DISPLACEMENT-AMPLIFYING COMPLIANT MECHANISMS FOR SENSOR APPLICATIONS

Transcription

1 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 INDIA DECEMBER 2006

2 Dedicated To My Parents

3 TABLE OF CONTENTS ABSTRACT v AKNOWLEDGEMENT vi LIST OF FIGURES viii LIST OF TABLES xiv 1. INTRODUCTION Displacement amplifying Compliant Mechanisms Evaluation and selection of DaCM topologies Topology Optimization for DaCMs High-Sensitivity Micro-g Resolution Accelerometers A Minute Force sensor Using a DaCM Organization of the thesis LITERATURE REVIEW Displacement amplifying Compliant Mechanisms (DaCMs) Compliant mechanisms Displacement amplifying compliant mechanisms (DaCMs) Optimal Design of Compliant Mechanisms Use of DaCMs for sensor applications Micro-g Accelerometers Introduction to Accelerometers Micro-g Accelerometers Force feedback in Accelerometers (Bao, 2000) Noise in Accelerometers Evolution of high-resolution, high sensitivity accelerometers Electronic Circuitry for capacitance detection Need for mechanical amplification for accelerometers 2.27 i

4 ii 2.3 Force sensor for micro-manipulation of Cells Introduction to force sensors for micro-manipulation Use of DaCMs as force sensors Closure OBJECTIVE COMPARISON OF VARIOUS DACMS FOR SENSOR APPLICATIONS Introduction DaCMs for Sensor applications Spring-mass-lever Model of a sensor with a DaCM Objective comparison of some DaCMs Comparison criteria Specification for Analysis Observations and insights Figure of merit Selection vs. Optimization Closure DESIGN OF A MICRO-G ACCELEROEMTER WITH A DACM Introduction General layout of an accelerometer with a DaCM Capacitance detection Selection of a DaCM for an accelerometer Selection of a DaCM for Chea et al. s (2004) micro-g accelerometer Selection of a DaCM for a bulk-micromachined accelerometer for the DRIE process Design of an accelerometer with a DaCM for a given chip area Suspension stiffness ( k s ) Proof mass ( M ) Size of the mechanism ( l mech ) 4.16

5 iii Optimization of the accelerometer for a given chip-area Design of sense combs and external suspension Cross-axis Sensitivities Design of the sense combs External sense-comb suspension Analysis of the accelerometer designs Closure SYSTEM LEVEL SIMULATION OF A MICRO-G BULK-MICROMACHINED ACCELEROMETER Introduction Mechanical components: Mode-Summation Method Calculating the Damping Coefficients Mechanical noise in the system Capacitance detection circuit Closed-loop response Feedback combs Design of the PID controller (Kraft, 1997) Closure TOPOLOGY OPTIMIZATION OF DACMS FOR SENSORS Introduction Topology optimization for sensor applications Objective functions and constraints used for topology optimization Optimality Criterion with non-linear constraints Sensitivity analysis for the Objective functions and constraints Numerical Examples Topology optimization of DaCMs with constraints on cross-axis displacement and natural frequency 6.14

6 iv Topology optimization of DaCMs for accelerometer applications Closure A DISPLACEMENT-AMPLIFYING COMPLIANT MECHANISM AS A MECHANICAL FORCE SENSOR Introduction Use of DaCMs as force sensors Vision based force sensing in micromanipulation of cells Force sensor for laparoscopic surgery Topology optimization of DaCMs for force sensor application Fabrication Process of the mechanism FEM Analysis of the mechanism using COMSOL Displacement Sensing Technique (Hall-effect Sensor) Experimental set-up to calibrate the force sensor and the DaCM Force required to rupture an inflated balloon Closure CONCLUSIONS AND FUTURE WORK Summary Contributions 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

7 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 forcebalance 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-dischargemachining. 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. v

8 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 vi

9 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.

10 LIST OF FIGURES 2.1 Design domain and problem specification for a compliant mechanism problem Principle of operation of an accelerometer (a)the frequency response curve for an accelerometer (b) Resolution vs Bandwidth required for the accelerometer An open loop accelerometer system System representation of a closed loop accelerometer Lumped characterization of the mechanical noise Bulk micromachined piezo-resistive accelerometer by Roylance and Angell Capacitive Z-axis accelerometer with quad beam configuration Capacitive torsional silicon accelerometer Surface Micromachined Z-axis accelerometer by Lu et al. (1995) ADXL 50 surface micromachined lateral accelerometer A high sensitivity z -axis accelerometer using combined bulk and surface micromachining by Chae et al. (2004) A high sensitivity In-plane accelerometer using combined bulk and surface micromachining by Chae et al. (2004) A DRIE in-plane accelerometer Using compliant mechanisms to measure force An accelerometer (sensor) with a DaCM and a suspension at the sensing port Spring-mass-lever model of a sensor combined with (a) an inverting DaCM and (b) a non-inverting DaCM viii

11 ix 3.3 Proposed method to calculate the lumped stiffness of a DaCM (a) input stiffness k ci (b) output stiffness k co Nine symmetric DaCMs labeled M1-M9 along with their deformed configurations of the right half Performance of DaCMs M1-M9 based on six criteria of comparison. The units of U d (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 A plot of the geometric amplification with respect to the characteristic length of the mechanism A plot of the maximum stress with respect to the characteristic length of M1- M (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) kci and (c) k co with respect to the length of the mechanism l mech A plot of the natural frequency of the mechanisms with respect to the characteristic length of M1-M A plot of the net geometric advantage ( NA) with respect to the sensor stiffness ( k s ) (a) Geometric amplification (GA ) (b) Maximum stress (c) Stiffness of all the mechanisms vs. applied force based on geometrically nonlinear analysis A plot of the maximum force that the mechanism can handle before failure Layout of an accelerometer with a DaCM. 4.3

12 x 4.2 (a) A single capacitance and (b) a differential capacitance arrangement (c) Circuit representation of the differential capacitance 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 Arrangement of both the configurations through out the sense-comb mass Optimization of the mechanisms for Chae et al. s (2004) accelerometer Deformed configurations of (a) M1 and (b) M2 that are combined with the accelerometer of Chae et al. (2004) Proof-mass and the suspension of the accelerometer The optimized mechanism size and suspension-length of accelerometer with DaCMs (a) M1 (b) M2 and (c) M 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 The sense capacitance at the output of the DaCM 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 Sense combs with the external suspension. l pos indicates the position of the suspension along the comb holder Effect of the external suspension length l susp2 on the cross-axis sensitivities and resolution of accelerometers with DaCMs (a) M1 (b) M2 and (c) M A plot of cross-axis sensitivity with respect to the position of the external suspension l pos Accelerometer designs with DaCMs (a) M1 (b) M2 and (c) M

13 xi 5.1 A conventional accelerometer represented as a spring-mass-dashpot system and its equivalent in Simulink A state space (Simulink) representation of the mechanical structure using modal-summation method Block diagram of the capacitance detection circuit (Boser et al., 1994) Simulink representation of the electronic capacitive sensing circuit The feedback combs with dc bias Simplified system design of the closed loop accelerometer Mathematical model of the analog closed loop accelerometer Bode plot of the open loop transfer function for design M1 for increasing integral gain k i Bode plot of the closed loop transfer function for increasing proportional gain k p Complete system representation of the accelerometer and the electronics Closed loop response of accelerometer M1.(a) A step signal of 40 µ g at 0.1 s (b) Output at the PID controller (c) Displacement of the sense-combs Closed loop response of accelerometer M2.(a) A step signal of 20 µ g at 0.1 s (b) Output at the PID controller (c) Displacement of the sense-combs Closed loop response of accelerometer M8. (a) A step signal of 60 µ g at 0.1s (b) Output at the PID controller (c) Displacement of the sense-combs 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 Design domain for generating complaint mechanisms with cross-axis displacement constraint Schematic of the optimality criteria with non-linear constraints (a) Ground structure used for optimization for the formulation given by Eq.

