design and analysis of micro-electro-mechanical …

DESIGN AND ANALYSIS OF MICRO-ELECTRO-MECHANICAL

SYSTEM (MEMS) BASED DISPLACEMENT AMPLIFICATION

MECHANISM

Sohail Iqbal

(140961)

Department of Mechanical & Aerospace Engineering

Institute of Avionics and Aeronautics

Air University

Islamabad, Pakistan

(2019)

DESIGN AND ANALYSIS OF MICRO-ELECTRO-MECHANICAL (MEMS) SYSTEM BASED DISPLACEMENT AMPLIFICATION

MECHANISM

A Dissertation Presented to

The Academic Faculty

Sohail Iqbal

In Partial Fulfillment of the Requirements for the Degree PhD Mechanical Engineering in the

Department of Mechanical and Aerospace Engineering Institute of Avionics and Aeronautics

Air University [June 2019]

COPYRIGHT © 2019 BY SOHAIL IQBAL

Author’s Declaration

I, Sohail Iqbal hereby state that my PhD thesis titled “Design and Analysis of Micro-Electro-Mechanical (MEMS) System based Displacement Amplification Mechanism” is my own work and has not been submitted

previously by me for taking any degree from Air University, Islamabad or

anywhere else in the country / world.

At any time, if my statement is found to be incorrect even after my

graduation, the University has the right to withdraw my PhD degree.

Sohail Iqbal Jan, 2019

Plagiarism Undertaking

I solemnly declare that the research work presented in the thesis titled

“Design and Analysis of Micro-Electro-Mechanical System based Displacement Amplification Mechanism,” is solely my research work with no

significant contribution from any other person. Small contributions / help wherever

taken, has been duly acknowledged and that complete thesis has been written by

me.

I understand the zero tolerance policy of the HEC and Air University

towards plagiarism. Therefore, I, as an Author of the above titled thesis, declare

that no portion of my thesis has been plagiarized and any material used as

reference is properly referred / cited.

I undertake that, if I am found guilty of any formal plagiarism in the above

titled thesis even after award of PhD degree, the University reserves the rights to

withdraw / revoke my PhD degree and that HEC and Air University have the right

to publish my name on the HEC / Air University websites on which names of

students who submitted plagiarized thesis are placed.

Sohail Iqbal January, 2019

DESIGN AND ANALYSIS OF MICRO-ELECTRO-MECHANICAL (MEMS) SYSTEM BASED DISPLACEMENT AMPLIFICATION

MECHANISM Approved by:

Dr. Afzaal.M Malik, Advisor Department of Mechanical and Aerospace Engineering, IAA Air University, Islamabad, Pakistan

Dr. Rana Iqtidar Shakoor, Co-Advisor Department of Mechatronics Engineering Air University, Islamabad, Pakistan

Dr. Babar Saeed Department of Mechanical and Aerospace Engineering, IAA Air University, Islamabad, Pakistan

Dr. Shafaat Ahmed Bazaz Dean Academics, Center for Advanced Studies in Engineering, Islamabad, Pakistan

Dr. Ibraheem Haneef Department of Mechanical and Aerospace Engineering, IAA Air University, Islamabad, Pakistan

Date Approved: [June 27, 2019]

To my dear Parents, beloved wife Rabia, cute daughter

Maryam & sweet son in heavens Ahmed

ACKNOWLEDGEMENTS

I would like to express my deepest gratitude to Prof. Afzaal Malik at Air University,

Pakistan for introducing me to the amazing culture and philosophy behind research and

development in the field of MEMS. I don’t have enough words to thank Dr. Rana Iqtidar

Shakoor at Air University, Pakistan for his continuous guidance, encouragement and

support. Both of my supervisors provided me an ideal environment to explore ideas in this

emerging field of MEMS. I would also like to thank Prof. Yongjun Lai at MEMS Labs,

Queens University, Kingston, ON, Canada for valuable discussions and advice, and sharing

his expertise and resources with me during my research attachment at Queen’s. Thanks to

Dr. Shafaat Bazaz for helping me in establishing these collaborations through his network

of MEMS professionals in North America. Furthermore, I’d like to thank Air University

and Higher Education Commission, Islamabad, Pakistan. Without their financial support,

this PhD study as well as the research attachments at Queens could have not been possible.

I am thankful to Dr. Jehanzeb Masud, DG IAA, Chair Department of Mechanical and

Aerospace Engineering for providing me the administrative support and relieving me from

my duties for this study period.

During my research attachments at Queens, I enjoyed the company of Jino, Ayan, Rick,

Hanni and Angelica. Here I would like to mention especially my younger brother Salman

Ghaffar at Toronto, ON for his generous hospitality whenever I visited Toronto (which I

did very frequently, mostly because of home sickness). Furthermore, I am also very

grateful to Dr. Ahmed and his family at Kingston for yummy dinners and making my Eid

day special.

I am extremely grateful to Prof. Yongjun Lai, Prof. Michael Kraft, Prof. Karl F.

Böhringer, Prof. Ronald A. Coutu, Prof. Brian D. Jensen and Asst. Prof. Mubasher

Saleem for evaluating my PhD thesis. My heartiest thanks to Prof. Shafaat Bazaz, Prof.

Babar Saeed and Prof. Ibraheem Haneef for their guidance and support.

iv

TABLE OF CONTENTS

ACKNOWLEDGEMENTS iv

LIST OF TABLES viii

LIST OF FIGURES ix

SUMMARY xii

LIST OF PUBLICATIONS xiii

CHAPTER 1. Introduction 1 1.1 Literature Review 2 1.2 Motivation 16 1.3 Objectives and Methodology 21 1.4 Outline of Dissertation 23

CHAPTER 2. Design of Micro Displacement Amplification Mechanism 26 2.1 Introduction 26 2.2 Modeling of Micro Displacement Amplification Mechanism 26 2.3 Configurations of Micro Displacement Amplification Mechanism 27

2.3.1 Configuration 1 28 2.3.2 Configuration 2 29 2.3.3 Configuration 3 30

2.4 Analytical Modeling 30 2.4.1 Mathematical Model 30 2.4.2 Matrix Method 31 2.4.3 Strain Energy Approach 33

2.5 Fabrication of Prototypes 35 2.5.1 Material Selection 35 2.5.2 Fabless Strategy 37 2.5.3 Proposed MUMPs Process 37

2.6 Conclusions 38

CHAPTER 3. Numerical Analysis of Micro Displacement Amplification Mechanism 39 3.1 Introduction 39 3.2 Parametric Analysis 39

3.2.1 Kinematic Analysis 39 3.2.2 Kinetic Analysis 43

3.3 Finite Element Analysis using Direct Stiffness Method 47 3.3.1 Disintegration of Designed Mechanism 49 3.3.2 Element Stiffness Matrices 49 3.3.3 Element External Force Matrices 54 3.3.4 Assemblage of Element Stiffness Matrices 60

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3.3.5 Assemblage of Element Load Matrices 62 3.3.6 Finding Unknown Displacements 62

3.4 Conclusions 64

CHAPTER 4. Finite Element Analysis Using Intellisuite® 66 4.1 Introduction 66 4.2 ABAQUS 67 4.3 Extended Finite Element Method 67 4.4 Static Analysis of Three Configurations 70


4.5 Modal Analysis 79 4.6 Comparison of Amplification Factor 79 4.7 Conclusions 80

CHAPTER 5. Displacement Amplification Mechanism Fabrication and Characterization 81 5.1 Introduction 81 5.2 PolyMUMPs 81 5.3 Kinematic Analysis 83


5.4 Force Analysis 89 5.4.1 Configuration 1 90 5.4.2 Configuration 2 91 5.4.3 Configuration 3 92

5.5 Natural Frequency Calculation 92 5.6 Comparison of Amplification Factors 98 5.7 Conclusions 101

CHAPTER 6. Variant of Displacement Amplification Mechanism 103 6.1 Introduction 103 6.2 Design Description 104 6.3 Analytical Modelling 106

6.3.1 Mathematical Model 106 6.4 Numerical Analysis 107 6.5 Parametric Analysis 111 6.6 Finite Element Analysis 123 6.7 Experimentation 126

6.7.1 Kinematic Analysis 127 6.7.2 Force Analysis 130 6.7.3 Comparison of Results 130

6.8 Conclusions 131

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CHAPTER 7. Micro Displacement Amplification Mechanism incorporated with Chevron Shaped Thermal Actuator 132 7.1 Introduction 132 7.2 Design Description 132 7.3 Mathematical Modeling 135 7.4 Modal Analysis 136 7.5 Steady State Static Thermal Electrical Analysis 137 7.6 Conclusions 145

CHAPTER 8. Conclusions and Recommendations 146 8.1 Conclusions 146 8.2 Recommendations 149

REFERENCES 151

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LIST OF TABLES

Table 1-1 Review of performance parameters of different amplification mechanisms explored by researchers ..................................................................................................... 15 Table 3-1 Geometric parameters and material properties of the designed mechanism .... 45 Table 4-1 Frequencies and associated modes of three configurations of the mechanism 79 Table 4-2 Comparison of displacement amplification factors .......................................... 80 Table 5-1 Natural frequencies of three configurations of the displacement amplification mechanisms ....................................................................................................................... 97 Table 5-2 Comparison of natural frequencies of three configurations of displacement amplification mechanism .................................................................................................. 98 Table 5-3 Comparison of displacement amplification factors .......................................... 99 Table 5-4 Comparison of FEM and experimental amplification factors of configuration 1, 2 & 3 ................................................................................................................................. 99 Table 6-1 Physical properties and dimensions of device ................................................ 123 Table 6-2 Comparison of analytical, FEM and Experimental results ............................. 131 Table 7-1 Natural frequencies of the proposed device ................................................... 137 Table 7-2 Parametric values of variables used in steady state static thermal electrical analysis ........................................................................................................................... 138

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LIST OF FIGURES

Figure 1-1 Schematic diagram of bridge-type mechanism (Lobontiu and Garcia, 2003) .. 3 Figure 1-2 A schematic of three flexure pivots levers in series (Jouaneh and Yang, 2003)............................................................................................................................................. 4 Figure 1-3 Schematic diagram of linear micropositioner (1) Piezoelectric actuator, (2) lever guide device, (3) lever mechanism, (4) toggle amplification mechanism, (5) flexible pivot, (6) parallel guide spring, (7) friction tip, (8) fixed screw holes, (9) preload spring and adjustment screw, (10) cut-out slot to form flexure hinges, (11) flexure hinges and (12) frame (Chu and Fan, 2006) ......................................................................................... 5 Figure 1-4 Schematic diagram of bridge-type mechanism (Ma et al., 2006) ..................... 6 Figure 1-5 FEM model and deformation pattern of the Scott–Russell mechanism (Tian et al., 2009) ............................................................................................................................. 7 Figure 1-6 (a) Artist view of the accelerometer designed by Ya’Akobovitz (b) Deformed shape of the device obtained from finite element simulation (Ya’Akobovitz and Krylov, 2010) ................................................................................................................................... 8 Figure 1-7 Schematic of the single-stage mechanically amplified capacitive accelerometer. The direction of motion is along the y-axis (Zeimpekis et al., 2012) ...... 10 Figure 1-8 A flexure-based bridge-type displacement amplifier with a backward output (Xu and Li, 2011) .............................................................................................................. 11 Figure 1-9 Schematic diagram of Bridge-type mechanism (Qi et al., 2015) .................... 12 Figure 1-10 Schematic diagram of bridge-type mechanism (Lai and Zhu, 2017) ............ 13 Figure 1-11 (a) Schematics of microgripper design (b) Complete microgripper design with integrated capacitive contact sensor (Bazaz, Khan and Shakoor, 2011) .................. 20 Figure 2-1 (a) Schematic diagram of the proposed amplification mechanism (b) Half model of the proposed amplification mechanism at two positions ................................... 27 Figure 2-2 Schematic diagram of proposed amplification mechanism showing functioning having configuration no. 1 ............................................................................. 28 Figure 2-3 Schematic diagram of proposed amplification mechanism showing functioning having configuration no. 2 ............................................................................. 29 Figure 2-4 Schematic diagram of proposed amplification mechanism showing functioning having configuration no. 3 ............................................................................. 30 Figure 2-5 Quarter model of displacement amplification mechanism .............................. 31 Figure 2-6 Spring model of displacement amplification mechanism ............................... 32 Figure 3-1 Schematic diagram of the mechanism ............................................................. 40 Figure 3-2 Plot of Input displacement vs. Amplification Factor showing a decrease in amplification factor from 26.75 to 15.50 .......................................................................... 41 Figure 3-3 Parametric analysis showing change in amplification factor with varying angle made by micro-flexure ...................................................................................................... 42 Figure 3-4 Parametric analyses showing significant increase in amplification factor with increasing length from 250 μm to 600 μm but after 600 μm the change in amplification factor becomes insignificant ............................................................................................. 43 Figure 3-5 Plot showing linear relationship between output force along (a) Y-axis (b) Z-axis and input displacement produced by application of force to the side masses ........... 44

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Figure 3-6 (a) Plot showing a decrease in output force with increasing micro-flexure length (b)Plot showing a decrease in output force with increasing micro-flexure angle .. 46 Figure 3-7 Schematic diagram of the beam element ........................................................ 48 Figure 3-8 Schematic diagram of the disintegrated mechanism ....................................... 49 Figure 4-1 Solid model of micro displacement amplification mechanism ....................... 66 Figure 4-2 (a) Input displacement of 1 μm given to side mass along X-axis (b) Output displacement of 9.17 μm produced along Y-axis ............................................................. 72 Figure 4-3 (a) Reaction force produced along X-axis (b) Y-axis, with an application of input displacement of 1 μm along X-axis ......................................................................... 73 Figure 4-4 (a) Input displacement of 1 μm given to side masses along X-axis (b) Output displacement of 17.09 μm produced along Y-axis ........................................................... 75 Figure 4-5 (a) Reaction force produced along X-axis (b) Y-axis, with an application of input displacement of 1 μm along X-axis ......................................................................... 76 Figure 4-6 (a) Input displacement of 1 μm given to side masses along X-axis (b) Output displacement of 17.04 μm produced along Y-axis ........................................................... 77 Figure 4-7 Reaction forces produced along (a) X-axis (b) Y-axis, with an application of 1 μm input displacement ...................................................................................................... 78 Figure 5-1 Seven layers of the PolyMUMPs process (Cowen et al., 2013) ..................... 82 Figure 5-2 Reference image to find pixel density per µm ................................................ 84 Figure 5-3 Microscopic image of the configuration 1 of micro displacement amplification mechanism ........................................................................................................................ 85 Figure 5-4 Microscopic image of the prototype after application of force along X-axis to the side mass producing output displacement of 8.44 μm along Y-axis .......................... 85 Figure 5-5 Microscopic image of the configuration 2 of micro displacement amplification mechanism ........................................................................................................................ 86 Figure 5-6 (a) Microscopic image of the output before application of force (b) Microscopic image of the output after application of force .............................................. 86 Figure 5-7 Microscopic image of the configuration 3 of micro displacement amplification mechanism ........................................................................................................................ 87 Figure 5-8 (a) Microscopic image of the output before application of force (b) Microscopic image of the output after application of force .............................................. 88 Figure 5-9 Deflection vs. input force plot of configuration 1 ........................................... 91 Figure 5-10 Deflection vs. input force plot of configuration 2 ......................................... 91 Figure 5-11 Deflection vs. input force plot of configuration 3 ......................................... 92 Figure 5-12 Comparison of FEM and experimental results of force analysis of configuration 1 ................................................................................................................ 100 Figure 5-13 Comparison of FEM and experimental results of force analysis of configuration 2 ................................................................................................................ 100 Figure 5-14 Comparison of FEM and experimental results of force analysis of configuration 3 ................................................................................................................ 101 Figure 6-1 Schematic diagram of displacement amplification mechanism .................... 104 Figure 6-2 Spring model of displacement amplification mechanism ............................. 105 Figure 6-3 Input displacement U1y vs. AF1 ..................................................................... 108 Figure 6-4 Input displacement U1y vs. output displacement U4x .................................... 108 Figure 6-5 Input displacement U4x vs. AF2 ..................................................................... 109 Figure 6-6 Input displacement U4x vs. output displacement U8y .................................... 109

