIE etch of the DaCM and the suspension

Figure B.1 Etching of proof-mass and suspension on the structural silicon layer of the SOI wafer using DRIE.

A pyrex glass 1 mm thick is taken. Aluminium layer is deposited on it by

sputtering an patterned to obtain metal interconnects as shown in Fig B.2a-b. The

The suspension and the DaCM are carved

Metal layer 0.5 mµ thick

A 5 mµ trench is etched on the structural layer


structural layer of the SOI wafer is then bonded on the glass wafer such that the support

structures formed by the 5 mµ etch is directly in contact with the glass.

Figure B.2 Pyrex glass used as the base for the accelerometer

The SOI wafer is then bonded on the glass wafer such that the support structures

formed by the 5 mµ etch in the structural layer are directly in contact with the glass as

shown in Fig. B.3a. The base layer of silicon, which is 250 mµ thick is then etched to

define the proof-mass. This is shown in Fig. B.3b. The etching is wafer thick but stops

short of the oxide layer. The oxide layer connects the structural silicon layer to the base

layer. To release the suspension and the DaCM, the oxide layer above these structures

should be dissolved. However, the oxide layer sandwiched between the proof-mass and

the structural silicon layer should not be etched as this would detach the proof-mass from

the suspension and the mechanism. Once the selective oxide etch is performed, the final

released structure is obtained as in Fig. B.3c.

a) Pyrex glass

b) Aluminium sputtering


(a) Bonding of the structural layer on glass wafer

(b) DRIE etch of the base layer to define the proof-mass

(c) Selective etch of the oxide

Figure B.3. Anodic bonding and defining the proof-mass

Selective etch of the oxide to release the suspension and the DaCM

DRIE etch of the base layer

Anodic of silicon on glass wafer

R.1

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displacement-amplifying compliant mechanisms for sensor

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