14 xii 6.25 (b) Symmetric half of the optimized topology (c) Deformed plot of the mechanism A plot of the (a) objective function and (b) constraint history during optimization for the Case (a) Ground structure used for optimization for case (2) formulation (b) Optimized topology (c) Deformed plot of the optimized topology (a) Ground structure used for optimization for case (3b) formulation (b) Optimized topology (c) Deformed plot of the optimized topology (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 (a) Ground structure used for optimization for formulation given by Eq (b) Optimized topology (c) Deformed plot of the optimized topology Optimized mechanism in conjunction with a proof-mass and suspensions Size optimization of the mechanisms for the force sensor application Vision based force sensing of cells Topology of a DaCM. (a) Ground structure made of grillages (b) Optimized topology Shape and size optimization of the DaCM. (a) Skeletal topology from topology optimization (b) Final mechanism from shape and size optimization SOLID WORKS model of the mechanism showing the dimensions in mm The Spring steel sheet shown along with the fabricated mechanism Wire cut EDM machining the mechanism Force vs. Displacement relation for the mechanism by FE analysis using COMSOL software Analysis in COMSOL indicating the maximum stress of 700 Mpa Hall Effect Sensor surface mounted on a PCB Experimental setup for calibrating the sensor with a DaCM Magnified view of the sensor and the magnet

15 xiii 7.13 Experimental Calibration of the Sensor Figure showing the initial and final output reading before and after the rupture of the balloon A.1 An accelerometer proof-mass with suspension, DaCM and the combdrives.. 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

16 LIST OF TABLES 2.1 Various types of accelerometers and their sensing techniques A comparison of various capacitive accelerometers discussed in the section above Summary of the effect of scaling only the length dimension of the mechanisms on various attributes Net amplifications of the mechanisms optimized for the Chae et al. (2004) accelerometer Weights associated with the four criteria to choose a DaCM for Chae et al. (2004) accelerometer Specifications of a high-resolution accelerometer by Chae et al. (2004) Specifications of the M2 and M8 that are added to the accelerometer of Table Performance of the accelerometer by Chae et al. (2004) of Table 4.3 integrated with M2 and M Net amplifications of the mechanisms optimized for the DRIE with SOI accelerometer Weights associated with the four criteria to choose a DaCM for the DRIE with SOI accelerometer Analysis of accelerometers with DaCMs M1, M2 and M Details of the feedback combs for the three designs discussed in Chapter Specifications of the three designs of the accelerometer Weights associated with a quantity for the computation of the overall figure of merit for a forced sensor application A.1 A comparison of the sensitivity and resolution of various accelerometers with different fabrication limitations.. A.2 xiv

17 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 singlepiece 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). 1

18 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

19 Chapter 1: Introduction 3 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 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 µ g resolution. We demonstrate in this work that similar resolution is possible with a simpler process by using a DaCM in the

20 Chapter 1: Introduction 4 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-oninsulator 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 µ g (0.03%), 40 µ g (0.01%), and 70 µ g (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 displacementamplifying 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

21 Chapter 1: Introduction 5 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.

22 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) 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 2.1

23 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). Displacementamplifying 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 piezoelectric 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

24 Chapter 2: Literature Review 2.3 constant effort to make most mechanisms compliant when it is beneficial. A notable recent real-life application is a compliant wheel (Michelin, 2006) 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 piezoelectric 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 µ m 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.

25 Chapter 2: Literature Review 2.4 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 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 T σ ε dd and MSE = Ω 1 T SE = d 2 σ ε Ω (2.1)

26 Chapter 2: Literature Review 2.5 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 F in ; MSE is the mutual strain energy due to applied and unit dummy loads, F in and F d ; ε and σ are the strain and the stress fields due to the load F in applied at point P 1 ; and ε d is the strain field due to the unit dummy load F d applied at point P 2. It should be noted that MSE is numerically equal to the displacement of point P 2 in the direction of F d due to the applied load, F in. 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. F and u in in P 1 Ω P 2 F and u d out Figure 2.1 Design domain and problem specification for a compliant mechanism problem. A large deflection at P 2 is desired due to the applied load at P 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

27 Chapter 2: Literature Review 2.6 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 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 1) Piezo-resistive ( g) Bending of suspension beams due to proofmass 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. 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. 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. Applications Advantages Disadvantages Air bag deployment in automobiles; High g accelerometers are used for impact testing etc. Air bag deployment in automobiles, Inertial navigation, micro gravity detection, Inertial navigation, zero-gravity measurement in space. a) Simplicity in structure and fabrication. b) Simple readout circuitry generating a low impedance voltage. High sensitivity, good dc response, good noise performance, low drift, low power dissipation, low temperature sensitivity, amenability for feedback. High sensitivity, excellent resolution, linearity can be maintained by operating in the closed loop mode. a) Large temperature sensitivity. b) Small overall sensitivity of 1-2 mv/g and thus requires a huge proof mass. Parasitic capacitance, electromagnetic interference. Difficult to fabricate sharp tips; nonlinear variation of tunneling current with distance; high noise levels

28 Chapter 2: Literature Review 2.7 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. 6) Resonant ( g s) This type of acceleration measurement makes use of the shift in the natural frequency of the structure with applied acceleration. 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. Inertial navigation, detecting small vibrations in machines. Vibration sensors in machine tools. Inclination sensing (Dual axis), automotive, electronic and gaming applications. High resolution, low noise, can operate in places where other principles fail (e.g., when electromagnetic fields are present, we cannot use capacitance sensors.) High dynamic range, sensitivity, bandwidth, adaptable to digital circuits. Low cost, low noise, less drift, simple signal conditioning circuitry. Large weight, complex fabrication process and detection circuitry. Leakage of signal, high noise levels. 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

29 Chapter 2: Literature Review 2.8 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. k ma k a m m y x z = y x Figure 2.2 Principle of operation of an accelerometer A lumped spring mass damper model schematically represents an accelerometer as shown in Fig 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 my && + k( y - x) + d( y& - x& ) = 0 (2.2) For base excitations of the form x = X sin( ωt + ψ ) where ψ is the phase difference between the input and output signals. By substituting this in the above equation, we get 2 2 [(- mω k) sin( ωt) dωcos( ωt)] Z mxω sin( ωt ψ) + + = + (2.3) By considering only the amplitudes in the solution of the above equations, we get

30 Chapter 2: Literature Review Z mω = X ( k mω ) + ( dω) Now, by defining terms of the form recast Eq. 2.4 as Z X = 2 ω ω 2 n 2 ω 2 ω (1 ) + (2 ξ ) 2 ωn ωn ω 2 n (2.4) k d = and ξ = where dc = 4km, we m d c (2.5) When operated at frequencies much less than the resonance frequency, i.e., when ω << ω n, the denominator approaches unity. We then get Z = ω. But 2 2 X / ωn ω 2 X = A (equal to acceleration) gives A Z = (2.6) 2 ωn 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 Q = 1/ 2ξ. 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

31 Chapter 2: Literature Review 2.10 X / A Amplitude Ratio Working range (a) 1 Frequency ratio ω / ω n Range in g Bandwidth in Hz (b) Figure 2.3 a) The frequency response curve for an accelerometer. b) Resolution vs bandwidths required for various applications 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

32 Chapter 2: Literature Review 2.11 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 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 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. 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 ApMa Vout = k From the model shown in Fig. 2.4, we get Vout Figure 2.4 An open loop accelerometer system = 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

33 Chapter 2: Literature Review 2.12 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 Fm = Ma F = F F m f x = F / k Ff = Apqx + Spring-massdamper system with a steady state gain of 1/k Change of capacitance. Gain p A Electrostatic feedback q Vout = Apqx 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 V = Ap ( Ma- Apqx)/ k (2.9) out and the feed back force for small displacements is proportional to the output voltage and can be given by F = Apq( Ma - Apqx)/ k (2.10) f But from the Fig. 2.5, Ff = Apqx. By equating the two expressions we get x = Ma /( Apq + k) (2.11)

34 Chapter 2: Literature Review 2.13 We note that the effective stiffness term has increased. Here, A, p and q are the electronic stiffness parameters. By defining a quantity β = Apq / k as the ratio between the electronic and the mechanical stiffness parameters. The displacement x is now given by x = Ma/ k(1 + β ) (2.12) The value of β could be made much greater than unity, i.e. β >> 1, 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 V out pma β Ma = = k(1 + β ) q(1 + β ) Using the approximation of β >> 1, the output voltage becomes V out (2.13) Ma = (2.14) q 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 mx && + cx& + k(1 + β ) x = 0 (2.15) The natural frequency of such a system is now given by ω β ω β nff = k(1 + ) / m = n (1+ ). (2.16) So, the natural frequency has increased by a factor (1+ β ). The damping ratio now becomes ξ = C/ C = C/ 2 k(1 + β) m = ξ / (1 + β) (2.17) c 0

35 Chapter 2: Literature Review 2.14 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 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. F f = 4 K DT (2.18) B F = Fluctuating force in N / Hz f K B = Boltzmann's constant in N - m/ K D = Damping coefficient in N - s/ m T = absolute temperature in K K F f = 4K DT B M D 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

36 Chapter 2: Literature Review 2.15 signal are within the operating range, then we can express the displacement of the mass to be 4K DT B X n = 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 A X = in m/ Hz (2.20) s s 2 ω0 Now, the signal to noise ratio ( SNR ) is as shown below. SNR s s s s 2 n 4 B ω0 4 B 4 B ω0 X KA MA MA Q = = = = X K DT K DT K T (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 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 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

37 Chapter 2: Literature Review 2.16 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. Piezo resistors Mass Cantilever suspension l m l b 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 K st 12EI = l l l l l b 2 2 b(4 b + 6 m b + 3 m ) (2.22) where I b is the area moment of inertia of the beam, E is the elastic modulus and l m and lb are as shown in Fig 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.

38 Chapter 2: Literature Review 2.17 Electrodes Beams Sli 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.

39 Chapter 2: Literature Review 2.18 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 A capacitive accelerometer using Pressure sensor fabrication technology - Rudolf et al. (1983) Electrodes Suspension Mass 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 3 T 2Gdh α = 3L (2.23) The torque is related to the acceleration by the formula T = a rdm 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.