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Figure 6-7 Input displacement U1y vs. AF3 ..................................................................... 110 Figure 6-8 Input displacement U1y vs. Output displacement U8y ................................... 110 Figure 6-9 Input displacement U1y vs. AF1 ..................................................................... 112 Figure 6-10 Input displacement U1y vs. output displacement U4x .................................. 113 Figure 6-11 Input displacement U4x vs. AF2 ................................................................... 114 Figure 6-12 Input displacement U4x vs. output displacement U8y .................................. 115 Figure 6-13 Input displacement U1y vs. AF3 ................................................................... 116 Figure 6-14 Input displacement U1y vs. output displacement U8y .................................. 117 Figure 6-15 Input displacement U1y vs. AF1 ................................................................... 118 Figure 6-16 Input displacement U1y vs. output displacement U4x .................................. 119 Figure 6-17 Input displacement U4x vs. AF2 ................................................................... 120 Figure 6-18 Input displacement U4x vs. output displacement U8y .................................. 120 Figure 6-19 Input displacement U1y vs. AF3 ................................................................... 121 Figure 6-20 Input displacement U1y vs. output displacement U8y .................................. 122 Figure 6-21 Displacement produced along (a) X-axis and (b) Y-axis by the proposed mechanism with input displacement of 1 µm applied to four side masses ..................... 124 Figure 6-22 Reaction forces produced by the proposed mechanism with input displacement of 1 µm applied to four side masses along (a) X-axis and (b) Y-axis ...... 126 Figure 6-23 Microscopic image of proposed micro displacement amplification mechanism ...................................................................................................................... 128 Figure 6-24 (a) Microscopic image of the output before application of force (b) Microscopic image of the output after application of force ............................................ 129 Figure 6-25 Displacement vs. input force plot of micro displacement amplification mechanism ...................................................................................................................... 130 Figure 7-1 Schematic diagram of thermally actuated displacement amplification mechanism ...................................................................................................................... 133 Figure 7-2 Displacement produced along (a) X-axis (b) Y-axis when a biased voltage of 2V is applied to thermal chevrons .................................................................................. 139 Figure 7-3 Displacement produced along (a) X-axis (b) Y-axis when a biased voltage applied to thermal chevrons is varied from 1V to 6V ..................................................... 140 Figure 7-4 Plot of decreasing amplification factor with varying voltages from 1V to 6V......................................................................................................................................... 141 Figure 7-5 Reaction produced along (a) X-axis and (b) Y-axis by application of voltage bias of 2V to thermal chevrons ....................................................................................... 142 Figure 7-6 Reaction forces produced along (a) X-axis (b) Y-axis when a biased voltage applied to thermal chevrons is varied from 1V to 6V ..................................................... 143 Figure 7-7 Temperature distribution of proposed device at 2V bias voltage ................. 144 Figure 7-8 Plot shows an increase in temperature with increasing bias voltage ............ 145

xi

SUMMARY

This PhD dissertation reports design and analysis of Micro-Electro-Mechanical system

(MEMS) based displacement amplification mechanism. The proposed displacement

amplification mechanism has been designed and tested in three different configurations

giving experimental amplification factors of 7.6, 16 and 16.14 respectively. The

mechanism has been analytically modeled using kinematic and Direct Force Displacement

Method. Numerical simulations are carried out by writing a code using MATLAB. Results

are found in good agreement with that of the simulated results using FEM based software

IntelliSuite® and experimental results substantiating the viability of this displacement

amplification mechanism. Parametric analysis of the proposed mechanism evaluated the

effect of different geometric parameters, hence finalizing the design geometry. The

analysis predicts that length and the angle of flexure are the two key geometric parameters

that significantly affect amplification factor and have direct relationship with length and

inverse relationship with angle of micro-flexure. Natural frequencies and associated mode

shapes of all three configurations of mechanism during free vibrations have been

determined from modal analysis. Polysilicon based Multi-User MEMS Processes

(PolyMUMPs) has been used for the fabrication of all micro displacement amplification

mechanism prototypes. A micro mechanical probe station equipped with tungsten and

acupuncture needles has been used to test the functionality of prototypes. The amplification

mechanism can be incorporated with micro gripers for increasing displacement at jaw tip.

Design and analysis of a variant of displacement amplification mechanism consisting of

two configurations of designed amplification mechanisms having experimental

amplification factor of 5.8 has been proposed. Thermal and static analysis simulations of

configuration 3 of the designed displacement amplification mechanism incorporated with

thermal chevron actuator and electrostatic comb drives have been carried out and results

discussed in detail at the end.

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https://en.wikipedia.org/wiki/Vibration

LIST OF PUBLICATIONS

1. Iqbal, S., Malik, A. A., & Shakoor, R. I. (2019). Design and analysis of novel micro displacement amplification mechanism actuated by chevron shaped thermal actuators. Microsystem Technologies, 25(3), 861–875. https://doi.org/10.1007/s00542-018-4078-9

2. Iqbal, S., Shakoor, R. I., Lai, Y., Malik, A. M., & Bazaz, S. A. (2019). Experimental evaluation of force and amplification factor of three different variants of flexure based micro displacement amplification mechanism. Microsystem Technologies. https://doi.org/10.1007/s00542-019-04313-6

3. Iqbal, S., Malik, A., & Shakoor, R. I. (2018). Kinematic sensitivity analysis of a novel micro-mechanism for displacement amplification. Transactions of the Canadian Society for Mechanical Engineering, 42(4), 436–443. https://doi.org/10.1139/tcsme-2017-0149

4. Iqbal, S., Shakoor, R. I., Gilani, H. N., Abbas, H., & Malik, A. M. (2018). Performance Analysis of Microelectromechanical System Based Displacement Amplification Mechanism. Iranian Journal of Science and Technology, Transactions of Mechanical Engineering. https://doi.org/10.1007/s40997-018-0213-6

xiii

CHAPTER 1. INTRODUCTION

Microelectromechanical system (MEMS) is a field comprising of micro size

electromechanical devices and structures that are fabricated using special fabrication

procedures. MEMS has played a vital role in revolutionizing inertial sensors and actuators.

System miniaturization, easy integration, small size and weight make MEMS attractive for

many commercial applications. MEMS technology has a remarkable capability of

integrating sensors, actuators and signal processing on a single chip and their integration

has constructive effects upon performance, reliability and cost (Bell et al., 2005).

Amplification mechanisms based on compliant structures are gaining importance in

MEMS applications where motion precision, reliability, accuracy, and compactness are

needed (Bharanidaran and Srikanth, 2016). These displacement amplification mechanisms

govern the sensitivity and performance of micro-devices. Large voltage stroke ratio (µm/V)

is of primary concern for MEMS devices like micro-actuators, micro-gyroscopes and

micro-pumps (Shakoor et al., 2011). So, the role of an amplified displacement with

improved voltage-stroke ratio becomes imperative.

In this dissertation, design, analysis and characterization of a displacement amplification

mechanism for MEMS based micro grippers is presented. In this chapter, an introduction

to micro displacement amplification mechanisms is given. A literature review in the area

of micro displacement amplification mechanisms has been carried out in detail. The

motivation and objectives of the research are presented followed by description of the

general layout of this dissertation.

1

1.1 Literature Review

Producing substantial translational movements has been challenging and of primary

significance in field of MEMS especially in case of sensors and actuators. Performance of

these actuators can be enhanced by incorporating an intermediate mechanism that amplifies

the actuated displacement at the output. These miniaturized amplification mechanisms

should consume less power, be light weight and easily manufacturable with standard

MEMS fabrication techniques.

Typically, for micro displacement amplification mechanism designs, micro-flexures and

micro-hinges are used because they do not need lubrication, have no backlash, have

repeatable motion and are vacuum compatible (Choi, Sreenivasan and Choi, 2008; Liaw

and Shirinzadeh, 2008; Yue et al., 2010; Zongyue Ni ; Dawei Zhang; Yingjun Wu ;

Yanling Tian ; Ming Hu, 2010). Enhancing seismic mass displacement plays an important

role in enhancing the sensitivity of MEMS device by sustaining a high compliance and

low damping coefficients along the sense axis (Yazdi and Najafi, 2000; Amini, Abdolvand

and Ayazi, 2006; Abdolvand, Amini and Ayazi, 2007; Xue et al., 2008; Hsu et al., 2010).

To suspend proof mass, compliant mechanisms are used which are typically flexures. They

not only act as suspensions but also act as joints (L. L. Howell, 2001).

Pokines and Garcia (Pokines and Garcia, 1998) developed a MEMS based micro

amplification device through integration of smart material with MEMS theory. Lobontiu

and Garcia (Lobontiu and Garcia, 2003) used the principle based on strain energy and

Castigliano’s second theorem to analyze the relationship between the ideal displacement

amplification ratio of bridge-type mechanism and input displacement as shown in Figure

2

1-1. Authors formulated an analytical method for displacement and stiffness calculations

of planar compliant mechanisms with single-axis flexure hinges. A parametric study of the

mechanism performance was performed based on the mathematical model. Neither the

mechanism was analyzed dynamically nor was experimentation carried out. Only

numerically and FEM results were compared.

An efficient approach for the modeling and simulation of single-axis, flexure-hinge type,

translational, piezo driven micro positioning stages was reported by Jouaneh and Yang in

2003 as shown in Figure 1-2 (Jouaneh and Yang, 2003).

Figure 1-1 Schematic diagram of bridge-type mechanism (Lobontiu and Garcia, 2003)

3

Figure 1-2 A schematic of three flexure pivots levers in series (Jouaneh and Yang, 2003)

A novel long-travel piezoelectric-driven linear nano-positioning stage capable of operating

in stepping mode as well as in a scanning mode was reported in 2006, as shown in Figure

1-3. In the stepping mode, the stick–slip friction effect between a linear micro-positioner

and a sliding stage was used to drive the stage step-by-step. In the scanning mode, the

micro-positioner acts as an elastic deformation-type linear displacement amplification

device and drives the stage through displacements in the micrometer level range (Chu and

Fan, 2006). However, non-linearity effects of elastic deformations were not considered.

Moreover, the stress increases in higher values for smaller displacements leading to the

failure of mechanism.

4

Figure 1-3 Schematic diagram of linear micropositioner (1) Piezoelectric actuator, (2) lever guide device, (3) lever mechanism, (4) toggle amplification mechanism, (5) flexible pivot, (6) parallel guide spring, (7) friction tip, (8) fixed screw holes, (9) preload spring and adjustment screw, (10) cut-out slot to form flexure hinges, (11) flexure hinges and (12) frame (Chu and Fan, 2006)

Ma et al. (Ma et al., 2006), using kinematic theory and virtual work principle, derived the

ideal displacement amplification ratio of bridge-type mechanism, as shown in Figure 1-4.

Authors presented ideal amplification ratio against angular change, but the formulations

were not suitable for smaller angles which are useful for gaining large amplification ratios.

Theoretical amplification factor of 40 and that obtained by Finite Element Method (FEM)

was 20. FEM was used for comparison with the mathematical model, and to analyze the

mode shapes of the structure (Ma et al., 2006). Effect of thickness on displacement

amplification ratio was discussed in detail but length and width were not discussed. No

experimentation was carried out to validate the design.

5

Figure 1-4 Schematic diagram of bridge-type mechanism (Ma et al., 2006)

A flexure-based Scott–Russell mechanism was designed and developed in 2009 for the

nano-manipulation applications using piezoelectric actuator that improved mechanical

performance, as shown in Figure 1-5. Finite element analysis (FEA) was used to investigate

the mechanical structure of the Scott–Russell mechanism. A lumped parameter dynamic

model of the flexure based Scott–Russell mechanism was developed to investigate the

vibration and dynamic responses (Tian et al., 2009).

6

Figure 1-5 FEM model and deformation pattern of the Scott–Russell mechanism

(Tian et al., 2009)

Assaf Ya’akobovitz and Slava Krylov used proof mass displacement for mechanical

amplification, as shown in Figures 1-6 (a & b). Using their proposed mechanical

amplification principle, authors presented an operational principle of MEMS inertial sensor

having improved sensitivity. Silicon-on-insulator (SOI) technology was used to fabricate

the devices. The devices were operated using electrostatic and inertial actuation of the

proof mass combined with the optical sensing (Ya’Akobovitz and Krylov, 2010). Bending

compliance was not taken into consideration that has major effect on length of sensing

element.

7

Figure 1-6 (a) Artist view of the accelerometer designed by Ya’Akobovitz (b) Deformed shape of the device obtained from finite element simulation

(Ya’Akobovitz and Krylov, 2010)

8

Authors presented an operational principle based on the out-of-plane linear to angular

motion transformation that was achieved using an integrated compliant motion amplifier.

Out-of-plane deflections of the proof mass are in nanometers that are amplified 12.1 times

under inertial forces. Since this is a capacitive sensing device that may result in improved

sensitivity because acceleration was extracted from measured large tilting deflections of

the sensing element (Ya’Akobovitz and Krylov, 2010).

Dimensions of suspension mechanism act as a limiting factor for sensitivity by restricting

the motion according to their stiffness(Zeimpekis, Sari and Kraft, 2012). Performance of

MEMS sensors, that are entirely dependent on deflection to produce an output signal, was

enhanced by incorporating mechanical motion amplifiers. A single-axis mechanically

amplified capacitive accelerometer, using a system of micromechanical levers, with

amplification factor of 40 was designed, simulated, and fabricated by Zeimpekis et al., in

2012 with improved sensitivity, as shown in Figure 1-7. Authors showed that the addition

of the mechanical amplifier provides higher deflection without sacrificing bandwidth and

does not alter the noise floor of the sensor. The mechanical motion amplifier produces large

displacement that improves the capacitive signal at electronic readout stage as compared

to the conventional accelerometer design. Mechanical amplifier implemented with large

sensing element does not add significant Brownian noise nor does it influence the

electronic noise level of the interface circuitry. Hence, the overall signal-to-noise ratio

(SNR) of the sensor is improved (Zeimpekis, Sari and Kraft, 2012). Implementation of

system of microlevers to an accelerometer produced statically indeterminate mechanism

and bulky equations, which were not analyzed.

9

Figure 1-7 Schematic of the single-stage mechanically amplified capacitive accelerometer. The direction of motion is along the y-axis (Zeimpekis et al., 2012)

Some other types of amplification mechanisms are also used such as lever type mechanism

(Jouaneh and Yang, 2003; Chu and Fan, 2006), bridge-type, four bar linkage and Scott-

Russell mechanisms (Xu and King, 1996; Kim, Kim and Kwak, 2004; Ma et al., 2006;

Tian et al., 2009; Xu and Li, 2011).

A flexure based compound bridge-type displacement amplifier having large amplification

ratio, compact size, higher lateral stiffness values was reported in 2011, as shown in the

Figure 1-8 (Xu and Li, 2011). One drawback of this amplifier was its low lateral stiffness

value. If the amplifier observed a large lateral load, the low lateral stiffness will not be

sufficient to tolerate it. To resolve this issue, double bridges were constructed making the

mechanism structure complex.

10

Figure 1-8 A flexure-based bridge-type displacement amplifier with a backward output (Xu and Li, 2011)

Ye et al. (Ye et al., 2010) used kinematic principle and elastic beam theory to analyze the

theoretical displacement amplification ratio of bridge-type mechanism. Authors

established a relationship between length and angle of flexural hinge with the amplification

ratio. Amplification ratio first increases then decreases with increase in flexure hinge length

giving maximum value at 0.688°. An amplification factor of 45 was obtained. Authors did

not considered tensile deformation and deflection of the hinges.

Ke-qi Qi et al. (Qi et al., 2015) analyzed kinematics of bridge type mechanism and derived

displacement amplification ratio formula using elastic beam theory. It was concluded that

the displacement amplification ratio of bridge-type mechanism was not related to the

material and thickness of the structure, and it was verified by FEA. Authors compared

different existing analytical models with their own. Results of linear and non-linear static

analysis were presented for different input displacements. An amplification factor of

approximately 7 was obtained. Analytical results were not verified by experiments.

11

Figure 1-9 Schematic diagram of Bridge-type mechanism (Qi et al., 2015)

The static and dynamic characteristics of the mechanically coupled and electrostatically

actuated H resonator with amplified output using polyimide as the structural layer was

presented by Ilyas et al. (Ilyas, Jaber and Younis, 2016). The electrode network allowed

the resonator to be actuated from either of the C-C beams or the coupler.

The effective actuation stroke of piezoelectric actuator was enhanced by Lai and Zai (Lai

& Zhu, 2017) with the help of displacement amplification mechanism as shown in Figure

1-10. The mechanism operational frequencies with and without actuator mounted were 155

Hz and 178 Hz respectively (Lai & Zhu, 2017).

12

Figure 1-10 Schematic diagram of bridge-type mechanism (Lai and Zhu, 2017)

Based on the elastic beam theory, a better analytical model of displacement amplification

ratio was derived for bridge-type amplification mechanism (Lin, Shen, Wu, & Yu, 2018).

The analytical model was compared with existing models, FEA and prototype. Kinematic

model for 6 DOF micro/nano positioning stage based on the improved analytical model

was developed.

A 3-D bridge-type mechanism was improved and compared with the previous designs

having advantages of compact size, simple design and symmetric structure (Chen et al.,

2018). Kim et al., (Kim, Kim and Kwaka, 2003) combined two bridge type mechanisms

13

and proposed a three dimensional amplification mechanism. It was expected that the

amplification will increase but it decreased from 98 to 10. Chen et al. (Chen et al., 2018)

presented that the reason of decrease in amplification factor was non configuration of

stiffness distribution in the two bridges. Authors improved the design by properly

configuring the overall stiffness of the mechanism. In addition to evaluating amplification

ratio, amplification rate was also analyzed that predicted the displacement loss in the

mechanism. Performance of the amplifier with amplification ratio of 41 and relative

amplification rate of 0.93 was confirmed with FEA simulation and experiments. Four types

of flexure, leaf, cartwheel, right circular and leaf hinges were compared and V type proved

to be the one giving maximum amplification (Chen et al., 2018).

A comparison of different characteristics of displacement amplification mechanisms is

shown in the Table 1-1. Most of the designed amplifiers produce in-plane amplified

displacements. Accelerometers, piezoelectric and electrostatically operated devices can be

benefited by mechanically amplifying signals. Drawbacks of displacement amplifiers is

that overall stiffness of the device increases that ultimately affects the modal frequencies.

Incorporation of these amplifiers increases the overall size of the micro device. When it

comes to electrostatic actuation or sensing an amplifier benefits by increasing output

deflections without affecting the noise level.

14

Table 1-1 Review of performance parameters of different amplification mechanisms explored by researchers

Reference In-Plane/ Out of plane

Displacement amplification

factor

Mechanism type

Theory Operating frequency

Manufacturing process

(Lobontiu and Garcia,

2003)

In-Plane

10 Flexure based

compliant

Castigliano’s theorem and elastic beam

theory

_

_

(Kim, Kim and Kwaka,

2003)

Out of plane

41 Flexure based

compliant

screw theory _ Wire-EDM

(Ma et al., 2006)

In-Plane

20 Flexure based

compliant

Elastic beam theory

_ _

(Chu and Fan, 2006)

In-Plane

13.7 Flexure based

compliant

Lever principle 1060 Hz Electrical discharge machining

(EDM) (Tian et al.,

2009) In-

Plane 10 Flexure

based compliant

Kinematics by geometry and Hookes law

114Hz Wire EDM

(Ya’Akobovitz and Krylov, 2010)

Out of plane

12.1 Flexure based

compliant

Lumped mass 2.87kHz SOI - DRIE

(Ye et al., 2010)

In-Plane

43 Flexure based

compliant

Elastic beam theory

_ Wire EDM

(Xu and Li, 2011)

In-Plane

4.5 Flexure based

compliant

Euler–Bernoulli beam theory

377.9 Hz

(Zeimpekis, Sari and

Kraft, 2012)

In-Plane

40 Flexure based

compliant with

micromechanical levers

Mass spring ODE’s

749Hz SOI-DRIE

(Qi et al., 2015)

In-Plane

7 Flexure based

compliant

Elastic beam theory

_ _

(Lai and Zhu, 2017)

In-Plane

27.5 Lever type Mass spring ODE’s

155 Hz EDM

(Lin et al., 2018)

Out of plane

12 Flexure based

compliant

Energy principle and elastic beam

theory

_

Wire EDM

Design and development of MEMS based devices having small size, capable of producing

large displacement with low power consumption, has always been prime focus of

designers. Electrostatic, electro-thermal, piezoelectric and electro-magnetic are the

conventional actuation mechanisms.

15

Thermal actuators using thermal expansion effects, activated by Joule heating, can provide

a large force and actuation in directions parallel and perpendicular to the substrate and

fabricated using micro-micromachining technology. Thermal actuators are categorized as

hot/cold arm thermal actuators and ‘V’ or ‘Chevron’ shaped actuators (Hickey et al., 2003;

Y. Lai, J. McDonald, M. Kujath, 2004). In-plane thermal actuators are classified as

bimorph design (Comtois, Bright and Phipps, 1995) and the chevron design (Que, Park

and Gianchandani, 1999; Rubbeca Cragun, 1999). Sinclair and Wang stated that chevron

shaped actuator as more efficient since “all bending beams are force-producing, even when

they are arrayed to increase output force” (Sinclair and Wang, 2003).

Vertical displacement of a chevron actuator is unwanted in design. Sinclair (Sinclair, 2000)

proposed measures to prevent these types of side effects. There is no vertical displacement

if conventional thermal chevrons is used because of constant width of micro beams and

constant pre-bending angle (Varona, Tecpoyotl-Torres and Hamoui, 2009a).

This dissertation presents design, analysis and experimentation of a new micro

displacement amplification mechanism. The viability of design is validated through

comparison of analytical, FEA (using commercial software IntelliSuite®) and

experimental results.

1.2 Motivation

One challenge that researchers have faced, while designing MEMS based inertial sensors

and actuators, is the production of significant output motions against smaller input

displacements. The challenge has been handled with thought provoking ideas in the area

of micro displacements amplification mechanisms (Bolzmacher et al., 2010; Lin et al.,

16

2018). These mechanisms, when incorporated into sensors and actuators, showed

remarkable increase in performance of devices without consuming high power and

manufacturing cost as they are easily made through standard MEMS fabrication techniques

(Millet et al., 2004; Ya’Akobovitz and Krylov, 2010).

A compliant based displacement amplifier has been proven to be helpful whether it’s piezo-

electric actuation or electrostatic sensing because of its adjustable compliant properties

merely based on micro flexures and flexure hinges. Wish to get more and more

displacement amplification has led to complex geometries generating nonlinearities in the

whole device design. Complicated designs and flexure hinges become source of stress

concentration that lead to failure especially when the fabricated material is brittle like

polysilicon. Amplification mechanisms designed having complex geometry and flexure

hinges showed low lateral stiffness value that will not tolerate large lateral load. In this

scenario the mechanism is to be strengthened either by adding more flexures or by

changing the dimensions (Xu and Li, 2011). An attempt to increase the amplification factor

using two bridge-type mechanisms resulted into decrease in amplification factor. Reason

for this decrease was complex geometry resulted in non-configuration of stiffness

distribution in the two bridges. The design was improved by making size more compact,

simple and symmetric. Simple designs without complex geometries and complex flexural

hinge designs will solve the nonlinearity issues and will be helpful in amplifying small

linear or angular displacements which are useful for gaining large amplification ratios.

In-plane devices that are being operated using inertial actuation and electrostatic sensing

shows a significant improvement in capacitive signal with the incorporation of amplifier

(Han et al., 2015). However, undesired out of plane motions are not taken seriously that

17

can be handled with simple geometric changes and boundary conditions taking bending

compliance into consideration that has major effect on length of sensing element.

Designers have proposed amplification mechanisms specialized to the device design like

micro grippers, micro mirror positioners, micro accelerometers, micro actuators etc. (Millet

et al., 2004; Chen, Schneider and Wallrabe, 2010; Ya’Akobovitz and Krylov, 2010; Lin et

al., 2018) . Mechanical amplifiers do not add noise and have no influence on readout

circuitry, thus improving signal-to-noise ratio. However, adding a series of these amplifiers

make the system statically indeterminate.

Extensive research has been carried out to improve the performance of piezo-electric

actuators, capacitive actuators and accelerometers by incorporating displacement

amplifiers. However, application of amplifiers to thermal actuators, micro grippers and

energy harvesters is yet to be explored. Besides improving the overall performance of

device, displacement amplifiers increases overall size and stiffness of the device that

ultimately affects the modal frequencies.

Motivation of this research work is to design a MEMS based displacement amplification

mechanism to enhance the total displacement at the tip of jaws of already designed micro

grippers, by Bazaz et al. (Bazaz, Khan and Shakoor, 2011). An electrostatically actuated

microgripper with novel capacitive contact sensor was reported by Bazaz et al. (Bazaz,

Khan and Shakoor, 2011) as shown in Figures 1-11(a & b). The microgripper prototype

was fabricated using standard SOI-MUMPs process with total size of 5.03mm×6.5mm.

FEA of microgripper showed a total displacement of 15.5µm at the tip of jaws when

voltage of 50Vdc was applied at the actuator. When tested experimentally, a total

18

displacement of 17 µm at the tip of microgripper jaws with displacement amplification of

approximately 4.0 times was achieved. To enhance the output displacement at ends of jaws

an intermediate displacement amplification mechanism with high amplification factor is

required. So an intermediate displacement amplification mechanism, incorporated with the

microgripper design that could produce 20 times amplified displacement at the tip of jaws

with application of same voltage at the actuator is designed.

(a)

19

(b)

Figure 1-11 (a) Schematics of microgripper design (b) Complete microgripper design

with integrated capacitive contact sensor (Bazaz, Khan and Shakoor, 2011)

The proposed amplification mechanism gives different amplification factors with different

boundary conditions making it a versatile mechanism that can also be incorporated with

MEMS applications like micro accelerometers, micro actuator and micro energy

harvesters. This gives a freedom to the designer to utilize this mechanism in his/her design

as per requirements. The mechanism has been designed according to commercial available

fabrication process PolyMUMPs unlike devices that need customized fabrication process.

20

Apart from getting intellectual joy by doing some creative work, personal motivation of

doing this research is to gain knowledge in the wide area of MEMS especially compliant

based displacement amplification mechanisms and to be a part of an interdisciplinary

research team that consists of members with different professional and cultural

backgrounds.

1.3 Objectives and Methodology

Objective of this research is to design an efficient micro displacement amplification

mechanism, with a target amplification factor of 20, which can be incorporated with a

micro gripper. Apart from amplifying displacement 20 times, the proposed mechanism can

be utilized in three configurations giving different amplification factors. Three

configurations are achieved by changing only the boundary conditions.

Design methodology flow chart is shown below. Numerical analysis of the mechanism will

be done by using kinematic model followed by parametric analysis by coding the kinematic

model using MATLAB. Geometric parameters will finalized by taking into consideration

parametric analysis results, desired amplification factor, overall size of the mechanism and

PolyMUMPs design rules. Same parameters will be used for carrying out numerical

simulations with direct stiffness method that is an implementation of FEM. Configuration

2 will be our prime focus and will be numerically analyzed using kinematic model and

direct stiffness method. However, all the three configurations will be analyzed while doing

FEM static simulations using commercial software IntelliSuite®.

21

Problem Statement

Analytical modeling

Finite Element Analysis

Direct Stiffness Method

(FEM) using MATLAB

Results Comparison

Prototype Fabrication

Prototype Testing

Literature Review

Design Specifications

Parametric Analysis

Design parameters finalization

Numerical Analysis Kinematic analysis

using MATLAB

Refine the design

22

To carry out simulation on IntelliSuite®, the amplification mechanism will be designed in

a way that input displacement will be provided by thermal chevrons and output

displacement sensed by electrostatic comb drives. Same designs will be fabricated and

tested. To restrict out of plane motion, supporting flexures, attached to the ends of combs,

will be incorporated. Static analysis of displacement amplification mechanism having three

configurations will be carried out under unidirectional input displacement load conditions.

Prototypes of three configurations of displacement amplification mechanism will be

designed and fabricated according to Polysilicon-Multi User MEMS Processes

(PolyMUMPs) from MEMSCAP, USA. All the experimentation will be conducted in the

MEMS Lab, Queen’s University, Kingston, ON, Canada. Experimental results using the

developed model, simulated results using FEM based software IntelliSuite® and analytical

results will be compared substantiating the viability of this amplification mechanism.

A variant of displacement amplification mechanism consisting of configuration 2 and

configuration 3, of already designed and tested amplification mechanism, will be designed

and analyzed. Mathematical model of the mechanism will be developed using kinematic

approach followed by parametric analysis to optimize the design and finalize the design

parameters. FEA will be performed using commercial software IntelliSuite®. Prototype

will be fabricated and tested using same testing and fabrication facilities.

1.4 Outline of Dissertation

In chapter 1, literature review focusing on micro displacement amplification mechanisms

is presented. The review covered operating principles and application in particular.

Moreover, objective and motivation behind this research work is discussed briefly.

23

Chapter 2 broadly covers the kinematics and kinetics of displacement amplification

mechanism with its mechanical design implementation. Design of proposed displacement

amplification mechanism along with its three configurations has been discussed in detail.

After design implementation, we briefly discussed implementation of fabless strategy

using PolyMUMPs process.

Numerical results from analytical model of the mechanism have been presented in chapter

3. Results generated from kinematic and kinetic analysis have been discussed in detail.

Results obtained by parametric analysis of the designed mechanism have been

comprehensively concluded in this chapter. This chapter also presents implementation of

direct stiffness matrix method, a finite element analysis technique on the designed

mechanism. Main emphasis of this work is to demonstrate the implementation of FEA

technique. FEA model developed is simulated using a code giving nodal displacements and

forces produced on application of distributed load and the given boundary conditions.

In chapter 4, FEA of the designed mechanism is presented. Finite element simulations

including kinematic analysis, kinetic analysis and modal analysis are carried out to predict

the performance of the proposed mechanism.

In chapter 5, fabrication and characterization of proposed mechanism is presented.

Prototypes of the amplification mechanism are fabricated from MEMSCAP, USA through

CMC Microsystems, Canada using commercially available Multi-User MEMS Processes

PolyMUMPs. All the experimentation is conducted in the MEMS Lab, Queen’s University,

Kingston, ON, Canada. Manual Electric Probe System – SÜSS MicroTec PM5 has been

used to test prototypes. Prototypes are designed keeping in view design rules for

24

PolyMUMPs. IntelliMask has been used for the designing of mask layouts and design rule

checks. Experimental results and their comparison with FEM and numerical results is

presented in detail.

A variant of displacement amplification mechanism consisting of two configurations of

mechanism is discussed in chapter 6. FEA has been carried out using IntelliSuite® to verify

the analytical results from mathematical models. Based on the parametric analysis, design

has been optimized. Prototype has been fabricated using PolyMUMPs process and tested.

Chapter 7 presents static and thermal simulations carried out using IntelliSuite® software

of configuration 3 designed with thermal chevrons and electrostatic comb drives. Steady

state static thermal electrical analysis has been performed under variable resistivity and

voltage bias of 2V. Time domain simulations of device have been carried out with constant

electrical resistivity, variable voltage and convective boundary conditions.

In chapter 8, the conclusion of this research in the area of MEMS based displacement

amplification mechanisms is presented along with recommendations for future work.

25

CHAPTER 2. DESIGN OF MICRO DISPLACEMENT

AMPLIFICATION MECHANISM

2.1 Introduction

In previous chapter literature review, motivation and objective of research has been

discussed. In this chapter, design of micro displacement amplification mechanism is

presented. As a design part, three configurations, mathematical modeling and different

materials available for micro-fabrication have been described in detail. Mathematical

model of the mechanism is developed using a geometric approach for solving kinematics

and Castigliano’s approach to develop kinetic model. In the end, the fabless strategy and

proposed fabrication MUMPS process (i.e. PolyMUMPs) is discussed.

2.2 Modeling of Micro Displacement Amplification Mechanism

Figures 2-1(a & b) demonstrates the design and functioning of the proposed amplification

mechanism. The amplification mechanism consists of micro-flexures having one end

constraint and other ends are free to move. The displacement applied is amplified in the

direction perpendicular to the direction of applied displacement.

The input displacement applied along x-axis will produce deflections of flexures which is

then amplified in the direction perpendicular to the input displacement direction. Figure 2-

1(b) shows the half model of the suspension mechanism at two positions after application

of displacement. The bold line shows the position before application of displacement.

26

(a)

(b)

Figure 2-1 (a) Schematic diagram of the proposed amplification mechanism (b) Half model of the proposed amplification mechanism at two positions

2.3 Configurations of Micro Displacement Amplification Mechanism

The proposed mechanism can produce three different amplification factors, when used in

three different configurations, depending on application. These configurations have same

geometric properties, only boundary conditions are changed. Three configurations are

discussed below:

27

2.3.1 Configuration 1

When link A is constrained and displacement is applied at link C, the amplified output of

the mechanism can be extracted from both links D & B as shown in Figure 2-2.

Figure 2-2 Schematic diagram of proposed amplification mechanism showing functioning having configuration no. 1

28


When the link D is constrained, and input displacement is applied to links A & C, an

amplified output can be extracted from the top link B as shown in the Figure 2-3.


29


In this configuration, input force applied to links A & C is amplified at links B & D of the

mechanism as shown in Figure 2-4. In this configuration, mechanism is to be supported

using external flexures.


2.4 Analytical Modeling

2.4.1 Mathematical Model

The proposed mechanism is symmetric, so a quarter model has been used to make

kinematic analytical model as shown in Figure 2-5. The kinematic equations of the micro

displacement amplification mechanism are derived using geometric approach assuming

flexures as rigid bodies.

30

Figure 2-5 Quarter model of displacement amplification mechanism

Uout = L(sinα’ – sinα) (2.1)

α’ = cos-1 �𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄 − 𝑼𝑼𝒊𝒊𝒊𝒊𝑳𝑳

� (2.2)

Here Uin and Uout are input and output displacements respectively, L is length of flexure, α

is the angle made by the flexure, α’ is the angle after deflection of flexure that will change

with the applied displacement.

2.4.2 Matrix Method

The displacement amplification mechanism consists of four micro-flexures. These micro-

flexures undergo elastic deformations under application of external forces. Four micro-

flexures act as spring element each having same stiffness, K, as shown in Figure 2-6. The

node point 1 and 2 have two branches in between consisting of two identical springs

31

connected in series. According to series connection rule of springs each branch has a

stiffness of K/2. Hence the total stiffness of the mechanism is found by adding spring

stiffness’s in parallel that comes out to be K.