40 Chapter 2: Literature Review 2.19 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 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 proofmass. 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.

41 Chapter 2: Literature Review 2.20 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 squeezefilm 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 a 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 The L-shaped beams give a very low stiffness in the z -direction and low cross-axis sensitivity. If l b is the length of the beam and I b its area moment of inertia, the stiffness of the entire structure with four beams is given by: K str 24EIb = (2.24) l 3 b 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. Proof mass Suspension Figure 2.10 Surface Micromachined z-axis accelerometer by Lu et al. (1995)

42 Chapter 2: Literature Review 2.21 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 b Surface micromachined lateral ( x y ) commercial accelerometer (ADXL-50) - (Kuehnel and Sherman, 1994) This accelerometer shown in Fig 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 l b is the length of each beam, and I b is the area moment of inertia of the cross section of the beam then the stiffness of the suspension is given by K str 48EIb = (2.25) l 3 b 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 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

43 Chapter 2: Literature Review 2.22 machining processes where huge proof-mass was obtained by bulk micromachining while small gaps and suspension stiffness were realized by surface micromachining. Static combs Moving combs Suspension Proof mass Figure 2.11 ADXL 50 surface micromachined lateral accelerometer a 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 m 2 µ. The thickness of the proof mass is 450 m There are eight suspension springs each 700 µ having a mass of 2.07 mg. µ m long, 3 µ m thick and 40 µ m wide providing a stiffness of 14 N / m. The electrodes are at the top and bottom of the proof mass maintaining a gap of µ m. 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.

44 Chapter 2: Literature Review 2.23 Figure 2.12 A high sensitivity z -axis accelerometer using combined bulk and surface micromachining by Chae et al. (2004) b 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 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 µ m. The stiffness of the suspension is given by K = 96 EI / l (2.26) 3 stiff b b where l b is the length of the suspension length and I b 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.

45 Chapter 2: Literature Review 2.24 Figure A high sensitivity In-plane accelerometer using combined bulk and surface micromachining by Chae et al. (2004) 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 µ m 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 µ m. 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 highresolution 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 µ g 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 basecapacitance 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.

46 Chapter 2: Literature Review 2.25 Figure A DRIE in-plane accelerometer 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 capacitancedetection 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 noisecancellation 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

47 Chapter 2: Literature Review 2.26 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) ( ) Bulk micromachined capacitive accl. (Rudolf et al.,1990) ( ) Surface micromachined accelerometers (Lu et al., 1994) ( ) Acceleromet ers with combination of bulk and surface micron. (Chae et al., ) ( ) Accelerometer s with high aspect ratio etching using DRIE (Chae et al., 2001) ( ) Mass in kg 1e-9 4e-6 0.5e e e-6 Stiffness N/m Natural frequency in Hz Base Capacitance in pf Sensitivity in C/ C / g ( ) Resolution (assuming a circuit with 10 ppm resolution) µ g 650 µ g 1.9 mg 0.4 µ g 400 µ g 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.

48 Chapter 2: Literature Review 2.27 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 semiconductor 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 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 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 micromanipulation 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

49 Chapter 2: Literature Review 2.28 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. δ Figure Using compliant mechanisms to measure force A lot of work has gone into developing elastic-force sensors for micromanipulation. 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 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

50 Chapter 2: Literature Review 2.29 forces applied to elastic mechanisms. This method requires effective noise reduction algorithms for correct force estimation 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 noncontact 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.

51 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 Forceamplifying 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 visionbased sensing of forces in the manipulation of single biological cells. This chapter aims at 3.1

52 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.

53 Chapter 3: Objective Comparison of various DaCMs for Sensor Applications 3.3 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 k s and m s. These can be determined with either an analytical lumped model using the

54 Chapter 3: Objective Comparison of various DaCMs for Sensor Applications 3.4 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 k ci 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 k co is calculated by applying a unit force F out 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 k co is different from that of k ci. The application of force at the output results in an output displacement that is influenced by k ci 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 k ext and m ext. It should be noted that most sensors would not require output spring stiffness k ext. 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). k s m s Sensor ( kci, kco, mci, mci, n) DaCM m ext k ext External (suspension) Figure 3.1 An accelerometer (sensor) with a DaCM and a suspension at the sensing port

55 Chapter 3: Objective Comparison of various DaCMs for Sensor Applications 3.5 By denoting the input and output forces by F in and F, we can obtain the amplified out output displacement in either case as x ( ) 2 Fout ( ks + kcon + kci ) m Fin nkco 2 = 2 kco ks + kci + kext n kco + kci + ks ( ) ( ) (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 x = F k, the net amplification factor, NA, is given by 3 in / NA s 2 = = 3 2 { ( + + ) m ( )} x Fout ks kcon kci Fin nkco ks x k k k k n k k k F 2 { ( + ) + ( + + )} co s ci ext co ci s in (3.2) x 1 k s n k ci x 2 k co k ext (a) x 1 k s n k ci x2 kco k ext (b) Figure 3.2 Spring-mass-lever model of a sensor combined with (a) an inverting DaCM and (b) a non-inverting DaCM

56 Chapter 3: Objective Comparison of various DaCMs for Sensor Applications 3.6 k ci n k ci n x in F in x out k co y in k co F out yout x in F in y in x out k ci = F x in in y out F out k co Foutkci = k y F n ci out out 2 (a) (b) Figure 3.3 Proposed method to calculate the lumped stiffness of a DaCM (a) input stiffness k ci (b) output stiffness k co 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 F out = 0, k ext = 0, Eq. 3.2 simplifies to the form nks n > p kci < ks ( 1) ( k + k ) p s ci (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< that NA must be smaller than n. p, k ci is negative. This means

57 Chapter 3: Objective Comparison of various DaCMs for Sensor Applications 3.7 The natural frequency of the system, which is useful in characterizing the dynamic behaviour, is given as follows f = λ1 λ1 λ2 (3.4) 2π 2 where M m m 2 kco + kext ks + kci + n k co λ1 = + m M 4( k + k )( k + k ) 4n k k λ2 = + Mm Mm 2 s ci co s co ext = s + ci and m mext mco = +, m ci and input and the output sides respectively., m co are the inertia of the DaCM at the 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, m ci and m co. But when they cannot be ignored, we can obtain the approximations for these two quantities, by applying a uniform body load 1 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 k s = 0, m s = 0, k ext = 0, m ext = 0, Fin = mcig and Fout = mco g. Another equation is obtained by evaluating the first natural frequency ( f b ) of the mechanism from the FEA model and equating this with Eq. 3.4 by substituting k s = 0, m s = 0, k ext = 0 and m ext = 0. Eq. 3.6 and Eq. 3.7 are two non-linear equations which can be solved iteratively to get estimates of m ci and m co. y b ( ) m k n k m nk = k k 2 co( co + ci ) m ci co co ci (3.6) co ci + co co ci + co ci co 1 k k nk 1 k k nk 4( k )( k ) fb = + + 2π mco mci 2 mco mci mcomci (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.

58 Chapter 3: Objective Comparison of various DaCMs for Sensor Applications Objective comparison of some DaCMs 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 It is dependent on the stiffnesses ( k s and k ext ) and inertias ( m s and m ext ) 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 ( k cross ): 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.

59 Chapter 3: Objective Comparison of various DaCMs for Sensor Applications 3.9 Thus, to make effective use of a DaCM in practical situations, cross-axis stiffness should be given due consideration. Unloaded Output Displacement ( U d ): Unloaded output displacement ( U d ) is simply the output displacement under an applied input force at the respective points when k s = 0. This quantity is a measure of the flexibility of the DaCM. The mode shapes corresponding to the natural frequencies also matter. The modemagnification factor ( n m or NA m ) 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 ( n m or NA m ) 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 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.

60 Chapter 3: Objective Comparison of various DaCMs for Sensor Applications 3.10 (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.

61 Chapter 3: Objective Comparison of various DaCMs for Sensor Applications 3.11 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 Specification for Analysis Five criteria noted above and the inherent amplification ( n ) are computed for the nine DaCMs shown in Fig In order to compare the topologies of these mechanisms on 6 2 equal grounds, all the mechanisms were modified to fit in an area of µ m. 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 k s while m s, k ext, m ext 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 µ m. The overall thickness of the mechanism is 25 µ m. There is no constraint on the maximum beam thickness provided that the slender beam theory remains valid. That is, the largest crosssection dimensions is less than 1/15 th 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.

62 Chapter 3: Objective Comparison of various DaCMs for Sensor Applications 3.12 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 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 µ N for all the mechanisms. Figure 3.5 shows the six criteria ( NA, n, U c, FS, f, and K cross ) 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 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 size 2, 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.