Kt = K/2 + K/2 = K (2.3)

Figure 2-6 Spring model of displacement amplification mechanism

The stiffness matrix of the micro-flexure is given below:

𝐊𝐊 =

⎣⎢⎢⎢⎡

𝐄𝐄𝐄𝐄𝐄𝐄𝐋𝐋

𝟎𝟎 𝟎𝟎

𝟎𝟎 𝐆𝐆𝐄𝐄𝐄𝐄𝐄𝐄𝟑𝟑

(𝟒𝟒𝐆𝐆𝐋𝐋𝟑𝟑 + 𝛂𝛂𝐬𝐬 𝐄𝐄𝐋𝐋𝐄𝐄𝟐𝟐 ) 𝐄𝐄𝐄𝐄𝐄𝐄𝟑𝟑

𝟔𝟔𝐋𝐋𝟐𝟐

𝟎𝟎 𝐄𝐄𝐄𝐄𝐄𝐄𝟑𝟑

𝟔𝟔𝐋𝐋𝟐𝟐 𝐄𝐄𝐄𝐄𝐄𝐄𝟑𝟑

𝟏𝟏𝟐𝟐𝐋𝐋 ⎦⎥⎥⎥⎤

(2.4)

Here E is the modulus of elasticity, G is the modulus of rigidity, b and t denote the width

and thickness of the micro-beam, L denotes the length of the micro-beam, μ is the Poisson

32

ratio and αs is the shear coefficient of the material. The shear coefficient αs of micro-beam

having rectangular cross section was presented by Cowper (Cowper, 1966; Zhu et al.,

2014) as shown in the equation 2.5.

𝛂𝛂𝐬𝐬 = 𝟏𝟏𝟐𝟐+𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟎𝟎(𝟏𝟏+𝟏𝟏)

(2.5)

According to Castigliano’s 1st theorem (Lobontiu, Nicolae, Garcia, 2005), force matrix for

micro-flexure is calculated to be:

�𝑭𝑭𝒙𝒙𝑭𝑭𝒚𝒚𝑴𝑴

�=

⎣⎢⎢⎢⎡

𝑬𝑬𝑬𝑬𝑬𝑬𝑳𝑳

𝟎𝟎 𝟎𝟎

𝟎𝟎 𝑮𝑮𝑬𝑬𝑬𝑬𝑬𝑬𝟑𝟑

(𝟒𝟒𝑮𝑮𝑳𝑳𝟑𝟑 + 𝒄𝒄𝒊𝒊 𝑬𝑬𝑳𝑳𝑬𝑬𝟐𝟐 ) 𝑬𝑬𝑬𝑬𝑬𝑬𝟑𝟑

𝟔𝟔𝑳𝑳𝟐𝟐

𝟎𝟎 𝑬𝑬𝑬𝑬𝑬𝑬𝟑𝟑

𝟔𝟔𝑳𝑳𝟐𝟐 𝑬𝑬𝑬𝑬𝑬𝑬𝟑𝟑

𝟏𝟏𝟐𝟐𝑳𝑳 ⎦⎥⎥⎥⎤

�𝒖𝒖𝒙𝒙𝒖𝒖𝒚𝒚𝜽𝜽𝒙𝒙

� (2.6)

Here Fx and Fy are forces in x and y direction respectively, M is the moment about z-axis,

Ux and Uy are displacements in x and y direction respectively and θx is the angular

displacement.

2.4.3 Strain Energy Approach

Strain energy approach can be used to develop the analytical model of the displacement

amplification mechanism. The load-displacement relationships for the mechanisms were

formulated using Castigliano’s displacement theorem. The total strain energy, Utotal, of the

mechanism is combination of axial, shearing and bending loading.

Utotal = Uaxial +Ubending + Ushear (2.7)

33

Strain energy given for axial, shearing and bending loading is expressed as follows.

Uaxial = ∑ �∫ 𝑭𝑭𝒊𝒊𝒊𝒊𝟐𝟐

𝟐𝟐𝑬𝑬𝟐𝟐𝒅𝒅𝒙𝒙𝒊𝒊

𝒍𝒍𝒊𝒊𝟎𝟎 �𝒊𝒊

𝒊𝒊=𝟏𝟏 (2.8)

Ushear = ∑ �∫ 𝒄𝒄𝑭𝑭𝒊𝒊𝒄𝒄𝟐𝟐

𝟐𝟐𝑮𝑮𝟐𝟐𝒅𝒅𝒙𝒙𝒊𝒊


𝒊𝒊=𝟏𝟏 (2.7)

Ubending = ∑ �∫ 𝑴𝑴𝒊𝒊𝟐𝟐

𝟐𝟐𝑬𝑬𝟐𝟐𝒅𝒅𝒙𝒙𝒊𝒊


𝒊𝒊=𝟏𝟏 (2.8)

Here Fa, Fs are axial and shearing forces respectively, M denotes bending moment, A is the

cross sectional area and I is moment of inertia.

Fundamental equations of the Castigliano’s displacement theorem are given below.

𝒖𝒖𝒊𝒊𝒙𝒙 = 𝝏𝝏𝑼𝑼𝑬𝑬𝝏𝝏𝑭𝑭𝒊𝒊𝒙𝒙

(2.9)

𝒖𝒖𝒊𝒊𝒚𝒚 = 𝝏𝝏𝑼𝑼𝑬𝑬𝝏𝝏𝑭𝑭𝒊𝒊𝒚𝒚

(2.10)

𝛉𝛉𝒊𝒊𝒙𝒙 = 𝝏𝝏𝑼𝑼𝑬𝑬𝝏𝝏𝑴𝑴𝒊𝒊𝒊𝒊

(2.11)

Here uix , uiy and θix are in-plane translation and rotations at any point on a micro flexure

based compliant mechanism respectively.

The displacement and rotations at each node can be found using equations 2.12 - 2.14.

34

𝒖𝒖𝒊𝒊𝒙𝒙 = 𝝏𝝏𝑼𝑼𝑬𝑬𝝏𝝏𝑭𝑭𝒊𝒊𝒙𝒙

= ∑ �∫ 𝑭𝑭𝒊𝒊𝒊𝒊𝟐𝟐

𝑬𝑬𝟐𝟐𝝏𝝏𝑭𝑭𝒊𝒊𝒊𝒊𝝏𝝏𝑭𝑭𝒊𝒊𝒙𝒙

𝒅𝒅𝒙𝒙𝒊𝒊𝒍𝒍𝒊𝒊

𝟎𝟎 + ∫ 𝒄𝒄𝑭𝑭𝒊𝒊𝒄𝒄𝟐𝟐

𝑮𝑮𝟐𝟐𝝏𝝏𝑭𝑭𝒊𝒊𝒄𝒄𝝏𝝏𝑭𝑭𝒊𝒊𝒙𝒙


𝟎𝟎 +𝒊𝒊𝒊𝒊=𝟏𝟏

∫ 𝑴𝑴𝒊𝒊𝟐𝟐

𝑬𝑬𝟐𝟐𝝏𝝏𝑴𝑴𝒊𝒊𝝏𝝏𝑭𝑭𝒊𝒊𝒙𝒙


𝟎𝟎 � (2.12)

𝒖𝒖𝒊𝒊𝒚𝒚 = 𝝏𝝏𝑼𝑼𝑬𝑬𝝏𝝏𝑭𝑭𝒊𝒊𝒚𝒚

= ∑ �∫ 𝑭𝑭𝒊𝒊𝒊𝒊𝟐𝟐

𝑬𝑬𝟐𝟐𝝏𝝏𝑭𝑭𝒊𝒊𝒊𝒊𝝏𝝏𝑭𝑭𝒊𝒊𝒚𝒚


𝟎𝟎 + ∫ 𝒄𝒄𝑭𝑭𝒊𝒊𝒄𝒄𝟐𝟐

𝑮𝑮𝟐𝟐𝝏𝝏𝑭𝑭𝒊𝒊𝒄𝒄𝝏𝝏𝑭𝑭𝒊𝒊𝒚𝒚


𝟎𝟎 +𝒊𝒊𝒊𝒊=𝟏𝟏

∫ 𝑴𝑴𝒊𝒊𝟐𝟐

𝑬𝑬𝟐𝟐𝝏𝝏𝑴𝑴𝒊𝒊𝝏𝝏𝑭𝑭𝒊𝒊𝒚𝒚


𝟎𝟎 � (2.13)

𝛉𝛉𝒊𝒊𝒙𝒙 = 𝝏𝝏𝑼𝑼𝑬𝑬𝝏𝝏𝑴𝑴𝒊𝒊𝒊𝒊

= ∑ �∫ 𝑴𝑴𝒊𝒊𝑬𝑬𝟐𝟐

𝝏𝝏𝑴𝑴𝒊𝒊𝝏𝝏𝑴𝑴𝒊𝒊𝒊𝒊


𝟎𝟎 �𝒊𝒊𝒊𝒊=𝟏𝟏 (2.14)

The displacement amplification (D.A), that is ratio of output displacement to input

displacement, is given below:

D.A = 𝒖𝒖𝒄𝒄𝒖𝒖𝑬𝑬𝒖𝒖𝒊𝒊𝒊𝒊

(2.15)

2.5 Fabrication of Prototypes

2.5.1 Material Selection

Polysilicon has been used as a structural layer to fabricate the prototypes using

PolyMUMPs process. Polysilicon having Young's modulus E = 130–188GPa, Shear

modulus G = 51–80GPa) is a mechanically stable semiconductor material suitable for use

in the structures of electromechanical devices. Polysilicon is Hookean material. When it is

flexed there is virtually no hysteresis and hence almost no energy dissipation during elastic

35

deformation. Metals having low young’s modulus like gold with E=79GPa, G=27GPa,

aluminum with E=70GPa, G=26GPa, copper with E=110–128GPa, G=48GPa, and silver

with E=83GPa, G=30GPa can also be used as structural material for MEMS fabrication.

Low Young’s modulus makes the structure more compliant and flexible which increases

the deflection. Low young’s modulus materials are mostly ductile that saves them from

undesirable brittle fracture with weak resistance to impact loading. However, endurance

limit of the material decreases with decrease in modulus of elasticity that ultimately affects

the reliability and the factor of safety(Richard G. Budynas, 2011). Polysilicon on the other

hand is very stable in fatigue loadings. Materials with a low Young’s modulus cannot

maintain a linear relationship between applied load and the induced deformations. They

have a low melting temperature that restricts their usage at elevated temperatures. Melting

temperature of silicon is 1400oC that makes it dimensionally stable even at elevated

temperature (Hsu, 2008). Polysilicon is an elastic material with low plasticity or creep

below 800°C.

Use of polyimide with E= 8.0-15.0GPa, G= 1.0-10.0GPa as a structural layer for

fabrication of MEMS devices using Polyimide MEMS Multi-User Process” (PiMMPs) has

been reported(Arevalo et al., 2015; Ilyas, Jaber and Younis, 2016). Low coefficient of

thermal expansion, low film stress, lower cost than metals and semiconductors and high

temperature stability compared to other polymers makes it attractive for MEMS

fabrication(Arevalo et al., 2015). However, it is dimensionally unstable with high moisture

absorption that is, it shows decrease up to 30 % of tensile strength and 50% of tensile

modulus. Its electrical and mechanical properties are also influenced by moisture content.

36

2.5.2 Fabless Strategy

To develop our prototypes, we adopted fabless strategy. This strategy focuses on designing

and characterizing the device using in house resources, while using external processing and

manufacturing facilities. Complete validation of design is very important before sending

the design for production. It is economical to develop design centers, get expertise in design

and use commercial processing and fabrication facilities. Using this approach researchers

can validate the technical viability and productivity of new design in short time.

2.5.3 Proposed MUMPs Process

Prototypes of displacement amplification mechanism were designed and fabricated

according to Polysilicon-Multi User MEMS Processes (PolyMUMPs) for verification of

the design concept. PolyMUMPs is micromachining process available for developing

MEMS devices available from MEMSCAP, USA. The process consists of three layer of

Polysilicon that are micro-machined and used as the structural material for device

development. Sacrificial layer is deposited oxide and for electrical isolation silicon nitride

used (Cowen et al., 2013). It was developed by Berkeley Sensor & Actuator Center. The

basic process includes 8 lithography levels, and 7 physical layers including two mechanical

layers of Polysilicon, one electrical layer of Polysilicon, two sacrificial layers, one

electrical conduction layer and one electrical isolation layer.

37

2.6 Conclusions

In this chapter, design, analytical modeling and prototype development of the mechanism

is presented. Kinematic and force equation are developed using geometric approach, matrix

method and strain energy approach. Moreover, implementation of fabless strategy using

PolyMUMPs process is also discussed. Numerical analysis of the kinematic model

developed is carried out in the next chapter. Based on the numerical results geometric

parameters will finalized keeping in view the amplification factor to be achieved.

38

CHAPTER 3. NUMERICAL ANALYSIS OF MICRO

DISPLACEMENT AMPLIFICATION MECHANISM

3.1 Introduction

In this chapter, parameters of the displacement amplification mechanism are finalized

keeping in view the amplification factor, overall size of the mechanism and PolyMUMPs

process design rules by carrying out a parametric analysis using kinematic equations

developed in previous chapter. Same parameters are used for carrying out numerical

simulations with Direct Stiffness Method (DSM) that is an implementation of FEM.

3.2 Parametric Analysis

3.2.1 Kinematic Analysis

Before carrying out parametric analysis some necessary assumptions are finalized. The

mechanism shown in Figure 3-1 is fixed at link D and input displacement varying from 0.5

μm to 2 μm is applied at links A and C. Design amplification factor of 20 is desired.

Minimum feature size and thickness in PolyMUMPs process is 2 μm, so width is chosen

to be 2 μm and thickness is as per PolyMUMPs fabrication process rule. Effects of change

in length of micro-flexure, from 250 µm to 750 µm and angle of micro-flexure, from 1o to

10 o is discussed in detail in the coming section. Based on these parametric analysis,

dimensions of the proposed amplification mechanism are finalized and accordingly

fabricated.

39

Parametric analysis of micro displacement amplification mechanism has been carried out

using MATLAB programming. Change in amplification factor with changing input

displacement is plotted as shown in Figure 3-2. For input displacement varying from 0.5

μm to 2 μm, an amplification factor decreases from 26.75 to 15.50. Our point of focus is 1

μm input displacement, at which we are getting an amplification factor of 18.75. Results

show that the amplification factor decreases with increasing input displacement, that

depicts that the amplification factor is limited to a specific range of input displacement

beyond which increasing input displacement will decrease amplification factor.

Figure 3-1 Schematic diagram of the mechanism

40

Figure 3-2 Plot of Input displacement vs. Amplification Factor showing a decrease in amplification factor from 26.75 to 15.50

To carryout parametric analysis of the proposed mechanism, one parameter is varied

keeping all others parameters constant. First of all, angle is varied from 1o to 10o, input

displacement is plotted against amplification factor as shown in Figure 3-3. The results

show that with increasing angle of micro-flexure, amplification factor decreases and

becomes constant at higher angles with increasing input displacement.

41

Figure 3-3 Parametric analysis showing change in amplification factor with varying angle made by micro-flexure

Length of micro-flexure is varied from 250 μm to 750 μm. Input displacement vs.

amplification factor plot for varying micro-flexure length is shown in Figure 3-4. With

increasing length of micro-flexure, amplification factor increases but decreases with

increasing input displacement. Parametric analysis further shows a significant change in

amplification factor with increasing length from 250 μm to 600 μm but after 600 μm the

change in amplification factor becomes small restricting the design to micro-flexure length

values within certain limit.

42

Figure 3-4 Parametric analyses showing significant increase in amplification factor with increasing length from 250 μm to 600 μm but after 600 μm the change in

amplification factor becomes insignificant

Results of parametric analysis show that the amplification factor is limited to certain range

of length and angle made by micro flexure. Displacement amplification depends on the

elastic deformation of flexures that amplify the input displacement. This elastic

deformation is limited depending on the material properties and dimensions of the structure

that cannot exceed beyond a critical range. So, the amplification is limited to range of

lengths and angles.

3.2.2 Kinetic Analysis

An optimal value of force varying from 1600 µN to 2000 µN has been taken from numerous

finite element simulations done in IntelliSuite® assuming that these magnitude of forces

will be enough to produce sufficient input displacement. Output forces produced in y and

z direction is potted against input displacement as shown in the Figures 3-5(a & b)

43

respectively. The graphs show that the output force along y axis increases and along z-axis

decreases.

(a)

(b)

Figure 3-5 Plot showing linear relationship between output force along (a) Y-axis (b) Z-axis and input displacement produced by application of force to the side masses

44

Parametric analysis has been carried out to investigate the effect of varying design

parameters on output force. Length of the micro-flexure is varied from 250 μm to 750 μm,

with input displacement of 2 µm, and plotted against the output force along Y-axis as

shown in Figure 3-6(a). With increasing length of micro-flexure output force decreases.

Angle of the micro-flexure is varied from 1o to 10o and plotted against the output force

along Y-axis as shown below in Figure 3-6(b). The plot shows a decrease in output force

with increasing angle.