63 Chapter 3: Objective Comparison of various DaCMs for Sensor Applications 3.13 Figure 3.5. Performance of DaCMs M1-M9 based on six criteria of comparison. The units of U d (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

64 Chapter 3: Objective Comparison of various DaCMs for Sensor Applications 3.14 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 ( k ci and k co ) with the length of the mechanism ( l mech ) can be approximated by the following equations below. k k ci co a b c = + + (3.8a) l l l 3 2 mech mech mech a b c = + + (3.8b) l l l mech mech mech 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 k ci and k co with the size of the mechanism l mech 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 µ N ). 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.

65 Chapter 3: Objective Comparison of various DaCMs for Sensor Applications 3.15 Figure 3.7 A plot of the maximum stress with respect to the characteristic length of M1- M8. (a)

66 Chapter 3: Objective Comparison of various DaCMs for Sensor Applications 3.16 (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) k ci and (c) k co with respect to the length of the mechanism l mech.

67 Chapter 3: Objective Comparison of various DaCMs for Sensor Applications 3.17 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 ( k s ).

68 Chapter 3: Objective Comparison of various DaCMs for Sensor Applications 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 w1α 1+ w2α wnαn P1 = (3.9) w + w w where 1 2 s s s w1α1 + w2α wnα n P2 = w1+ w wn n 1/ s (3.10) wi s denote the weights assigned to the various attributes, αi 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 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

69 Chapter 3: Objective Comparison of various DaCMs for Sensor Applications 3.19 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 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

70 Chapter 3: Objective Comparison of various DaCMs for Sensor Applications 3.20 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 Geometric Quantity advantage ( n ) Effect on scaling Remains constant if the proportions are fixed (Fig. 3.6). Input stiffness ( k ci ) Varies nonlinearly vs. the length scale (Fig. 3.8). Stress Increases linearly vs. the length scale (Fig. 3.7) Inertia (M 1 and m 1 ) Increases linearly. Natural frequency Decreases non-linearly (Fig. 3.9).

71 Chapter 3: Objective Comparison of various DaCMs for Sensor Applications 3.21 (a) (b)

72 Chapter 3: Objective Comparison of various DaCMs for Sensor Applications 3.22 (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 A plot of the maximum force that the mechanism can handle before failure.

73 Chapter 3: Objective Comparison of various DaCMs for Sensor Applications 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.

74 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 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 It consists of a proofmass and suspension along with a 4.1

75 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 crossaxis 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-offreedom 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.

76 Chapter 4: Design of a Micro-g Accelerometer with a DaCM 4.3 DaCM l susp2 l comb l mech Sense combs and external suspension Sense comb fingers DaCM 2 lmech l susp l mass Proof mass with suspension b mass Feedback comb-fingers l bc 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 C 0 ε A = (4.1) d 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

77 Chapter 4: Design of a Micro-g Accelerometer with a DaCM 4.4 b h C0 + C 2 1 C0 C2 d V + V + V (a) 1 (c) V out 3 + C C 0 2 d + x (b) C + C 0 1 d x 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 V out which simplifies to ( ) C0 C2 V = V 2C C + C ( C + C ) C + C Vout = V V 2C0 C1 C2 C0 where Cs = 2C0 is the sense capacitance. C + C V 1 2 = 2C s for C1 C2 0 (4.2)

78 Chapter 4: Design of a Micro-g Accelerometer with a DaCM 4.5 Changes in capacitance C1 and C2 are obtained by a change in the distance between the electrodes d. Referring to Fig. 4.2b, the output voltage is given by ε 0A ε0a ε0a ε0a xd x V d x d x d x d x out = + V = + V = V V for x<< d 2 2 2ε0A 2ε0A d x d d d (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 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 2x x ε A ε A ε A d d C = C C = = = C d x d + x d x x 1 1 d d For small displacements, i.e. x << d, we get (4.4)

79 Chapter 4: Design of a Micro-g Accelerometer with a DaCM 4.6 x d x C x C 0 C 2 0 = x d C 0 a d 1 d (4.5) Moving combs d x Static combs 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. _ + d + x + d x d + x Case ( a ) Case (b ) d1 x + + Case ( a ) Case (b ) 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 x x ε A ε A ε A ε A d d = + = C C0 C d x d1 x d + x d1+ x x 1 x 1 d d1 For small displacements, i.e. x << d, we get (4.6)

80 Chapter 4: Design of a Micro-g Accelerometer with a DaCM 4.7 x x d d1 x x C 0 C 2 0 C 2 0 C1 x d d1 1 x 1 d d1 (4.7) where C0 = 2 ε 0A/ d is the base capacitance between a pair of combs. And C1 = 2 ε 0A/ d1 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 x x 1 1 C0 C1 C d d d 1 1 d x( d1 d) x = = = C C + C d d 0 b d1 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 r a/ b ( / ) ( C/ C ) C C0 b d1 d = = d 0 a 1 d1 1 taking = α, r a / b = 1 (4.9) d α 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 combholder, i.e., l comb in Fig 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 The criteria relevant to the accelerometer application are net amplification ( NA), cross-axis sensitivity ( k cross ), natural frequency

81 Chapter 4: Design of a Micro-g Accelerometer with a DaCM 4.8 ( 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 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 µ m proof-mass and eight suspension beams that are 750 µ m long and 3.5 µ m thick. The minimum feature size that can be realized by this process is 3.5 µ m. The details of the mechanism are given in Table 4.3. This accelerometer has been fabricated with a combination of surface- and bulkmicromachining processes. Grooves along the proof-mass define capacitances with a sense-gap of 1.1 µ m. The capacitance detection arrangement is of the type (a) shown in the previous section with a sensitivity ( C/ C0 ) 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 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 inplane beam widths ( X i ) as the design variables as shown in Fig It should be noted that the topology and the area occupied by the mechanisms remain unchanged during

82 Chapter 4: Design of a Micro-g Accelerometer with a DaCM 4.9 optimization. A comparison of optimized mechanisms would then be a comparison of their topologies alone. The optimization formulation is stated as Max NA ({ X i }) given by Eq. 3.2 NA = 2 { ( + + ) m ( )} mk k n k M nk k s co ci co s 2 { ( + ) + ( + + )} k k k k n k k k F co s ci ext co ci s in Subject to Equilibrium equations And bounds X max < X < Xmin (4.10) i To compute NA from Eq. 3.2 the following quantities were used. M = Proof-mass = 4 mg m = Sense-comb mass = 0.04 mg k s = Sensor stiffness = 15 N / m k ext = External suspension stiffness = 0.01 N / m X min = minimum beam width = 3.5 µ m X i Area occupied by the mechanism is 2 mm 2 mm 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

83 Chapter 4: Design of a Micro-g Accelerometer with a DaCM 4.10 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 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 M M M M M M M M M Table 4.2 Weights associated with the four criteria to choose a DaCM for Chae et al. (2004) accelerometer M1 M2 M4 M7 M8 NA n FS n f n K crossn Case Case Case Weights NA n FS n f n K crossn Case Case Case

84 Chapter 4: Design of a Micro-g Accelerometer with a DaCM 4.11 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 2000 µ m long, 1000 µ m wide and Stiffness Natural frequency k s f 450 µ m thick weighing 2.07 mg Eight beams each 700 µ m long, 40 µ m wide and 3 µ m thick. Spring const. = N/m Hz Sense gap g 1.1 µ m Base capacitance C 0 8 pf Change in capacitance for 1g acceleration Cross axis sensitivity C/ C X % cross Table 4.4 Specifications of the M2 and M8 that are added to the accelerometer of Table 4.2 Mass, M e-9 kg Size of the mechanism 2000 µ m 2000 µ m Magnification -6.5 (M1), 5.8 (M2) Thickness (out of plane) 3.5 µ m Minimum width (in-plane) 3.5 µ m

85 Chapter 4: Design of a Micro-g Accelerometer with a DaCM 4.12 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) Natural Frequency in Hz Sensitivity per g C/ C Cross axis sensitivity X 0.002% 0.05% cross (a) (b) Figure 4.6. Deformed configurations of (a) M1 and (b) M2 that are combined with the accelerometer of Chae et al. (2004) 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 µ m thickness and a base layer 250 µ m 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 µ m. The details of the process are presented in Appendix II. This process requires only DRIE process for defining both thin

86 Chapter 4: Design of a Micro-g Accelerometer with a DaCM 4.13 suspensions as well as the thick proof mass. Furthermore, the tolerances and the minimum dimensions are as high as 5 µ m as compared to 1 µ m 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 bulkmicromachining 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 ({ X i }) given by Eq. 3.2 NA = 2 { ( + + ) m ( )} mk k n k M nk k s co ci co s 2 { ( + ) + ( + + )} k k k k n k k k F co s ci ext co ci s in Subject to Equilibrium equations And bounds X max < X < Xmin (4.11) i To compute NA from Eq. 3.2 the following quantities were used. M = Proof-mass = 4 mg m = Sense-comb mass = 0.04 mg k s = Sensor stiffness = 200 N / m k ext = External suspension stiffness = 1.5 N / m X min = minimum beam width = 5 µ m 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 crossaxis stiffness ( k cross ). In the next section, these mechanisms will be added on to a proof-

87 Chapter 4: Design of a Micro-g Accelerometer with a DaCM 4.14 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 M M M M M M M M M Table 4.7 Weights associated with the four criteria to choose a DaCM for the DRIE with SOI accelerometer. M1 M2 M7 M8 NA n FS n f n K crossn Case Case Case Weights NA n FS n f n K crossn Case Case Case

88 Chapter 4: Design of a Micro-g Accelerometer with a DaCM 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 proofmass 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 proofmass size to obtain high sensitivity Suspension stiffness ( k s ) The suspension chosen for the accelerometer application has a folded-beam design as shown in Fig 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 If l susp is the length of the beam, b susp the in-plane width and h 1 its thickness, the total stiffness, assuming linear-beam theory, is given by k s 3 2Ebsusp h1 3 lsusp = (4.12) where E is the Young s modulus of silicon. b susp and h 1 are determined by the fabrication procedure. In this case, b susp = 10 µ m and h 1 = 25 µ m. The variable l susp needs to be determined Proof mass ( M ) The proof mass M is determined by its geometrical quantities i.e., its length ( l mass ), width ( b mass ) and thickness ( h 2 ) shown in Fig. 4.7 and Fig. 4.1: M = l b h ρ (4.13) mass mass 2

89 Chapter 4: Design of a Micro-g Accelerometer with a DaCM 4.16 where ρ is the density of silicon. The thickness of the proof-mass is fixed by the fabrication process to 250µ m. The quantities l mass and b mass are determined by the total area occupied by the mechanism. This is explained in the next section. l susp b susp Proof mass l mass b mass Figure 4.7 Proof-mass and the suspension of the accelerometer Size of the mechanism ( l mech ) The attributes of the DaCM that are dependent on its size are its input stiffness k ci, output stiffness k co, input inertia m i, output inertia m o 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, l mech, 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 It was further shown in section 3 that k ci and k co can be approximated as a function of the characteristic length l mech as shown in Eq k ci a b c a1 b1 c1 = + + and kco = + + (4.14) 3 2 l l l l l l 3 2 mech mech mech mech mech mech where the constants a, b and c depend upon the Young s modulus of the material, inplane widths of the beams that make up the mechanisms and out of plane thickness which

90 Chapter 4: Design of a Micro-g Accelerometer with a DaCM 4.17 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 m i and to the proof-mass M and sense-comb mass m. m o can be considered negligible when compared The size of the mechanism l mech has to be optimized for a given area 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 2l mech and width ( l mech + l mass ). The total area is then given by Area = 2 l ( l + l ) (4.15) mech mech mass For a given Area, l mass can be determined from the above formula as l mass 2 ( Area 2 lmech ) = (4.16) 2l mech The width of the proof-mass ( b mass ) can be expressed in terms of the mechanism length ( l mech ) and suspension-length ( l susp ) with reference to Fig. 4.1 as b = 2( l l ) (4.17) mass mech susp We have now expressed all the quantities needed for calculating output displacement given by Eq. 3.1 in terms of two geometric parameters l mech and l susp. By substituting the values of all these quantities from Eqs into Eq. 3.1 for k ci, k co, M and k s, we get 2 Eb h ( Area 2 l ) 3 2 susp 1 2 mech m( + k 3 con + kci ) m ( lmech lsusp ) h2 ρ nkci lsusp lmech 2 = 3 3 2Ebsusp h1 2Eb 2 susp h1 kco ( + k ) ( ) 3 ci + kext n kco + kci + 3 lsusp lsusp x ( ) (4.18) where k ci and k co are given by Eq In the above formula, the values for the sense-comb mass m and external-suspension stiffness k ext are as used in Section 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

91 Chapter 4: Design of a Micro-g Accelerometer with a DaCM 4.18 fixed to 25 2 mm. Form the optimized value of l mech and l susp, the proof-mass dimensions can be obtained from Eq and Eq Design of sense combs and external suspension The need for having sense combs at the output of the DaCM was explained in Section 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 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 X cross xcross = 100 (4.19) x des This definition of the cross-axis sensitivity requires us to minimize the ratio of x cross and x des and not just x cross. 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.

92 Chapter 4: Design of a Micro-g Accelerometer with a DaCM 4.19 (a) (b)

93 Chapter 4: Design of a Micro-g Accelerometer with a DaCM 4.20 (c) Figure 4.8 The optimized mechanism size and suspension-length of accelerometer with DaCMs (a) M1 (b) M2 and (c) M3 x cross x des (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 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

94 Chapter 4: Design of a Micro-g Accelerometer with a DaCM 4.21 chosen for this application is 1 pf. The number of comb-fingers needed for this is determined by Eq ε Nbh d = 1 pf (4.20) Taking permittivity of air ε = / F m, thickness of the electrode h 1 = 25µ m and sense gap between electrodes d0 = 5µ m, we can fix the number of combs N and the length of each comb b. By noting that 4 Nb = , we fix N = 40 and b= 700µ m. The sense comb arrangement at the output of the DaCM is shown in Fig It consists of a comb holder of length l comb 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 d 0 = 5 µ m. The movable electrode of one pair and the static electrode of the other pair are isolated from each other by a large distance d = α d, where α > 1 as shown in Fig 4.3, Section 4.2. Larger the value of α, the 1 0 greater the sensitivity as given by Eq However, a large α also increases the combholder 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 l comb. 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 (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.

95 Chapter 4: Design of a Micro-g Accelerometer with a DaCM 4.22 b d 0 Comb holder d 1 l comb Static combs Moving combs Figure 4.10 The sense capacitance at the output of the DaCM. (a)

96 Chapter 4: Design of a Micro-g Accelerometer with a DaCM 4.23 (b) (c)

97 Chapter 4: Design of a Micro-g Accelerometer with a DaCM 4.24 (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 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 is determined by its length l susp2 as its in-plane width b susp2 and out-of-plane thickness h 1 are fixed by the fabrication process to 5µ m and 25µ m respectively. k ext Eb h = (4.21) l 3 susp 1 3 susp2 Figure 4.13 shows the resolution and cross-axis sensitivity for different beam lengths ( susp2 l ). It is seen that as l susp2 increases, the cross-axis sensitivity decreases and the resolution becomes better except for DaCM M8 shown in Fig (d) where an

98 Chapter 4: Design of a Micro-g Accelerometer with a DaCM 4.25 optimum length of l susp2 gives minimum cross-axis sensitivity. Figure 4.14 with reference to Fig shows the optimum position ( l pos ) 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 The suspension length l susp2 for all designs is taken to be 1000 µ m. b susp2 l susp2 l pos l comb Figure 4.12 Sense combs with the external suspension. l pos suspension along the comb holder. indicates the position of the (a)

99 Chapter 4: Design of a Micro-g Accelerometer with a DaCM 4.26 (b) Figure 4.13 Effect of the external suspension length lsusp2 on the cross-axis sensitivities and resolution of accelerometers with DaCMs (a) M1 (b) M2 and (c) M3 (c)

100 Chapter 4: Design of a Micro-g Accelerometer with a DaCM 4.27 Figure 4.14 A plot of cross-axis sensitivity with respect to the position of the external suspension l pos 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 (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.

101 Chapter 4: Design of a Micro-g Accelerometer with a DaCM 4.28 Table 4.8 Analysis of accelerometers with DaCMs M1, M2 and M8 Quantities M1 M2 M8 Area occupied by the structure in 2 mm Base-capacitance ( C 0 ) in pf Sense-gap in µ m Open-loop Resolution (in µ g ) Cross-axis sensitivity (expressed as % of the desired axis sensitivity) Natural frequency in Hz 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 µ g 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.

102 Chapter 4: Design of a Micro-g Accelerometer with a DaCM 4.29 (a) (b) (c) Figure Accelerometer designs with DaCMs (a) M1 (b) M2 and (c) M8

103 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 V/mg, V/mg, and 0.07 V/mg and a resolution of 40 µ g, 20 µ g, and 70 µ g respectively with a circuit noise of 2 mv. 5.1 Introduction An accelerometer with a DaCM has been designed in the previous chapter. The openloop sensitivity of the accelerometer was ascertained. However, it was stated in the literature review that high-sensitivity accelerometers operate in the closed loop (force rebalance) 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 5.1

104 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 systemlevel. The accelerometer-system components can be classified as open-loop and closedloop 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 proofmass 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 openloop 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

105 Chapter 5: System-level simulation of a Micro-g Bulk-micromachined accelerometer 5.3 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 Mx+Dx+Kx=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 N 1 vector containing the displacements of all the degrees of freedom and f the N 1 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-massdamper 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. KU=Λ MU, (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 where U KU =K and T d d d U MU = M (5.3) T d d d U d is a matrix of size N 4 that consists of the modal vectors corresponding to the first four modes, K d and M d 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 N 1 can also be converted to f d of size 4 1 by the following transformation T 1. T f = U f (5.4) 1 d T d

106 Chapter 5: System-level simulation of a Micro-g Bulk-micromachined accelerometer 5.4 K D M (a) x Figure 5.1 A conventional accelerometer represented as a spring-mass-dashpot system and its equivalent in Simulink (b) The same transformation is applied to the damping matrix too. With these four modal parameters ( M, d K d, f d and D d ) independent transient analysis is performed to obtain the state variable corresponding to each mode given by y, i = 1L 4. These individual modal displacements are converted back into the N -dimensional space by the following transformation. T2 u= Udy (5.5) where u is the N 1 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 In the representation below, f, i = 1L 4 i i

107 Chapter 5: System-level simulation of a Micro-g Bulk-micromachined accelerometer 5.5 are components of f d vector. Uri, i = 1L 4 is given by Ud ( r, i ), as we are interested at the displacement at the r th degree of freedom. K, i = 1 4, M, i = 1 4 and Di, i = 1 4 are the diagonal components of the K d, M d and i D d respectively. i Mode 1 Mode 2 Transformation T 1 Mode 3 Transformation T 2 Mode 4 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.