Based on the parametric analysis, overall size of the mechanism and PolyMUMPs process

following parameters are finalized as given in the Table 3-1. Same parameters have been

used for calculating amplification factor through DSM, IntelliSuite® simulations and

fabrication.

Table 3-1 Geometric parameters and material properties of the designed mechanism

Parameter Value

Elastic modulus (N/m2 ) 160 x 109

Shear modulus (N/m2 ) 65.25 x 109

Length of micro-flexure (μm) 250

Angle made by micro-flexure (Degrees) 3o

Width of micro-flexure (μm) 2

Thickness of micro-flexure (μm) 2

Cross sectional area A(µm2) 4

Moment of inertia (µm4) 1.33

Overall mechanism size (μm) 600x100x2

45

(a)

(b)

Figure 3-6 (a) Plot showing a decrease in output force with increasing micro-flexure length (b)Plot showing a decrease in output force with increasing micro-flexure

angle

2 3 4 5 6 7 8x 10-4

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2x 10-5 KINETICS(varying length)

LENGTH OF FLEXURE(meters)

OU

TPU

T FO

RC

E(N

ewto

ns)

1 2 3 4 5 6 7 8 9 100.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2 x 10-3 KINETICS(VARYING ANGLE)

ANGLE OF FLEXURE WITH HORIZONTAL(degrees)

OU

TPU

T FO

RC

E(N

ewto

ns)

46

3.3 Finite Element Analysis using Direct Stiffness Method

FEM is a numerical method for solving engineering problems having complicated

geometries, loadings, and material properties. The finite element formulation of the

problem results in a system of simultaneous algebraic equations for solution, rather than

requiring the solution of differential equations (Logan et al., 2007). These numerical

methods yield approximate values of the unknowns at discrete numbers of points in the

continuum. In FEM, equations are formulated for each finite element. These equations are

then combined to get the vector matrix form of equations which are then solved

numerically using the techniques of numerical linear algebra.

Beam element, as shown in Figure 3-7, is used for finite element modeling of the designed

mechanism. A general transformed global stiffness matrix for a beam element that includes

axial force, shear force, and bending moment effects is given by the equation 3.1 (Logan

et al., 2007). As global stiffness matrix is being used so the transformation from link to

link has already been implemented in the matrix. The element stiffness is functions of

Young’s Modulus, area, length, moment of inertia and the angle of orientation as shown

below

47

Figure 3-7 Schematic diagram of the beam element

(3.1)

The analysis of the mechanism, which is a rigid plane frame, is undertaken by using the

above stiffness matrix. The mechanism is a rigid plane frame consisting of a chain of beam

elements rigidly connected to each other. Joints are transmitting moments and forces from

one element to another. Applied loads and centroids share a common plane (x-y plane)

having moment continuity at rigid joints. To get numerical solution, code has been written

in MATLAB.

48

3.3.1 Disintegration of Designed Mechanism

The designed mechanism is disintegrated into seven elements having eight nodes as shown

in Figure 3-8. Node 1 and node 8 are fixed. Equations are formulated for each finite element

which are then assembled to obtain the solution of the whole mechanism. All calculations,

assemblage, application of boundary conditions and finding end solution is done using

MATLAB. The solution provides unknown displacements and forces at each node. A

positive horizontal distributed force of magnitude 2000 µN is applied to elements 2 and

negative horizontal distributed force of magnitude 2000 µN is applied at element 6. An

optimal value of 2000 µN has been taken from numerous finite element simulations done

in IntelliSuite® assuming that 2000 µN will be enough to produce significant input

displacement. With application of this force, input displacement is produced which is

amplified at the output element 4.

Figure 3-8 Schematic diagram of the disintegrated mechanism

3.3.2 Element Stiffness Matrices

Global stiffness matrix for each element is found using matrix shown in matrix 3.1 having

E = 160 × 109 N/m2, A= 4 µm2 and I = 1.33 µm4 for all elements.

49

3.3.2.1 Element 1

Angle between global coordinate axis, x, and local coordinate axis, x', is defined as θ . It is

assumed that x' is directed from node i to node i+1. For this element θ = 177o counter

clockwise positive. Using equation 3.1, the element 1 stiffness matrix is:

d1x d1y θ1 d2x d2 θ2

1276.28 -68.89 -2.76E-07 -1276.28 68.89 -2.76E-07

-68.89 3.73 -5.11E-06 68.89 -3.73 -5.11E-06

K(1) = -2.76E-07 -5.11E-06 1.71E-09 2.76E-07 5.11E-06 8.53E-10 N/m

-1276.28 68.89 2.76E-07 1276.28 -68.89 2.76E-07

68.89 -3.73 5.11E-06 -68.89 3.73 -5.11E-06

-2.76E-07 -5.11E-06 8.53E-10 2.76E-07 -5.11E-06 1.71E-09

50

3.3.2.2 Element 2

For element 2, θ = 90o and the element stiffness matrix is:

d2x d2y θ2 d3x d3y θ3

9.77 7.95 -0.0003 -9.77 -7.95 -0.0003

7.95 9999.99 2.49E-07 -7.95 -9999.99 2.49E-07

K(2) = -0.0003 2.49E-07 1.33E-08 0.0003 -2.49E-07 6.67E-09 N/m

-9.77 -7.95 0.0003 9.77 7.95 0.0003

-7.95 -9999.99 -2.49E-07 7.95 9999.99 2.49E-07

-0.0003 2.49E-07 6.67E-09 0.0003 2.49E-07 1.33E-08

3.3.2.3 Element 3

For element 3, θ = 3o. Using equation 3.1, the element 3 stiffness matrix is:


1276.49 66.86 -2.68E-07 -1276.49 -66.86 -2.68E-07

66.86 3.52 5.11E-06 -66.86 -3.52 5.11E-06

K(3) = -2.68E-07 5.11E-06 1.71E-09 2.68E-07 -5.11E-06 8.53E-10 N/m

-1276.49 -66.86 2.68E-07 1276.49 66.86 2.68E-07

-66.86 -3.52 -5.11E-06 66.86 3.52 5.11E-06

-2.68E-07 5.11E-06 8.53E-10 2.68E-07 5.11E-06 1.71E-09

51

3.3.2.4 Element 4

For element 4, θ = 0o.The element stiffness matrix for element 4 is:


10000 0 0 -10000 0 0

0 9.76 0.0003 0 -9.76 0.0003

K(4) = 0 0.0003 1.33E-08 0 -0.0003 6.67E-09 N/m

-10000 0 0 10000 0 0

0 -9.76 -0.0003 0 9.76 0.0003

0 0.0031 6.67E-09 0 0.0003 1.33E-08

3.3.2.5 Element 5

For element 5, θ = 3o.The element stiffness matrix for element 5 is:


1276.05 -70.91 2.84E-07 -1276.05 70.91 2.84E-07

-70.91 3.96 5.11E-06 70.91 -3.96 5.11E-06

K(5) = 2.84E-07 5.11E-06 1.71E-09 -2.84E-07 -5.11E-06 8.53E-10 N/m

-1276.05 70.91 -2.84E-07 1276.05 -70.91 -2.84E-07

70.91 -3.96 -5.11E-06 -70.91 3.96 5.11E-06

2.84E-07 5.11E-06 8.53E-10 -2.84E-07 5.11E-06 1.71E-09

52

3.3.2.6 Element 6

For element 6, θ = 270o and stiffness matrix for element 6 as:


9.82 23.86 0.0003 -9.82 -23.86 0.0003

23.86 9999.94 -7.47E-07 -23.86 -9999.94 -7.47E-07

K(6) = 0.0003 -7.47E-07 1.33E-08 -0.0003 7.47E-07 6.67E-09 N/m

-9.82 -23.86 -0.0003 9.82 23.86 -0.0003

-23.86 -9999.94 7.47E-07 23.86 9999.94 -7.47E-07

0.0003 -7.47E-07 6.67E-09 -0.0003 -7.47E-07 1.33E-08

3.3.2.7 Element 7

For element 7 θ = 183o. Using equation 3.1, element 7 stiffness matrix is:


1276.70 64.83 2.60E-07 -1276.70 -64.83 2.60E-07

64.83 3.31 -5.11E-06 -64.83 -3.31 -5.11E-06

K(7) = 2.60E-07 -5.11E-06 1.71E-09 -2.60E-07 5.11E-06 8.53E-10 N/m

-1276.70 -64.83 -2.60E-07 1276.70 64.83 -2.60E-07

-64.83 -3.31 5.11E-06 64.83 3.31 -5.11E-06

2.60E-07 -5.11E-06 8.53E-10 -2.60E-07 -5.11E-06 1.71E-09

53

3.3.3 Element External Force Matrices

External local force vector is found for all elements. A positive horizontal distributed force

of magnitude 2000 µN is applied to elements 2 and negative horizontal distributed force of

magnitude 2000 µN is applied at element 6.

3.3.3.1 External Force Matrix of Element 1

External local force matrix 𝑓𝑓′(1) is found by transformation of distributed loads to nodal

loads. For element 1, local forces and moments at node 1 are f1x = f1y = m1 =0 and at node

2 f2x = 0, f2y = F×L/2, m2 = F×L2/12. For F = 2000 µN, external local force matrix is given

below:

�𝒇𝒇′(𝟏𝟏)�=

⎩⎪⎪⎨

⎪⎪⎧

𝒇𝒇𝟏𝟏𝒙𝒙′

𝒇𝒇𝟏𝟏𝒚𝒚′

𝒎𝒎𝟏𝟏′

𝒇𝒇𝟐𝟐𝒙𝒙′

𝒇𝒇𝟐𝟐𝒚𝒚′

𝒎𝒎𝟐𝟐′ ⎭

⎪⎪⎬

⎪⎪⎫

=

⎩⎪⎨

⎪⎧

𝟎𝟎𝟎𝟎𝟎𝟎𝟎𝟎

𝟔𝟔. 𝟒𝟒𝟎𝟎𝐄𝐄 − 𝟓𝟓𝟔𝟔. 𝟖𝟖𝟑𝟑𝐄𝐄 − 𝟏𝟏𝟎𝟎⎭

⎪⎬

⎪⎫

N

To find force matrix,𝑓𝑓(1), external local load matrix is to be multiplied with transformation

matrix. Global element force matrix is found by using the relation given below.

�𝒇𝒇(𝟏𝟏)� = [𝑻𝑻]�𝒇𝒇′(𝟏𝟏)�

54

�𝒇𝒇(𝟏𝟏)� =

⎩⎪⎨

⎪⎧

𝒇𝒇𝟏𝟏𝒙𝒙𝒇𝒇𝟏𝟏𝒚𝒚𝒎𝒎𝟏𝟏𝒇𝒇𝟐𝟐𝒙𝒙𝒇𝒇𝟐𝟐𝒚𝒚𝒎𝒎𝟐𝟐⎭

⎪⎬

⎪⎫

⎣⎢⎢⎢⎢⎡−𝟎𝟎. 𝟗𝟗𝟗𝟗𝟖𝟖 −𝟎𝟎. 𝟎𝟎𝟓𝟓𝟑𝟑 𝟎𝟎 𝟎𝟎 𝟎𝟎 𝟎𝟎𝟎𝟎. 𝟎𝟎𝟓𝟓𝟑𝟑 −𝟎𝟎. 𝟗𝟗𝟗𝟗𝟖𝟖 𝟎𝟎 𝟎𝟎 𝟎𝟎 𝟎𝟎

𝟎𝟎 𝟎𝟎 𝟏𝟏 𝟎𝟎 𝟎𝟎 𝟎𝟎𝟎𝟎 𝟎𝟎 𝟎𝟎 −𝟎𝟎. 𝟗𝟗𝟗𝟗𝟖𝟖 −𝟎𝟎. 𝟎𝟎𝟓𝟓𝟑𝟑 𝟎𝟎𝟎𝟎 𝟎𝟎 𝟎𝟎 𝟎𝟎. 𝟎𝟎𝟓𝟓𝟑𝟑 −𝟎𝟎. 𝟗𝟗𝟗𝟗𝟖𝟖 𝟎𝟎𝟎𝟎 𝟎𝟎 𝟎𝟎 𝟎𝟎 𝟎𝟎 𝟏𝟏⎦

⎥⎥⎥⎥⎤

⎩⎪⎨

⎪⎧

𝟎𝟎𝟎𝟎𝟎𝟎𝟎𝟎

𝟔𝟔. 𝟒𝟒𝟎𝟎𝐄𝐄 − 𝟎𝟎𝟓𝟓𝟔𝟔. 𝟖𝟖𝟑𝟑𝐄𝐄 − 𝟏𝟏𝟎𝟎⎭

⎪⎬

⎪⎫

=

⎩⎪⎨

⎪⎧

𝟎𝟎𝟎𝟎𝟎𝟎

−𝟑𝟑. 𝟒𝟒𝟓𝟓𝐄𝐄 − 𝟎𝟎𝟑𝟑−𝟔𝟔. 𝟑𝟑𝟗𝟗𝐄𝐄 − 𝟎𝟎𝟓𝟓𝟔𝟔. 𝟖𝟖𝟑𝟑𝐄𝐄 − 𝟏𝟏𝟎𝟎 ⎭

⎪⎬

⎪⎫

N


For element 2, local forces and moments at node 2 are f2x = 0, f2y = F×L/2, m2 = F×L2/12

and at node 3 are f3x = 0, f3y = F×L/2, m3 = F×L2/12. External Local force matrix, 𝑓𝑓′(2), is

given below:

�𝒇𝒇′(𝟐𝟐)�=

⎩⎪⎪⎨

⎪⎪⎧

𝒇𝒇𝟐𝟐𝒙𝒙′

𝒇𝒇𝟐𝟐𝒚𝒚′

𝒎𝒎𝟐𝟐′

𝒇𝒇𝟑𝟑𝒙𝒙′

𝒇𝒇𝟑𝟑𝒚𝒚′

𝒎𝒎𝟑𝟑′ ⎭

⎪⎪⎬

⎪⎪⎫

=

⎩⎪⎨

⎪⎧

𝟎𝟎𝟔𝟔. 𝟒𝟒𝟎𝟎𝐄𝐄 − 𝟎𝟎𝟓𝟓𝟔𝟔. 𝟖𝟖𝟑𝟑𝐄𝐄 − 𝟏𝟏𝟎𝟎

𝟎𝟎𝟔𝟔. 𝟒𝟒𝟎𝟎𝐄𝐄 − 𝟎𝟎𝟓𝟓𝟔𝟔. 𝟖𝟖𝟑𝟑𝐄𝐄 − 𝟏𝟏𝟎𝟎⎭

⎪⎬

⎪⎫

N

To find global force matrix external local load matrix is to be multiplied with

transformation matrix. The global force matrix, 𝑓𝑓(2), is found by using the relation given

below:

�𝒇𝒇(𝟐𝟐)� = [𝑻𝑻]�𝒇𝒇′(𝟐𝟐)�

55

�𝒇𝒇(𝟐𝟐)� =

⎩⎪⎨

⎪⎧

𝒇𝒇𝟐𝟐𝒙𝒙𝒇𝒇𝟐𝟐𝒚𝒚𝒎𝒎𝟐𝟐𝒇𝒇𝟑𝟑𝒙𝒙𝒇𝒇𝟑𝟑𝒚𝒚𝒎𝒎𝟑𝟑⎭

⎪⎬

⎪⎫

=

⎣⎢⎢⎢⎢⎡−𝟎𝟎. 𝟎𝟎𝟓𝟓𝟏𝟏𝟓𝟓𝟏𝟏 −𝟎𝟎. 𝟗𝟗𝟗𝟗𝟖𝟖𝟔𝟔𝟗𝟗 𝟎𝟎 𝟎𝟎 𝟎𝟎 𝟎𝟎𝟎𝟎. 𝟗𝟗𝟗𝟗𝟖𝟖𝟔𝟔𝟗𝟗 −𝟎𝟎. 𝟎𝟎𝟓𝟓𝟏𝟏𝟓𝟓𝟏𝟏 𝟎𝟎 𝟎𝟎 𝟎𝟎 𝟎𝟎

𝟎𝟎 𝟎𝟎 𝟏𝟏 𝟎𝟎 𝟎𝟎 𝟎𝟎𝟎𝟎 𝟎𝟎 𝟎𝟎 −𝟎𝟎. 𝟎𝟎𝟓𝟓𝟏𝟏𝟓𝟓𝟏𝟏 −𝟎𝟎. 𝟗𝟗𝟗𝟗𝟖𝟖𝟔𝟔𝟗𝟗 𝟎𝟎𝟎𝟎 𝟎𝟎 𝟎𝟎 𝟎𝟎. 𝟗𝟗𝟗𝟗𝟖𝟖𝟔𝟔𝟗𝟗 −𝟎𝟎. 𝟎𝟎𝟓𝟓𝟏𝟏𝟓𝟓𝟏𝟏 𝟎𝟎𝟎𝟎 𝟎𝟎 𝟎𝟎 𝟎𝟎 𝟎𝟎 𝟏𝟏⎦