108 Chapter 5: System-level simulation of a Micro-g Bulk-micromachined accelerometer 5.6 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 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 proofmass 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) dsq 3 3 = nµ b l/ h (5.6) and the Couette flow damping coefficient is given by µ A d h where µ = viscosity of air = N- s/ m proof Cf = (5.7) n = no. of comb fingers = b = width of the comb fingers = 25 µ m l = length of the fingers = 700 µ m h = gap between the fingers = 5 µ m A proof = Area of the proof mass = µ m The squeezed-film damping coefficient d sq is calculated to be N s m / while the coefficient arising out of the Couette flow has a value of N s m / Thus the damping coefficient at the sense-comb end is -5 dsc = dsq = N s/ m while at the proof-mass end is -5 dpm dsq dcf N s/ m = + =.

109 Chapter 5: System-level simulation of a Micro-g Bulk-micromachined accelerometer Mechanical noise in the system The source of mechanical noise was discussed in Section It is further seen from Eq 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. where, The spectral density of the noise sources that act on M and m are given by Fn pm= 4KBTdpm and Fn sc= 4KBTdsc (5.8) K B T = Boltzman s constant = 1.238e-23 = Temperature in Kelvin = 300 K d pm = proof-mass damping co-efficient = JK N / d sc = Damping co-efficient at the sense-comb end = -12 F = N / Hz. n pm s m N / s m Fn sc = N / Hz For a given acceleration A s 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 x 2 4 K BTdsc ( ks + kcon + kci ±αnkco ) noise = 2 kco ( ks + kci ) + kext ( n kco + kci + ks ) (5.9) where α = d / d pm sc. Thus, the signal to noise ratio ( SNR ) is given by ( ) x ( ) 2 A M ks + k s con + kci m m nkco 2 noise 4 B sc ( s + co + ci mα co ) SNR = = x K Td k k n k nk (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.

110 Chapter 5: System-level simulation of a Micro-g Bulk-micromachined accelerometer Capacitance detection circuit Figures 5.3 and 5.4 shows the differential capacitance in a bridge arrangement with capacitance C s. The output voltage for such an arrangement is proportional to C/ Cs as shown in Section 4.2. This capacitance change is modulated with a high frequency pulse of opposite phase, shown as V mn and V mp respectively. This modulation voltage V m 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 V diff shown below. V diff C Vm = (5.11) C 2 s 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 noisereduction technique is called as chopper stabilization (Kshirasagar, 2006). The net output of the circuit is given by V out C VV m dm = A (5.7) C 4 s where V dm 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).

111 Chapter 5: System-level simulation of a Micro-g Bulk-micromachined accelerometer 5.9 Differential Capacitance Capacitance Detection Circuit 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

112 Chapter 5: System-level simulation of a Micro-g Bulk-micromachined accelerometer 5.10 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 V/mg, V/mg, and 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 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 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.

113 Chapter 5: System-level simulation of a Micro-g Bulk-micromachined accelerometer 5.11 Assuming an area of overlap of A and a sense gap of d, the electrostatic force between two electrodes is given by F ε AV d 2 0 el = (5.12) 2 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. d x d + x V b V f V b + V f Figure 5.5 The feedback combs with dc bias The feedback force is given by F fb Assuming d ε An ( V + V ) ε An ( V V ) = 2( d x) 2( d + x) fb b fb 0 fb b fb 2 2 >> x, the above equation simplifies to (5.13a) 2ε 0nfb AVV b fb Ffb = (5.13b) 2 d where n fb is the number of feedback combs. It can be seen from the above equation that the feedback force is proportional to the voltage V fb. It is further seen from Eq. 5.13a that the maximum value of the feedback voltage V fb is the bias voltage V b. 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 V b 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 Vfb = V ) b

114 Chapter 5: System-level simulation of a Micro-g Bulk-micromachined accelerometer 5.12 is 2V b, the bias voltage V b 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 V p pm and the sense-comb end V p sc are different. They are given by the formula: where 3 3 8kind0 V = p pm 27ε n A and 8koutd0 V = p sc 27ε n A (5.14) 0 pm 0 sc n pm and n sc 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, d 0 is the gap between the two combs, and k in and k out 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 nk k co ext kin = ( ks + kci ) + k + k co ext and k out ( k + k )( k + k ) + n k k = 2 s ci ext co co ext 2 ks + kci + n kco (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 sense-comb side is taken to be 250 µ m. µ m and that on the Table 5.1 Details of the feedback combs for the three designs discussed in Chapter 4 Accelerometer design M1 (Fig. 4.16(a)) M2 (Fig. 4.16(b)) M8 (Fig. 4.16(c)) Feedback combs in the Feedback combs in the Proof-mass bias voltage Proof-mass bias voltage Pull-in voltage at proof-mass sense-mass Vb pmin V Vb scin V the senseend Vp sc n pm n sc in V

115 Chapter 5: System-level simulation of a Micro-g Bulk-micromachined accelerometer 5.13 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 Vb sc arange = (5.16) sensitivity 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 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 V b sc. 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 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 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 where G f 2 Tm() s = ms + ds + k (5.17) G f signifies the gain of the mechanical amplification. The entire electronic circuitry can be considered as a gain PID controller can be represented by G c if the dynamics of the op-amp are ignored. The

116 Chapter 5: System-level simulation of a Micro-g Bulk-micromachined accelerometer 5.14 ki TPID = kp + + kd s (5.18) s For small deflections, the feedback force can be represented by a gain G fb. The transfer function of the system is represented symbolically in Fig. 5.7 Figure 5.6 Simplified system design of the closed loop accelerometer a + - f m 2 Gm T () s = Gc ms + ds + k I TPID = k k p + + kd s s G fb 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 F OP = 2 m c( p + i + d ) 3 2 fg G ks k ks ms + ds + ks (5.19a) The closed loop transfer function is given by

117 Chapter 5: System-level simulation of a Micro-g Bulk-micromachined accelerometer 5.15 F CL 2 fgmgc( k ps + ki + kds ) ( + fb m c d ) + ( + fb m c p ) + fb m c i = ms d G fggk s k G fggk s G fggk (5.19b) From the denominator of the expression in Eq. 5.19b, it can be seen that coefficients of s and 2 s which signify stiffness and damping respectively have been increased due to feedback. From Fig. 5.8, it can be seen that the increasing k i increases the sensitivity but does not affect the bandwidth. From Fig. 5.9, it can be seen that increasing k p increases the system gain thereby increasing the bandwidth. However, this makes the system unstable because the phase margin decreases. A small k p and a high k i would increase the sensitivity as well as the stability of the system. For this reason k p = 2 and k i = 1500 was chosen for the simulations. Figure 5.10 shows the combined accelerometer system with the electronics. Figures 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 Phase in deg Magnitude in db Frequency in rad/s Figure 5.8 Bode plot of the open loop transfer function for design M1 for increasing integral gain k i.

118 Chapter 5: System-level simulation of a Micro-g Bulk-micromachined accelerometer 5.16 Phase in deg Magnitude in db Increasing k p Figure 5.9 Bode plot of the closed loop transfer function for increasing proportional gain k p. Table 5.2 Specifications of the three designs of the accelerometer. Quantities M1 M2 M8 Base-capacitance ( C 0 ) in pf Sense-gap in µ m Cross-axis sensitivity (expressed as % of the desired axis sensitivity) Frequency in rad/s Natural frequency in Hz Open-loop sensitivity in V / mg Open-loop Resolution (in µ g ) Closed-loop Resolution (in µ g ) Pull-in voltage at the sense-comb end in V Maximum working range in mg Log 10 ( SNR ) for 1 µ g acceleration

119 Chapter 5: System-level simulation of a Micro-g Bulk-micromachined accelerometer 5.17 Figure 5.10 Complete system representation of the accelerometer and the electronics

120 Chapter 5: System-level simulation of a Micro-g Bulk-micromachined accelerometer 5.18 (a) (b)

121 Chapter 5: System-level simulation of a Micro-g Bulk-micromachined accelerometer 5.19 (c) Figure 5.11 Closed loop response of accelerometer M1.(a) A step signal of 40 µ g at 0.1 s (b) Output at the PID controller (c) Displacement of the sense-combs (a)

122 Chapter 5: System-level simulation of a Micro-g Bulk-micromachined accelerometer 5.20 Figure 5.12 Closed loop response of accelerometer M2.(a) A step signal of 20 µ g at 0.1 s (b) Output at the PID controller (c) Displacement of the sense-combs

123 Chapter 5: System-level simulation of a Micro-g Bulk-micromachined accelerometer 5.21 (a) (b)

124 Chapter 5: System-level simulation of a Micro-g Bulk-micromachined accelerometer 5.22 (c) Figure 5.13 Closed loop response of accelerometer M8. (a) A step signal of 60 µ g 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 V/mg, V/mg, and 0.07 V/mg; a resolution of 40 µ g, 20 µ g, and 70 µ g ; 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.