⎥⎥⎥⎥⎤

⎩⎪⎨

⎪⎧

𝟎𝟎𝟔𝟔. 𝟒𝟒𝟎𝟎𝐄𝐄 − 𝟎𝟎𝟓𝟓𝟔𝟔. 𝟖𝟖𝟑𝟑𝐄𝐄 − 𝟏𝟏𝟎𝟎

𝟎𝟎𝟔𝟔. 𝟒𝟒𝟎𝟎𝐄𝐄 − 𝟎𝟎𝟓𝟓𝟔𝟔. 𝟖𝟖𝟑𝟑𝐄𝐄 − 𝟏𝟏𝟎𝟎⎭

⎪⎬

⎪⎫

=

⎩⎪⎨

⎪⎧

−𝟔𝟔. 𝟑𝟑𝟗𝟗𝐄𝐄 − 𝟎𝟎𝟓𝟓−𝟑𝟑. 𝟑𝟑𝟎𝟎𝐄𝐄 − 𝟎𝟎𝟔𝟔𝟔𝟔. 𝟖𝟖𝟑𝟑𝐄𝐄 − 𝟏𝟏𝟎𝟎

−𝟔𝟔. 𝟑𝟑𝟗𝟗𝐄𝐄 − 𝟎𝟎𝟓𝟓−𝟑𝟑. 𝟑𝟑𝟎𝟎𝐄𝐄 − 𝟎𝟎𝟔𝟔𝟔𝟔. 𝟖𝟖𝟑𝟑𝐄𝐄 − 𝟏𝟏𝟎𝟎 ⎭

⎪⎬

⎪⎫

N


For element 3 local forces and moments at node 3 are f3x = 0, f3y = F×L/2, m3 = F×L2/12

and node 4 are f4x = f4y = m4 = 0. External Local force matrix is given below:

�𝒇𝒇′(𝟑𝟑)�=

⎩⎪⎪⎨

⎪⎪⎧

𝒇𝒇𝟑𝟑𝒙𝒙′

𝒇𝒇𝟑𝟑𝒚𝒚′

𝒎𝒎𝟑𝟑′

𝒇𝒇𝟒𝟒𝒙𝒙′

𝒇𝒇𝟒𝟒𝒚𝒚′

𝒎𝒎𝟒𝟒′ ⎭

⎪⎪⎬

⎪⎪⎫

=

⎩⎪⎨

⎪⎧

𝟎𝟎𝟔𝟔. 𝟒𝟒𝟎𝟎𝐄𝐄 − 𝟎𝟎𝟓𝟓𝟔𝟔. 𝟖𝟖𝟑𝟑𝐄𝐄 − 𝟏𝟏𝟎𝟎

𝟎𝟎𝟎𝟎𝟎𝟎 ⎭

⎪⎬

⎪⎫

N

The force matrix is found by using the relation given below:

�𝒇𝒇(𝟑𝟑)� = [𝑻𝑻]�𝒇𝒇′(𝟑𝟑)�

56

�𝒇𝒇(𝟑𝟑)� =

⎩⎪⎨

⎪⎧

𝒇𝒇𝟑𝟑𝒙𝒙𝒇𝒇𝟑𝟑𝒚𝒚𝒎𝒎𝟑𝟑𝒇𝒇𝟒𝟒𝒙𝒙𝒇𝒇𝟒𝟒𝒚𝒚𝒎𝒎𝟒𝟒⎭

⎪⎬

⎪⎫

=

⎣⎢⎢⎢⎢⎡−𝟎𝟎. 𝟎𝟎𝟓𝟓𝟏𝟏𝟓𝟓𝟏𝟏 −𝟎𝟎. 𝟗𝟗𝟗𝟗𝟖𝟖𝟔𝟔𝟗𝟗 𝟎𝟎 𝟎𝟎 𝟎𝟎 𝟎𝟎𝟎𝟎. 𝟗𝟗𝟗𝟗𝟖𝟖𝟔𝟔𝟗𝟗 −𝟎𝟎. 𝟎𝟎𝟓𝟓𝟏𝟏𝟓𝟓𝟏𝟏 𝟎𝟎 𝟎𝟎 𝟎𝟎 𝟎𝟎

𝟎𝟎 𝟎𝟎 𝟏𝟏 𝟎𝟎 𝟎𝟎 𝟎𝟎𝟎𝟎 𝟎𝟎 𝟎𝟎 −𝟎𝟎. 𝟎𝟎𝟓𝟓𝟏𝟏𝟓𝟓𝟏𝟏 −𝟎𝟎. 𝟗𝟗𝟗𝟗𝟖𝟖𝟔𝟔𝟗𝟗 𝟎𝟎𝟎𝟎 𝟎𝟎 𝟎𝟎 𝟎𝟎. 𝟗𝟗𝟗𝟗𝟖𝟖𝟔𝟔𝟗𝟗 −𝟎𝟎. 𝟎𝟎𝟓𝟓𝟏𝟏𝟓𝟓𝟏𝟏 𝟎𝟎𝟎𝟎 𝟎𝟎 𝟎𝟎 𝟎𝟎 𝟎𝟎 𝟏𝟏⎦

⎥⎥⎥⎥⎤

⎩⎪⎨

⎪⎧

𝟎𝟎𝟔𝟔. 𝟒𝟒𝟎𝟎𝐄𝐄 − 𝟎𝟎𝟓𝟓𝟔𝟔. 𝟖𝟖𝟑𝟑𝐄𝐄 − 𝟏𝟏𝟎𝟎

𝟎𝟎𝟎𝟎𝟎𝟎 ⎭

⎪⎬

⎪⎫

=

⎩⎪⎨

⎪⎧

−𝟔𝟔. 𝟑𝟑𝟗𝟗𝐄𝐄 − 𝟎𝟎𝟓𝟓−𝟑𝟑. 𝟑𝟑𝟎𝟎𝐄𝐄 − 𝟎𝟎𝟔𝟔𝟔𝟔. 𝟖𝟖𝟑𝟑𝐄𝐄 − 𝟏𝟏𝟎𝟎

𝟎𝟎𝟎𝟎𝟎𝟎 ⎭

⎪⎬

⎪⎫

N


For element 4 local forces and moments at node 4 are f4x = f4y = m4 = 0 and node 5 are f5x

= f5y = m5 = 0. External Local force matrix is given below. Since it’s a zero matrix so no

need to do transformations and external local and global force matrices will be same.

�𝒇𝒇(𝟒𝟒)� = �𝒇𝒇′(𝟒𝟒)�=

⎩⎪⎪⎨

⎪⎪⎧

𝒇𝒇𝟒𝟒𝒙𝒙′

𝒇𝒇𝟒𝟒𝒚𝒚′

𝒎𝒎𝟒𝟒′

𝒇𝒇𝟓𝟓𝒙𝒙′

𝒇𝒇𝟓𝟓𝒚𝒚′

𝒎𝒎𝟓𝟓′ ⎭

⎪⎪⎬

⎪⎪⎫

=

⎩⎪⎨

⎪⎧

𝟎𝟎𝟎𝟎𝟎𝟎𝟎𝟎𝟎𝟎𝟎𝟎⎭

⎪⎬

⎪⎫

N


For element 5, local forces and moments at node 5 are f5x = f5y = m5 = 0 and node 6 are f6x

= 0, f6y = F×L/2, m6 = F×L2/12. External local force matrix is given below:

57

�𝒇𝒇′(𝟓𝟓)�=

⎩⎪⎪⎨

⎪⎪⎧

𝒇𝒇𝟓𝟓𝒙𝒙′

𝒇𝒇𝟓𝟓𝒚𝒚′

𝒎𝒎𝟓𝟓′

𝒇𝒇𝟔𝟔𝒙𝒙′

𝒇𝒇𝟔𝟔𝒚𝒚′

𝒎𝒎𝟔𝟔′ ⎭

⎪⎪⎬

⎪⎪⎫

=

⎩⎪⎨

⎪⎧

𝟎𝟎𝟎𝟎𝟎𝟎𝟎𝟎

−𝟔𝟔. 𝟒𝟒𝟎𝟎𝐄𝐄 − 𝟎𝟎𝟓𝟓−𝟔𝟔. 𝟖𝟖𝟑𝟑𝐄𝐄 − 𝟏𝟏𝟎𝟎⎭

⎪⎬

⎪⎫

N


�𝒇𝒇(𝟓𝟓)� = [𝑻𝑻]�𝒇𝒇′(𝟓𝟓)�

�𝒇𝒇(𝟓𝟓)� =

⎩⎪⎨

⎪⎧

𝒇𝒇𝟓𝟓𝒙𝒙𝒇𝒇𝟓𝟓𝒚𝒚𝒎𝒎𝟓𝟓𝒇𝒇𝟔𝟔𝒙𝒙𝒇𝒇𝟔𝟔𝒚𝒚𝒎𝒎𝟔𝟔⎭

⎪⎬

⎪⎫

=

⎣⎢⎢⎢⎢⎡−𝟎𝟎. 𝟗𝟗𝟗𝟗𝟖𝟖 −𝟎𝟎. 𝟎𝟎𝟓𝟓𝟑𝟑 𝟎𝟎 𝟎𝟎 𝟎𝟎 𝟎𝟎𝟎𝟎. 𝟎𝟎𝟓𝟓𝟑𝟑 −𝟎𝟎. 𝟗𝟗𝟗𝟗𝟖𝟖 𝟎𝟎 𝟎𝟎 𝟎𝟎 𝟎𝟎

𝟎𝟎 𝟎𝟎 𝟏𝟏 𝟎𝟎 𝟎𝟎 𝟎𝟎𝟎𝟎 𝟎𝟎 𝟎𝟎 −𝟎𝟎. 𝟗𝟗𝟗𝟗𝟖𝟖 −𝟎𝟎. 𝟎𝟎𝟓𝟓𝟑𝟑 𝟎𝟎𝟎𝟎 𝟎𝟎 𝟎𝟎 𝟎𝟎. 𝟎𝟎𝟓𝟓𝟑𝟑 −𝟎𝟎. 𝟗𝟗𝟗𝟗𝟖𝟖 𝟎𝟎𝟎𝟎 𝟎𝟎 𝟎𝟎 𝟎𝟎 𝟎𝟎 𝟏𝟏⎦

⎥⎥⎥⎥⎤

⎩⎪⎨

⎪⎧

𝟎𝟎𝟎𝟎𝟎𝟎𝟎𝟎

−𝟔𝟔. 𝟒𝟒𝟎𝟎𝐄𝐄 − 𝟎𝟎𝟓𝟓−𝟔𝟔. 𝟖𝟖𝟑𝟑𝐄𝐄 − 𝟏𝟏𝟎𝟎⎭

⎪⎬

⎪⎫

=

⎩⎪⎨

⎪⎧

𝟎𝟎𝟎𝟎𝟎𝟎

𝟑𝟑. 𝟒𝟒𝟓𝟓𝑬𝑬 − 𝟎𝟎𝟔𝟔𝟔𝟔. 𝟑𝟑𝟗𝟗𝑬𝑬 − 𝟎𝟎𝟓𝟓

−𝟔𝟔. 𝟖𝟖𝟑𝟑𝐄𝐄 − 𝟏𝟏𝟎𝟎⎭⎪⎬

⎪⎫

N


For element 6 local forces and moments at node 6 are f6x = 0, f6y = F×L/2, m6 = F×L2/12

and node 7 are f7x = 0, f7y = F×L/2, m7 = F×L2/12. External Local force matrix is given

below:

58

�𝒇𝒇′(𝟔𝟔)�=

⎩⎪⎪⎨

⎪⎪⎧

𝒇𝒇𝟔𝟔𝒙𝒙′

𝒇𝒇𝟔𝟔𝒚𝒚′

𝒎𝒎𝟔𝟔′

𝒇𝒇𝟗𝟗𝒙𝒙′

𝒇𝒇𝟗𝟗𝒚𝒚′

𝒎𝒎𝟗𝟗′ ⎭

⎪⎪⎬

⎪⎪⎫

=

⎩⎪⎨

⎪⎧

𝟎𝟎𝟔𝟔. 𝟒𝟒𝟎𝟎𝐄𝐄 − 𝟎𝟎𝟓𝟓𝟔𝟔. 𝟖𝟖𝟑𝟑𝐄𝐄 − 𝟏𝟏𝟎𝟎

𝟎𝟎𝟔𝟔. 𝟒𝟒𝟎𝟎𝐄𝐄 − 𝟎𝟎𝟓𝟓𝟔𝟔. 𝟖𝟖𝟑𝟑𝐄𝐄 − 𝟏𝟏𝟎𝟎⎭

⎪⎬

⎪⎫

N


�𝒇𝒇(𝟔𝟔)� = [𝑻𝑻]�𝒇𝒇′(𝟔𝟔)�

�𝒇𝒇(𝟔𝟔)� =

⎩⎪⎨

⎪⎧

𝒇𝒇𝟔𝟔𝒙𝒙𝒇𝒇𝟔𝟔𝒚𝒚𝒎𝒎𝟔𝟔𝒇𝒇𝟗𝟗𝒙𝒙𝒇𝒇𝟗𝟗𝒚𝒚𝒎𝒎𝟗𝟗⎭

⎪⎬

⎪⎫

=

⎣⎢⎢⎢⎢⎡−𝟎𝟎. 𝟗𝟗𝟗𝟗𝟖𝟖 −𝟎𝟎. 𝟎𝟎𝟓𝟓𝟑𝟑 𝟎𝟎 𝟎𝟎 𝟎𝟎 𝟎𝟎𝟎𝟎. 𝟎𝟎𝟓𝟓𝟑𝟑 −𝟎𝟎. 𝟗𝟗𝟗𝟗𝟖𝟖 𝟎𝟎 𝟎𝟎 𝟎𝟎 𝟎𝟎

𝟎𝟎 𝟎𝟎 𝟏𝟏 𝟎𝟎 𝟎𝟎 𝟎𝟎𝟎𝟎 𝟎𝟎 𝟎𝟎 −𝟎𝟎. 𝟗𝟗𝟗𝟗𝟖𝟖 −𝟎𝟎. 𝟎𝟎𝟓𝟓𝟑𝟑 𝟎𝟎𝟎𝟎 𝟎𝟎 𝟎𝟎 𝟎𝟎. 𝟎𝟎𝟓𝟓𝟑𝟑 −𝟎𝟎. 𝟗𝟗𝟗𝟗𝟖𝟖 𝟎𝟎𝟎𝟎 𝟎𝟎 𝟎𝟎 𝟎𝟎 𝟎𝟎 𝟏𝟏⎦

⎥⎥⎥⎥⎤

⎩⎪⎨

⎪⎧

𝟎𝟎𝟔𝟔. 𝟒𝟒𝟎𝟎𝐄𝐄 − 𝟎𝟎𝟓𝟓𝟔𝟔. 𝟖𝟖𝟑𝟑𝐄𝐄 − 𝟏𝟏𝟎𝟎

𝟎𝟎𝟔𝟔. 𝟒𝟒𝟎𝟎𝐄𝐄 − 𝟎𝟎𝟓𝟓𝟔𝟔. 𝟖𝟖𝟑𝟑𝐄𝐄 − 𝟏𝟏𝟎𝟎⎭

⎪⎬

⎪⎫

=

⎩⎪⎨

⎪⎧

𝟔𝟔. 𝟑𝟑𝟗𝟗𝐄𝐄 − 𝟎𝟎𝟓𝟓𝟑𝟑. 𝟑𝟑𝟎𝟎𝐄𝐄 − 𝟎𝟎𝟔𝟔

−𝟔𝟔. 𝟖𝟖𝟑𝟑𝐄𝐄 − 𝟏𝟏𝟎𝟎𝟔𝟔. 𝟑𝟑𝟗𝟗𝐄𝐄 − 𝟎𝟎𝟓𝟓𝟑𝟑. 𝟑𝟑𝟎𝟎𝐄𝐄 − 𝟎𝟎𝟔𝟔

−𝟔𝟔. 𝟖𝟖𝟑𝟑𝐄𝐄 − 𝟏𝟏𝟎𝟎⎭⎪⎬

⎪⎫

N


The element 7 local forces and moments at node 7 are f7x = 0, f7y = F×L/2, m7 = F×L2/12

and node 8 are f8x = f8y= m8 = 0. External Local force matrix is given below:

�𝒇𝒇′(𝟔𝟔)�=

⎩⎪⎪⎨

⎪⎪⎧

𝒇𝒇𝟗𝟗𝒙𝒙′

𝒇𝒇𝟗𝟗𝒚𝒚′

𝒎𝒎𝟗𝟗′

𝒇𝒇𝟖𝟖𝒙𝒙′

𝒇𝒇𝟖𝟖𝒚𝒚′

𝒎𝒎𝟖𝟖′ ⎭

⎪⎪⎬

⎪⎪⎫

=

⎩⎪⎨

⎪⎧

𝟎𝟎−𝟓𝟓. 𝟎𝟎𝟎𝟎𝐄𝐄 − 𝟎𝟎𝟒𝟒−𝟒𝟒. 𝟏𝟏𝟗𝟗𝐄𝐄 − 𝟎𝟎𝟖𝟖

𝟎𝟎𝟎𝟎𝟎𝟎 ⎭

⎪⎬

⎪⎫

N

59


�𝒇𝒇(𝟗𝟗)� = [𝑻𝑻]�𝒇𝒇′(𝟗𝟗)�

�𝒇𝒇(𝟗𝟗)� =

⎩⎪⎨

⎪⎧

𝒇𝒇𝟗𝟗𝒙𝒙𝒇𝒇𝟗𝟗𝒚𝒚𝒎𝒎𝟗𝟗𝒇𝒇𝟖𝟖𝒙𝒙𝒇𝒇𝟖𝟖𝒚𝒚𝒎𝒎𝟖𝟖⎭

⎪⎬

⎪⎫

=

⎣⎢⎢⎢⎢⎡−𝟎𝟎. 𝟎𝟎𝟓𝟓𝟏𝟏𝟓𝟓𝟏𝟏 −𝟎𝟎. 𝟗𝟗𝟗𝟗𝟖𝟖𝟔𝟔𝟗𝟗 𝟎𝟎 𝟎𝟎 𝟎𝟎 𝟎𝟎𝟎𝟎. 𝟗𝟗𝟗𝟗𝟖𝟖𝟔𝟔𝟗𝟗 −𝟎𝟎. 𝟎𝟎𝟓𝟓𝟏𝟏𝟓𝟓𝟏𝟏 𝟎𝟎 𝟎𝟎 𝟎𝟎 𝟎𝟎

𝟎𝟎 𝟎𝟎 𝟏𝟏 𝟎𝟎 𝟎𝟎 𝟎𝟎𝟎𝟎 𝟎𝟎 𝟎𝟎 −𝟎𝟎. 𝟎𝟎𝟓𝟓𝟏𝟏𝟓𝟓𝟏𝟏 −𝟎𝟎. 𝟗𝟗𝟗𝟗𝟖𝟖𝟔𝟔𝟗𝟗 𝟎𝟎𝟎𝟎 𝟎𝟎 𝟎𝟎 𝟎𝟎. 𝟗𝟗𝟗𝟗𝟖𝟖𝟔𝟔𝟗𝟗 −𝟎𝟎. 𝟎𝟎𝟓𝟓𝟏𝟏𝟓𝟓𝟏𝟏 𝟎𝟎𝟎𝟎 𝟎𝟎 𝟎𝟎 𝟎𝟎 𝟎𝟎 𝟏𝟏⎦

⎥⎥⎥⎥⎤

⎩⎪⎨

⎪⎧

𝟎𝟎−𝟓𝟓. 𝟎𝟎𝟎𝟎𝐄𝐄 − 𝟎𝟎𝟒𝟒−𝟒𝟒. 𝟏𝟏𝟗𝟗𝐄𝐄 − 𝟎𝟎𝟖𝟖

𝟎𝟎𝟎𝟎𝟎𝟎 ⎭

⎪⎬

⎪⎫

=

⎩⎪⎨

⎪⎧

𝟒𝟒. 𝟗𝟗𝟗𝟗𝐄𝐄 − 𝟎𝟎𝟒𝟒𝟐𝟐. 𝟓𝟓𝟖𝟖𝐄𝐄 − 𝟎𝟎𝟓𝟓

−𝟒𝟒. 𝟏𝟏𝟗𝟗𝐄𝐄 − 𝟎𝟎𝟖𝟖𝟎𝟎𝟎𝟎𝟎𝟎 ⎭

⎪⎬

⎪⎫

N

3.3.4 Assemblage of Element Stiffness Matrices

All the stiffness matrices of element 1 to 7 are assembled to from the global stiffness matrix

of the whole mechanism shown below:

60

3.3.5 Assemblage of Element Load Matrices

All the external forces matrices of element 1 to 7 are assembled to form the global external

force matrix of the whole mechanism shown below:

{𝑭𝑭} =

⎩⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎧

𝒇𝒇𝟏𝟏𝒙𝒙𝒇𝒇𝟏𝟏𝒚𝒚𝒎𝒎𝟏𝟏𝒇𝒇𝟐𝟐𝒙𝒙𝒇𝒇𝟐𝟐𝒚𝒚𝒎𝒎𝟐𝟐𝒇𝒇𝟑𝟑𝒙𝒙𝒇𝒇𝟑𝟑𝒚𝒚𝒎𝒎𝟑𝟑𝒇𝒇𝟒𝟒𝒙𝒙𝒇𝒇𝟒𝟒𝒚𝒚𝒎𝒎𝟒𝟒𝒇𝒇𝟓𝟓𝒙𝒙𝒇𝒇𝟓𝟓𝒚𝒚𝒎𝒎𝟓𝟓𝒇𝒇𝟔𝟔𝒙𝒙𝒇𝒇𝟔𝟔𝒚𝒚𝒎𝒎𝟔𝟔𝒇𝒇𝟗𝟗𝒙𝒙𝒇𝒇𝟗𝟗𝒚𝒚𝒎𝒎𝟗𝟗𝒇𝒇𝟖𝟖𝒙𝒙𝒇𝒇𝟖𝟖𝒚𝒚𝒎𝒎𝟖𝟖⎭

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎫

=

⎩⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎧

𝟎𝟎𝟎𝟎𝟎𝟎

−𝟔𝟔. 𝟗𝟗𝟒𝟒𝐄𝐄 − 𝟎𝟎𝟓𝟓−𝟔𝟔. 𝟗𝟗𝟐𝟐𝐄𝐄 − 𝟎𝟎𝟓𝟓𝟏𝟏. 𝟑𝟑𝟗𝟗𝐄𝐄 − 𝟎𝟎𝟗𝟗

−𝟏𝟏. 𝟐𝟐𝟖𝟖𝐄𝐄 − 𝟎𝟎𝟒𝟒−𝟔𝟔. 𝟓𝟓𝟗𝟗𝐄𝐄 − 𝟎𝟎𝟔𝟔𝟏𝟏. 𝟑𝟑𝟗𝟗𝐄𝐄 − 𝟎𝟎𝟗𝟗

𝟎𝟎𝟎𝟎𝟎𝟎𝟎𝟎𝟎𝟎𝟎𝟎

𝟔𝟔. 𝟗𝟗𝟒𝟒𝐄𝐄 − 𝟎𝟎𝟓𝟓𝟔𝟔. 𝟗𝟗𝟐𝟐𝐄𝐄 − 𝟎𝟎𝟓𝟓

−𝟏𝟏. 𝟑𝟑𝟗𝟗𝐄𝐄 − 𝟎𝟎𝟗𝟗𝟓𝟓. 𝟔𝟔𝟑𝟑𝐄𝐄 − 𝟎𝟎𝟒𝟒𝟐𝟐. 𝟗𝟗𝟏𝟏𝐄𝐄 − 𝟎𝟎𝟓𝟓

−𝟒𝟒. 𝟐𝟐𝟑𝟑𝐄𝐄 − 𝟎𝟎𝟖𝟖𝟎𝟎𝟎𝟎𝟎𝟎 ⎭

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎫

N

3.3.6 Finding Unknown Displacements

A positive horizontal distributed force of magnitude 2000 µN is applied to elements 2 and

negative horizontal distributed force of magnitude 2000 µN is applied at element 6. Node

2 to 7 are free nodes while node 1 and 8 are fixed. Application of the boundary conditions

d1x= d1y=θ1 =0 and d8x= d8y=θ8 =0 at nodes 1 and 8 produce the condensed set of equations

for solution. Solving for displacements at free nodes give:

62

⎩⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎧

𝒅𝒅𝟐𝟐𝒙𝒙𝒅𝒅𝟐𝟐𝒚𝒚𝜽𝜽𝟐𝟐𝒅𝒅𝟑𝟑𝒙𝒙𝒅𝒅𝟑𝟑𝒚𝒚𝜽𝜽𝟑𝟑𝒅𝒅𝟒𝟒𝒙𝒙𝒅𝒅𝟒𝟒𝒚𝒚𝜽𝜽𝟒𝟒𝒅𝒅𝟓𝟓𝒙𝒙𝒅𝒅𝟓𝟓𝒚𝒚𝜽𝜽𝟓𝟓𝒅𝒅𝟔𝟔𝒙𝒙𝒅𝒅𝟔𝟔𝒚𝒚𝜽𝜽𝟔𝟔𝒅𝒅𝟗𝟗𝒙𝒙𝒅𝒅𝟗𝟗𝒚𝒚𝜽𝜽𝟗𝟗 ⎭

⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎫

=

⎩⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎧

𝟎𝟎. 𝟖𝟖𝟗𝟗𝐄𝐄 − 𝟎𝟎𝟔𝟔𝟎𝟎. 𝟎𝟎𝟏𝟏𝟔𝟔𝐄𝐄 − 𝟎𝟎𝟔𝟔

𝟎𝟎. 𝟎𝟎𝟎𝟎𝟓𝟓𝟑𝟑𝟑𝟑𝟎𝟎. 𝟗𝟗𝟔𝟔𝐄𝐄 − 𝟎𝟎𝟔𝟔

𝟎𝟎. 𝟎𝟎𝟏𝟏𝟔𝟔𝐄𝐄 − 𝟎𝟎𝟔𝟔𝟎𝟎. 𝟎𝟎𝟎𝟎𝟑𝟑𝟓𝟓𝟗𝟗

𝟎𝟎. 𝟏𝟏𝟑𝟑𝐄𝐄 − 𝟎𝟎𝟔𝟔𝟏𝟏𝟑𝟑. 𝟐𝟐𝐄𝐄 − 𝟎𝟎𝟔𝟔−𝟎𝟎. 𝟎𝟎𝟎𝟎𝟐𝟐𝟖𝟖𝟎𝟎𝟐𝟐𝟎𝟎. 𝟏𝟏𝟓𝟓𝐄𝐄 − 𝟎𝟎𝟔𝟔

𝟏𝟏𝟐𝟐. 𝟗𝟗𝟔𝟔𝐄𝐄 − 𝟎𝟎𝟔𝟔𝟎𝟎. 𝟎𝟎𝟎𝟎𝟏𝟏𝟔𝟔𝟐𝟐

−𝟎𝟎. 𝟗𝟗𝟑𝟑𝐄𝐄 − 𝟎𝟎𝟔𝟔𝟎𝟎. 𝟎𝟎𝟏𝟏𝟖𝟖𝐄𝐄 − 𝟎𝟎𝟔𝟔

−𝟎𝟎. 𝟎𝟎𝟎𝟎𝟓𝟓𝟎𝟎𝟖𝟖−𝟎𝟎. 𝟖𝟖𝟔𝟔𝐄𝐄 − 𝟎𝟎𝟔𝟔𝟎𝟎. 𝟎𝟎𝟏𝟏𝟒𝟒𝐄𝐄 − 𝟎𝟎𝟔𝟔

−𝟎𝟎. 𝟎𝟎𝟎𝟎𝟗𝟗𝟎𝟎𝟗𝟗 ⎭⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎫

These displacements are then multiplied with the stiffness matrix to find force reactions at

node 1 and 8 to get forces by using equation 3.2.

F = Ku (3.2)

63

{𝑭𝑭} =

⎩⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎧

𝑭𝑭𝟏𝟏𝒙𝒙𝑭𝑭𝟏𝟏𝒚𝒚𝑴𝑴𝟏𝟏𝑭𝑭𝟐𝟐𝒙𝒙𝑭𝑭𝟐𝟐𝒚𝒚𝑴𝑴𝟐𝟐𝑭𝑭𝟑𝟑𝒙𝒙𝑭𝑭𝟑𝟑𝒚𝒚𝑴𝑴𝟑𝟑𝑭𝑭𝟒𝟒𝒙𝒙𝑭𝑭𝟒𝟒𝒚𝒚𝑴𝑴𝟒𝟒𝑭𝑭𝟓𝟓𝒙𝒙𝑭𝑭𝟓𝟓𝒚𝒚𝑴𝑴𝟓𝟓𝑭𝑭𝟔𝟔𝒙𝒙𝑭𝑭𝟔𝟔𝒚𝒚𝑴𝑴𝟔𝟔𝑭𝑭𝟗𝟗𝒙𝒙𝑭𝑭𝟗𝟗𝒚𝒚𝑴𝑴𝟗𝟗𝑭𝑭𝟖𝟖𝒙𝒙𝑭𝑭𝟖𝟖𝒚𝒚𝑴𝑴𝟖𝟖 ⎭

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎫

=

⎩⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎧

−𝟏𝟏. 𝟖𝟖𝟒𝟒𝐄𝐄 − 𝟎𝟎𝟒𝟒𝟗𝟗. 𝟎𝟎𝟓𝟓𝐄𝐄 − 𝟎𝟎𝟓𝟓

−𝟏𝟏. 𝟐𝟐𝟖𝟖𝐄𝐄 − 𝟎𝟎𝟖𝟖−𝟐𝟐. 𝟓𝟓𝟏𝟏𝐄𝐄 − 𝟎𝟎𝟒𝟒−𝟗𝟗. 𝟏𝟏𝟐𝟐𝐄𝐄 − 𝟎𝟎𝟓𝟓 𝟏𝟏. 𝟔𝟔𝟔𝟔𝐄𝐄 − 𝟎𝟎𝟖𝟖−𝟔𝟔. 𝟗𝟗𝟒𝟒𝐄𝐄 − 𝟎𝟎𝟓𝟓−𝟔𝟔. 𝟗𝟗𝟐𝟐𝐄𝐄 − 𝟎𝟎𝟓𝟓𝟏𝟏. 𝟑𝟑𝟗𝟗𝐄𝐄 − 𝟎𝟎𝟗𝟗

−𝟏𝟏. 𝟐𝟐𝟖𝟖𝐄𝐄 − 𝟎𝟎𝟒𝟒−𝟔𝟔. 𝟓𝟓𝟗𝟗𝐄𝐄 − 𝟎𝟎𝟔𝟔𝟏𝟏. 𝟑𝟑𝟗𝟗𝐄𝐄 − 𝟎𝟎𝟗𝟗

−𝟏𝟏. 𝟗𝟗𝟖𝟖𝐄𝐄 − 𝟎𝟎𝟒𝟒𝟗𝟗. 𝟑𝟑𝐄𝐄 − 𝟎𝟎𝟔𝟔

−𝟐𝟐. 𝟗𝟗𝟐𝟐𝐄𝐄 − 𝟎𝟎𝟗𝟗𝟓𝟓. 𝟗𝟗𝐄𝐄 − 𝟎𝟎𝟓𝟓𝟔𝟔. 𝟐𝟐𝐄𝐄 − 𝟎𝟎𝟓𝟓

−𝟏𝟏. 𝟓𝟓𝟔𝟔𝐄𝐄 − 𝟎𝟎𝟗𝟗𝟑𝟑. 𝟗𝟗𝟒𝟒𝐄𝐄 − 𝟎𝟎𝟓𝟓𝟖𝟖. 𝟗𝟗𝟐𝟐𝐄𝐄 − 𝟎𝟎𝟓𝟓

−𝟏𝟏. 𝟑𝟑𝟗𝟗𝐄𝐄 − 𝟎𝟎𝟖𝟖−𝟐𝟐. 𝟎𝟎𝟐𝟐𝐄𝐄 − 𝟎𝟎𝟒𝟒𝟔𝟔. 𝟗𝟗𝟏𝟏𝐄𝐄 − 𝟎𝟎𝟓𝟓−𝟏𝟏. 𝟗𝟗𝐄𝐄 − 𝟎𝟎𝟖𝟖 ⎭

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎫

3.4 Conclusions

Numerical results of kinematic equations are discussed in this chapter. Detailed parametric

analysis shows the effect of different design parameters on amplification factor. Results

show that length and angle of micro-flexure are two major geometric parameters that

control the performance of amplification mechanism. Length is having a direct and angle

is having an inverse relationship with the amplification factor. Geometric parameters are

finalized based on parametric analysis, overall size and PolyMUMPs design rules. FEA of

the designed mechanism is performed using direct stiffness matrix method. The mechanism

is disintegrated in to seven elements having eight nodes. The mechanism is constrained at

64

node 1 and node 8. A distributed force of magnitude 2000 µN is applied to elements 2 and

6. This distributed load is transferred to nodes and assumed to give global load matrix.

Stiffness matrices are calculated for all elements and assembled into one global stiffness

matrix. This global stiffness matrix is reduced by application of boundary conditions.

Solving matrices provided displacements and forces at all nodes. Results show that the

input displacement at element 2 is amplified 15.2 times at the output element 4. This

amplification factor when compared with the kinematic analysis value that is 18.75 gives

error of 18.7%. Kinematic approach is a geometric approach without considering flexible

deflection of the links. However, DSM approach is based on matrix multiplication with

application of boundary conditions and finding unknown displacements.