125 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. 6.1

126 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

127 Chapter 6: Topology Optimization of DaCMs for sensors 6.3 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 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

128 Chapter 6: Topology Optimization of DaCMs for sensors 6.4 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 Maximize x subject to MSE / SE SEcross * SE and equilibrium equations. lb xi < ub, where xi x, i = 1 p 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 = u out = T vku in (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 u in. The design domain for this formulation is shown in Fig K is the global stiffness matrix. Strain energy ( SE ) is given by SE = u in 2 = 1 T uin Ku in (6.2) 2

129 Chapter 6: Topology Optimization of DaCMs for sensors 6.5 where u in is the displacement field due to the unit applied at the input point. Strain energy in the cross-axis direction ( SE displacement is given by cross ), which is indicative of the cross-axis where SE cross = u cross 2 = 1 T ucross Ku cross (6.3) 2 u cross 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 uout Fcross and u cross * SE. F in and u in Figure 6.2 Design domain for generating compliant mechanisms with cross-axis displacement constraint Case 2) Generating DaCMs with specified frequency constraints Maximize x MSE SE subject to the ω ω 0 and equilibrium equations. lb xi < ub, where xi x, i = 1 p 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 ω 0 is the square root of the eigen-values of the generalized eigen value problem given by ω 0 = 1 det( ) 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 modeshifts occurring during optimization. These are explained with later in this chapter.

130 Chapter 6: Topology Optimization of DaCMs for sensors 6.6 Case 3) Generating DaCMs for accelerometer applications Maximize x NA (see Eq. 3.2, Chapter 3) Subject to ω ω 0 and SE < SE * and equilibrium equations. cross lb xi < ub, where xi x, i = 1 p 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 It is important to note that this quantity depends upon proof-mass ( M ) dimensions, suspension-stiffness ( k s ), the comb-mass ( m ) and the external-suspension stiffness ( k ext ). 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 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 Maximize x Subject to * MSE SE SE < SE and SE < SE * cross along with equilibrium equations lb xi < ub, where xi x, i = 1 p 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 cross

131 Chapter 6: Topology Optimization of DaCMs for sensors 6.7 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 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 Minimize f ( x) x = { xi} i = 1 p x Subject to equilibrium equations and g( x ) < g* and lb < xi < ub (6.5) where x i, i = 1 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 f( x) g( x) +Λ = 0 x = { xi}, i = 1 p and x x j = 1 p, Λ 0 (6.6) j j

132 Chapter 6: Topology Optimization of DaCMs for sensors 6.8 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 x i, i = 1 p in the iteration as k+ 1 k f( x) g( x) xi = xi + sc +Λ i = 1 p (6.7) xi xi 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 x + i k 1 some elements of { } 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. g( x) g* g( x) xi x= x0 δx+ g( x ) g* where δx= x x g( x) g( x) x x0 g( x0) + g* or xi x x= x i 0 x= x0 g( x) Ax b where Α=, x = { x} = { xi}, i = 1 p xi x= x0 x= x0 g( x) b= { x0} g( x0) + g* xi 0 By substituting the value of Λ in the above equation, we get 0 (6.8)

133 Chapter 6: Topology Optimization of DaCMs for sensors 6.9 where T b Ax+ sc* Am Λ= T sc* A A ( A and m x )m m m f x m δ f are the components of A and f δ ( x) (6.9) whose corresponding elements have neither reached the upper nor the lower bound. The flowchart for the algorithm is shown in Fig 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 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 are presented. i) Ratio of mutual strain energy to the strain energy T MSE = vku in and SE =u T in Ku where the individual terms are explained in the Section 6.1 T MSE v T K T in = in + in + xi xi xi xi But we have Ku = F and where in Kv = F v in in MSE n = SE u Ku v u v K (6.10) F in is the force applied at the input of the structure and F v is the virtual load applied at a point in the structure where the output displacement is desired.

134 Chapter 6: Topology Optimization of DaCMs for sensors 6.10 Problem Specifications Objective f and constraint g Initial guess x 0 Evaluate f ( x) and g( x ) Calculate sensitivities f g and x x i i Guess value for Λ =Λ 0 Update design variables using Eq. 6.7 Update Λ using Eq.6.9 Is Λ Λprevious tol No Is No Yes k 1 k max( ) Yes x + x < tol STOP Figure 6.3 Schematic of the optimality criteria with non-linear constraints.

135 Chapter 6: Topology Optimization of DaCMs for sensors 6.11 Differentiating these two equations and assuming that the applied and the virtual loads are independent of the design variables, we get u K F or u K K + u = = 0 K = u in (6.11) x x x x x in in in in i i i i i Eq. 6.7 makes use of the symmetry of the stiffness matrix. Similarly, we have T T K v = K v and v K = v K (6.12) x x x x i i i i Substituting the above two equations in Eq. 6.10, we get MSE = xi Similarly we have i K T v u in (6.13) xi SE T K = uin u in (6.14) x x i MSE T K T K v uin SE + uin uin MSE SE xi xi So, = 2 x SE i (6.15) ii) Cross-axis displacement ( SE cross ) T SE cross =ucross Ku cross 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 SE x cross i T K = ucross u cross (6.16) x i iii) Natural Frequency ω The generalized eigen-value equation in dynamics is given as Ku =ω Mu (6.17) 2 i i i where ω i is the i th eigen value and the corresponding modal vector is u i. Premultiplying the above equation by T u i, we get the following equation T u Su = 0 and Su = 0 (6.18) i i i

136 Chapter 6: Topology Optimization of DaCMs for sensors 6.12 where S= M K is the dynamic compliance. 2 ( ω i ) Since most eigen-value problems have normalized mass matrices, we have T u Mu 1 (6.19) i i = Differentiating Eq with respect to the j th design variable, we get T T u i T S T u i Sui + ui ui + ui S = 0. x x x j j j Substituting the values for S, in the above equation, we get T S ui u i = 0 (6.20) x j Expanding S in terms of M and K in the above equation, we get ω M K u M u T i 2 i [2 ωi + ωi ] i = 0 xj xj xj Expanding the above equation, we get ω M K u Mu u u u u (6.21) i T 2 T T 2ωi i i + ωi i i i i = 0 xj xj xj Further, using Eq and re-arranging the terms, we get ωi ωi T M 1 T K = ui ui ui u i = 0 (6.22) x 2 x 2ω x j j i j 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 k ci, k co and n are dependent on the design variables. The input stiffness k ci is given by T k = 1/( u Ku ) (6.23) ci in in where u in is the displacement field for a unit input load applied. The derivative of u T in Ku in is as given by Eq The output stiffness co Chapter 3) T 2 2 co = ci /( in ) = ci /( ) k is given by (see Fig. 3.3, k k vku n k MSE n (6.24)

137 Chapter 6: Topology Optimization of DaCMs for sensors 6.13 where v is the displacement field due to a virtual load applied at the output point. Knowing the derivatives of the stiffness k ci, 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 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 u i as shown in Eq 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.

138 Chapter 6: Topology Optimization of DaCMs for sensors 6.14 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 Topology optimization of DaCMs with constraints on cross-axis displacement and natural frequency Case 1. Generating DaCMs with specified cross axis displacement. Maximize x MSE SE subject to SE < SE * and equilibrium equations cross lb xi < ub, where xi x, i = 1 p (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 µ m 500 µ m 3. Thickness of each element = 25 µ m 4. Upper bound for the element width = 10 µ m 5. Lower bound for the element width = 1e-5 µ m 6. Force at the input = 1 µ N 7. Value of SE * = 5e-4 µ m 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.

139 Chapter 6: Topology Optimization of DaCMs for sensors 6.15 (a) (b) (c) Figure 6.4 (a) Ground structure used for optimization for the formulation given by Eq (b) Symmetric half of the optimized topology (c) Deformed plot of the mechanism Case 2. Generating DaCMs with specified frequency constraints MSE Maximize x SE Subject to ω ω0 and equilibrium equations lb xi < ub, where xi x, i = 1 p (6.26) For the example solved here, 1. Ground structure is shown in Fig. 6.6 (a) with the input, output and fixed points. 2. Size of the grid = 500 µ m 500 µ m 3. Thickness of each element = 25 µ m 4. Upper bound for the element width = 10 µ m

140 Chapter 6: Topology Optimization of DaCMs for sensors Lower bound for the element width = 1e-5 µ m 6. Force at the input = 1 µ N 7. Value of ω * = 10e5 Hz 8. Maximized Objective function value = 1.3 (a) Figure 6.5 A plot of the (a) objective function and (b) constraint history during optimization for the Case 1. (b) 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)

141 Chapter 6: Topology Optimization of DaCMs for sensors 6.17 Figure 6.6 (a) Ground structure used for optimization for case (2) formulation (b) Optimized topology (c) Deformed plot of the optimized topology (c) 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 proofmass is around 200 times more than the minimum dimension of the DaCM. The modeshift 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

142 Chapter 6: Topology Optimization of DaCMs for sensors 6.18 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). Maximize x 2 { ( + + ) + ( )} Fout ks kcon kci Fin nkco ks NA( x ) = (6.27) k k k k n k k k F 2 { ( + ) + ( + + )} co s ci ext co ci s in * subject to ω 0 < ω and equilibrium constraints Where ω is given by Eq All the terms are explained in Chapter 3. As mentioned in Section 6.2, only k ci, k co 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 1. Size of the grid = 3000 µ m 3000 µ m 2. Thickness of each element = 25 µ m 3. Upper bound for the element width = 10 µ m 4. Lower bound for the element width = 1e-5 µ m 5. Force at the input = 1 µ N 6. Value of the Sensor stiffness ( k s ) = 197 N / m 7. Value of the proof-mass inertia ( M ) = 7.5e-6 kg 8. Value of the comb suspension ( k ext ) = 1.25 N / m * 9. Value of ω = 2500 Hz 10. Value of the maximized Objective = Value of inherent amplification ( n ) = 3.1

143 Chapter 6: Topology Optimization of DaCMs for sensors 6.19 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 µ m, 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.

144 Chapter 6: Topology Optimization of DaCMs for sensors 6.20 The formulation for the size optimization problem is shown below Maximize NA given by Eq. 3.2 x Subject to equilibrium equations and bounds on x, such that xmax < xi < xmin (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 x min is taken to be 5 µ m 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 µ g 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 x i (a) (b)

145 Chapter 6: Topology Optimization of DaCMs for sensors 6.21 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 (c) Case 1(b) Maximize x NA = NA 2 { ( + + ) + ( )} F k k n k F nk k out s co ci in co s 2 { ( + ) + ( + + )} k k k k n k k k F co s ci ext co ci s in subject to the equilibrium equations and constraints SEcross < SE * (6.29) Where SE cross is the strain energy for a load applied in the cross-axis direction. The crossaxis stiffness is given by Eq. 6.3 and is represented in Fig For the following problem, 1. Size of the grid = 3000 µ m 3000 µ m 2. Thickness of each element = 25 µ m 3. Upper bound for the element width = 10 µ m 4. Lower bound for the element width = 1e-5 µ m 5. Force at the input = 1 µ N 6. Value of the Sensor stiffness ( k s ) = 300 N / m 7. Value of the proof-mass inertia ( M ) = 7.5e-6 kg 8. Value of the comb suspension ( k ext ) = 1.25 N / m

146 Chapter 6: Topology Optimization of DaCMs for sensors Value of SE cross = 0.01 µ m 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 µ m. 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 (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 µ g with an open-loop circuit

147 Chapter 6: Topology Optimization of DaCMs for sensors 6.23 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 proofmass 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 µ g 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.

148 Chapter 6: Topology Optimization of DaCMs for sensors 6.24 Figure 6.10 Optimized mechanism in conjunction with a proof-mass and suspensions

149 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 intraplasmic 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 7.1

150 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 (U od ). This is directly obtained from the expression for output sensitivity given by Eq. 3.1 by equating F 2 = 0, k = 0, and k = 0. This gives x2 U od = F 1 ci s ± n = (7.1) k 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 ( F max ), cross-axis sensitivity ( K cross ), 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. ext

151 Chapter 7: A DaCM as a mechanical force sensor 7.3 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 µ m 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 intraplasmic live-cell injection. Typical forces that are exerted on the cell during manipulation are in the order of 1 µ N (Cappelleri et al., 2006). This usually generates a displacement of 5 µ m, 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 µ N. This corresponds to an unloaded output displacement U od 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 crossaxis sensitivity. To find the most suitable mechanism for this application all the mechanisms introduced in Chapter 3 are made fit into an area of mm 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 ( U od ) is performed. The formulation of optimization problem is given by Eq. 7.2 with reference to Fig Maximize U x given by Eq. 7.1

152 Chapter 7: A DaCM as a mechanical force sensor 7.4 U od = n k ci Subject to the equilibrium and bounds on the design variables x Such that xmax < xi < xmin (7.2) where x is the design-variable vector containing the in-plane widths of all the elements. The minimum width of the beam x min 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 x min was fixed to 5 µ m. The out-of-plane thickness of the mechanisms was fixed to 25 µ m. 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 µ m. Figure 7.2 shows mechanism M2 as a force sensor. The supports of the mechanism are fixed to the laparoscopic tube or the manipulator. x i Area occupied by the Mechanism is 3.5 mm 2.5 mm Figure 7.1 Size optimization of the mechanisms for the force sensor application.

153 Chapter 7: A DaCM as a mechanical force sensor 7.5 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 GA n U od K crossn f n F max Case Case GA n U od K crossn f n F max Weights Case 1 Case Input motion Point of contact Sensing port Input motion Cell 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.

154 Chapter 7: A DaCM as a mechanical force sensor 7.6 It was seen in Table 7.1 that mechanism M2 had greater sensitivity than any other mechanism. Since the sensitivity ( U od ) 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 µ m/ N. This means that output displaces by 0.3 µ m 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 µ m/ N with an output displacement of 12 µ m 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 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 ( k ci ). This can be expressed as U od = n k ci x x x F = out in in in x F = out in (7.3) where x out is the output displacement for a input load F in. Since the input load F in is constant, U od 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 uout MSE Maximize = with respect to the design variables x x u SE in Subject to equilibrium constraints and

155 Chapter 7: A DaCM as a mechanical force sensor 7.7 volume constraint Ax b and bounds xmin < xi < xmax (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 ( U od ). 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 Maximize y U od n = k ci Subject to equilibrium constraints And bounds ymin < yi < ymax (7.4) In the above equation y is the vector of design variables. It consists of a set of element lengths ( l i ), orientation of the element ( θ i ), and width of the element ( x i ). These are shown in Fig. 7.4a.

156 Chapter 7: A DaCM as a mechanical force sensor 7.8 x i l i θ i (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 Its sensitivity is 540 µ m/ N compared to 400 µ m/ N 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 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 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.

157 Chapter 7: A DaCM as a mechanical force sensor 7.9 Output port Fixed end Fixed end Input port 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 edgeundulations 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.

158 Chapter 7: A DaCM as a mechanical force sensor 7.10 Figure 7.6 The Spring steel sheet shown along with the fabricated mechanism EDM wire Figure 7.7 Wire cut EDM machining the mechanism 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 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

159 Chapter 7: A DaCM as a mechanical force sensor 7.11 factor for the mechanism is 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

160 Chapter 7: A DaCM as a mechanical force sensor 7.12 Figure 7.11 Analysis in COMSOL indicating the maximum stress of 700 Mpa 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

161 Chapter 7: A DaCM as a mechanical force sensor 7.13 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 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. Hall Effect Sensor 1cm Ground port Output port Input port 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 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 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

162 Chapter 7: A DaCM as a mechanical force sensor 7.14 Output reading Voltage supply with a regulator Figure 7.10 Experimental setup for calibrating the sensor with a DaCM Mechanism with Hall-effect sensor Magnet Hall Effect sensor Input Output Ground 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 N. Fig 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.

163 Chapter 7: A DaCM as a mechanical force sensor 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 is the output voltage reading before the contact, which is V. When the contact of the mechanism occurs with the balloon, the voltage changes and just when the balloon ruptures, the output voltage reads V. The change in voltage is V. Assuming a sensitivity of V / measured to be N. N, the force was Figure 7.12 Experimental Calibration of the Sensor

164 Chapter 7: A DaCM as a mechanical force sensor 7.16 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

165 Chapter 7: A DaCM as a mechanical force sensor 7.17 which can be used to measure in any direction within a plane. Furthermore, cross-axis stiffness constraints need to be included in topology optimization.

166 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 8.1

167 Chapter 8: Conclusions and Future Work 8.2 level. The designed accelerometers were found to have sensitivities of V / mg, V / mg, and 0.07 V / mg and a resolution of 40 µ g, 20 µ g, and 70 µ g 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 µ g.

168 Chapter 8: Conclusions and Future Work 8.3 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.

169 Chapter 8: Conclusions and Future Work 8.4 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.

170 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. b min h1 h2 g Figure A.1 An accelerometer proof-mass with suspension, DaCM and the comb-drives A.1