65

CHAPTER 4. FINITE ELEMENT ANALYSIS USING

INTELLISUITE®

4.1 Introduction

In this chapter, FEA of the proposed amplification mechanism using commercial software

IntelliSuite® is presented. In previous chapter amplification factor of the mechanism was

found out using direct stiffness method that is an implementation of FEM. Configuration

2, that is giving maximum amplification factor, was taken into consideration. However, all

the three configurations are analyzed here.

IntelliSuite® is a family of innovative CAD tools set for MEMS layout design, advanced

process simulation, FEA, parametric analysis, system simulation, fabrication and

packaging analysis. The model, as shown in Figure 4-1, is designed in 3D Builder module

and simulated in Thermo-Electro-Mechanical (TEM) module of IntelliSuite® software.

The simulated results are discussed in the upcoming sections. Static analysis of all three

configurations of the designed mechanism are performed and results discussed in detail.

Natural frequencies and associated mode shapes of the mechanism configurations during

free vibrations are also investigated.

Figure 4-1 Solid model of micro displacement amplification mechanism

66

4.2 ABAQUS

IntelliSuite® uses software technology from ABAQUS for numerical solutions of finite

element models. ABAQUS is well-known for handling highly nonlinear cases involving

high degrees of coupling between energy domains such as thermal, mechanical, and

electrical. ABAQUS utilizes Extended Finite Element Method (XFEM) approach for

handling discontinuities relating to moving boundaries and interfaces.

MEMS devices like micro-resonators, micro-rotors and RF-switches include mechanical

structures moving in an electrostatic field. For this type of problems, it is required to

evaluate accurately the electrostatic forces acting on the devices. XFEM can handle moving

boundaries and interfaces in the electrostatic domain very easily. Different XFEM

techniques were investigate to solve the electrostatic problem when the electrostatic

domain is bounded by a conducting material in 2010 (Rochus et al., 2011).

4.3 Extended Finite Element Method

The XFEM, also known as generalized finite element method (GFEM) or partition of unity

method (PUM) is a numerical technique that extends the solution space for solutions to

differential equations with discontinuous functions. XFEM was established to simplify

difficulties in solving problems with localized features that cannot be resolved by mesh

refinement.

In XFEM, discontinuous basis functions are added to standard polynomial basis functions

for nodes that belong to elements that are intersected by a crack to provide a basis that

included crack opening displacements. A basic advantage of XFEM is that in to track the

67

crack path we don’t need to update the finite element mesh every time. XFEM finds its

application for problems involving singularities, material interfaces, regular meshing of

micro structural features such as voids, and other problems where a localized feature can

be described by an appropriate set of basic functions.

Moreover, handling problems having discontinuities with XFEM suppresses the need to

mesh and re-mesh the discontinuity surfaces, thus lowering the computational costs and

projection errors associated with conventional FEM, at the cost of restricting the

discontinuities to mesh edges.

XFEM is multipurpose tool for analyzing problems having discontinuities and complex

geometries (Belytschko, Gracie and Ventura, 2009). These methods can handle problems

related to crack propagation, grain boundaries, dislocations and phase boundaries

(Belytschko, Gracie and Ventura, 2009).

In XFEM, mesh can be created which is not dependent on entities morphology.

Furthermore, we can add enrichment functions to approximate spaces. For example, while

investigating cracks in elastic medium, need for significant mesh refinement can be

reduced by incorporating asymptotic singular near-field solutions to crack fronts or

dislocation cores (Belytschko, Gracie and Ventura, 2009). Modeling of discontinuous

phenomena in structures and materials becomes easy with XFEM.

To deal with a crack problem in conventional FEM two conditions are to be satisfied. First

the meshed element edges should coincide properly with surface of the crack. Secondly to

place nodes on all side of cracks so that material can be separated. These two conditions

are very difficult to be fulfilled specially in case of 3D problems.

68

In XFEM, a discontinuous field is introduced by adding an extra basis functions. Similarly

XFEM facilitates modeling of phase boundaries. These advantages become more important

while dealing with the conditions where geometry is evolving, especially when crack is

growing crack or dynamic phase boundaries. XFEM uses local enrichments to handle

discontinuities. XFEM can be utilized to construct meshes that are structured and

unstructured. In material sciences, structured meshes are of interest, where unit cell

properties are to be determined. Engineering structures use unstructured meshes because

engineering problems require mesh to be conformed to external boundaries.

The development of XFEM was a result of the broad research in mesh free methods

(Belytschko et al., 1996). Previous surveys of XFEM have been given by Karihaloo and

Xiao (Karihaloo and Xiao, 2003) and by Abdelaziz and Hamouine (Abdelaziz and

Hamouine, 2008); a mathematical survey of XFEM was given by Babuska et al. (Babuka,

Banerjee and Osborn, 2003) and a recent review of XFEM for fracture is given by Rabczuk

et al. (Babuka, Banerjee and Osborn, 2003).

Partition of unity concept is the foundation of XFEM. This is used to enrich finite elements

or mesh-free approximations (Duarte and Oden, 1996). A partition of unity in a domain Ω

is a set of functions given below:

∑ 𝝋𝝋𝟐𝟐∀𝟐𝟐 (𝒙𝒙) = 𝟏𝟏, ∀𝒙𝒙 ∈ Ω (4.1)

In XFEM, any function can be regenerated by multiplication of partition of unity functions

with that function.

69

Furthermore, when the sum is modified by introducing parameters qI, the enrichment

function can be adjusted by these parameters by using the approximation. Using the

approximation given below, we can adjust the enrichment function in case a parameters qI

is added.

𝒖𝒖𝒉𝒉(𝒙𝒙) = ∑ 𝑵𝑵𝟐𝟐∀𝟐𝟐 (𝒙𝒙)𝒖𝒖𝟐𝟐 + ∑ 𝝋𝝋𝟐𝟐∀𝟐𝟐 (𝒙𝒙)𝜸𝜸(𝒙𝒙)𝒒𝒒𝟐𝟐 (4.2)

Here, NI is standard shape functions and uI is the standard nodal DOF.

4.4 Static Analysis of Three Configurations

The displacement amplification mechanism discussed in chapter 3 having parameters given

in the Table 3-1 has been analyzed here. Modeling and simulation of the designed

mechanism has been carried out using commercial software IntelliSuite®. Same models

were used to make masks for fabrication using IntelliMask. To test the mechanism,

controlled input and some sensing mechanism was desired. So, thermal chevrons were used

to provide input displacement because of their high force and large displacement

characteristics with less voltage and high power consumption. Electrostatic sensing was

used to sense the output displacement but unfortunately thermal chevrons did not produced

enough input displacement that can be sensed. So, design and functionality of thermal

chevrons and electrostatic comb drives is not discussed. Probe heads equipped with

tungsten needles were used to provide input displacement and image processing software

was used to calculate the output displacement produced. That’s why the three

configurations designs shown in the simulations have thermal chevrons at the input side

70

and electrostatic comb drives at the output end. To restrict out of plane motion, supporting

flexures were incorporated attached to the ends of comb drives.

Static analysis of displacement amplification mechanism having three configurations is

carried out under unidirectional input displacement load conditions. All the three

configurations are analyzed and discussed below in detail.


An input displacement of 1 μm along X-axis, as shown in Figure 4-2(a), is amplified to

9.17 μm along Y-axis as shown in Figure 4-2(b). Unidirectional motion will produce in-

plane reaction forces along X and Y axis as shown in Figures 4-3 (a & b).

71

(a)

(b)

Figure 4-2 (a) Input displacement of 1 μm given to side mass along X-axis (b) Output displacement of 9.17 μm produced along Y-axis

72

(a)

(b)

Figure 4-3 (a) Reaction force produced along X-axis (b) Y-axis, with an application of input displacement of 1 μm along X-axis

73


With unidirectional input displacement of 1 μm given along X-axis, as shown in Figure 4-

4(a), output displacement of 17.09 μm along Y-axis is produced as shown in Figure 4-4(b).

Figures 4-5 (a & b) show reaction forces produced with the application of motion.


With same input displacement of 1 μm given to side masses inward, as shown in the Figure

4-6(a), an amplified output along Y-axis comes out to be 17.04 μm as shown in the Figure

4-6(b). Figures 4-7(a & b) shows reaction forces produced when input displacement is

applied.

74

(a)

(b)

Figure 4-4 (a) Input displacement of 1 μm given to side masses along X-axis (b) Output displacement of 17.09 μm produced along Y-axis

75

(a)

(b)

Figure 4-5 (a) Reaction force produced along X-axis (b) Y-axis, with an application of input displacement of 1 μm along X-axis

76

(a)

(b)

Figure 4-6 (a) Input displacement of 1 μm given to side masses along X-axis (b) Output displacement of 17.04 μm produced along Y-axis

77

(a)

(b)

Figure 4-7 Reaction forces produced along (a) X-axis (b) Y-axis, with an application of 1 μm input displacement

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4.5 Modal Analysis

FEM has been used to perform modal analysis of the proposed amplification mechanism

to determine the natural frequencies and associated mode shapes during free vibration.

Natural frequencies of the proposed mechanism are given in Table 4-1. Since the proposed

mechanism moves in XY plane only, so the natural frequency at the desired mode will

exhibit pure in-plane motion.

Table 4-1 Frequencies and associated modes of three configurations of the mechanism

Mode no. Configuration 1

Frequency(kHz)

Configuration 2

Frequency(kHz)

Configuration 3

Frequency(kHz)

Mode 1 55.79 86.66 54.84

Mode 2 55.79 237.50 55.22

Mode 3 168.76 242.54 163.40

Mode 4 168.76 276.00(desired mode) 165.80

Mode 5 207.55 305.10 203.05

Mode 6 207.55 333.02 205.60

Mode 7 211.21 343.36 208.37

Mode 8 211.21(desired mode) 393.32 208.79(desired mode)

Mode 9 257.49 435.24 252.86

Mode 10 257.50 468.57 255.47

4.6 Comparison of Amplification Factor

Comparison of amplification factor achieved through kinematic equations, FEA using

direct stiffness method and FEA using IntelliSuite® with design amplification factor is

shown in Table 4-2. Reasons for error is because; the kinematic approach is a geometric

approach without considering flexible deflection of the links. However, DSM approach is

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https://en.wikipedia.org/wiki/Vibration

based on matrix multiplication with application of boundary conditions and finding

unknown displacements. While IntelliSuite® is using XFEM approach. Different

approaches may also increase the error percentages.

Table 4-2 Comparison of displacement amplification factors

Amplification Factor % Error

Kinematic analysis 18.75 -

FEM (DSM) 15.17 18.71

FEM(IntelliSuite®) 17.09 8.85

4.7 Conclusions

In this chapter, FEA of three configurations of the proposed mechanism is discussed in

detail. Static analysis of the mechanism depicts that amplification factors of 9.17, 17.09

and 17.04 can be achieved with three configurations of the mechanism. Amplification

factor of configuration 2 of the displacement amplification mechanism is compared with

FEA using DSM and kinematic model. Natural frequencies at desired modes of three

configurations are 221.21 kHz, 276 kHz and 208.79 kHz respectively.

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CHAPTER 5. DISPLACEMENT AMPLIFICATION

MECHANISM FABRICATION AND CHARACTERIZATION

5.1 Introduction

In the previous chapter, static and modal analysis of the displacement amplification

mechanism was carried out using commercial software IntelliSuite®. Prototypes are

designed, keeping in view design rules for PolyMUMPs, using IntelliMask for designing

of mask layouts. Prototypes of the designed amplification mechanisms are fabricated from

MEMSCAP, USA through CMC Microsystems, Canada using commercially available

Multi-User MEMS Processes PolyMUMPs. PolyMUMPs is a low cost, commercially

available, general purpose micromachining process for MEMS devices available from

MEMSCAP(Cowen et al., 2013). All the experimentation is conducted in the MEMS Lab,

Queen’s University, Kingston, ON, Canada. Static measurements were performed using

Manual Electric Probe System – SÜSS MicroTec PM5. High resolution, 2594x1944,

images were captured using PSM-1000 Laser ready Modular Microscope using ELWD

10X PA Plan Apochromat.

5.2 PolyMUMPs

PolyMUMPs is micromachining process available for developing MEMS devices available

from MEMSCAP, USA. The process consists of three layer of Polysilicon that are micro-

machined and used as the structural material for device development. Sacrificial layer is a

deposited oxide and for electrical isolation silicon nitride is used (Cowen et al., 2013). It

was developed by Berkeley Sensor & Actuator Center. The basic process includes 8

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lithography levels, and 7 physical layers as shown in Figure 5-1 including two mechanical

layers of Polysilicon, one electrical layer of Polysilicon, two sacrificial layers, one

electrical conduction layer and one electrical isolation layer.

Figure 5-1 Seven layers of the PolyMUMPs process (Cowen et al., 2013)

The process consists of 150 mm n-type (100) silicon wafers of 1-2 ohm-cm resistivity. The

surface of the wafer is heavily doped with phosphorus using a phosphor-silicate glass

(PSG) sacrificial layer trailed by a 600 nm low-stress LPCVD (low pressure chemical

vapor deposition) silicon nitride layer that acts as an electrical isolation layer. A LPCVD

Polysilicon film of 500nm is deposited-Poly 0 and patterned by photolithography. After

etching Poly 0 layer a 2.0 cm phosphor-silicate glass (PSG) sacrificial layer is deposited

by LPCVD and annealed at 1050°C for 1 hour in argon. This layer of PSG, known as First

Oxide, is removed to free the first mechanical layer of Polysilicon. The sacrificial layer is

lithographically patterned with the DIMPLES mask and the dimples are transferred into

the sacrificial PSG layer in an RIE (Reactive Ion Etch) system (Cowen et al., 2013).

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The wafers are patterned with the ANCHOR1 followed by reactive ion etching. After

etching, Poly 1 that is 1st structural layer of Polysilicon is deposited. A 200 nm PSG layer

is deposited over the substrate and annealed at 1050°C for 60 minutes. Polysilicon is doped

with phosphorus from the PSG layers on both sides by annealing. To get POLY1

Polysilicon is lithographically patterned using a mask designed followed by PSG layer

etched to produce a hard mask (Cowen et al., 2013).

2nd sacrificial PSG layer that is 750 nm thick is deposited, annealed and patterned using

two dissimilar etch masks. Etch holes in the 2nd Oxide down to the Poly 1 layer is delivered

by POLY1_POLY2_VIA level. This generates connection between the Poly 1 and Poly 2

layers. The POLY1_POLY2_VIA layer is lithographically patterned and etched by RIE.

Metal layer is final deposited layer having thickness of 0.5 cm and used for probing,

bonding and electrical routing (Cowen et al., 2013).

5.3 Kinematic Analysis

Prototypes of three configurations of the mechanism are designed such that thermal

chevrons have been used for providing input displacement and electrostatic comb drives

for sensing the output displacement. Thermal chevrons did not produced enough input

displacement that can be sensed by comb drives. So, design and functionality of thermal

chevrons and electrostatic comb drives are not discussed.

Probes Heads equipped with tungsten probes have been used for providing input

displacement and output displacement are measured using image processing software.

With image resolution of 2594x1944, pixels density, that is pixels per µm, has been

calibrated with the scale given by Polytec as shown in Figure 5-2. Images were captured

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keeping resolution, 2594x1944, fixed. Individual images of the mechanism were taken at

different displacements before and after application of forces. The displacements were

measured in pixels and converted to micrometers by image processing software.

Figure 5-2 Reference image to find pixel density per µm


An input displacement of 1.11 μm is applied manually through Probe Head placed on the

stable platen. The probe head, equipped with tungsten needles has been connected to platen

by vacuum. Figure 5-3 shows microscopic image of fabricated prototype of displacement

amplification mechanism having configuration 1. The input displacement is amplified at

the output giving displacement of 8.44 μm. Hence the amplification factor is 7.6. Figure 5-

4 shows the final position of the mechanism after application of force.

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Figure 5-3 Microscopic image of the configuration 1 of micro displacement amplification mechanism

Figure 5-4 Microscopic image of the prototype after application of force along X-axis to the side mass producing output displacement of 8.44 μm along Y-axis


Configuration 2 prototype is shown in Figure 5-5. Application of 0.22 μm displacement on

both side masses produces output displacement of 3.55 μm giving an amplification factor

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of 16. Figures 5-6 (a & b) shows initial and final output position before and after application

of force.


(a) (b)

Figure 5-6 (a) Microscopic image of the output before application of force (b) Microscopic image of the output after application of force

86


Microscopic image of the configuration 3 prototype is shown in Figure 5-7. An input

displacement of 0.55 μm displacement on both side masses produces output displacement

of 8.88 μm at both output ends giving an amplification factor of 16.14.


Figures 5-8(a & b) shows initial and final output position before and after application of

force.

87

(a)

(b)


88

5.4 Force Analysis

Force and stiffness produced by the designed mechanism on application of external force

applied is measured by using Manual Electric Probe System – SÜSS MicroTec PM5 having

an acupuncture micro-needle.

The micro-needle is placed at an angle of 30–45◦ at Y-axis with reference to the device

placed on horizontal bed of Manual Electric Probe System. The x, y and z positioning stages

have been calibrated before positioning needle tip and taking measurement. An

acupuncture needle and the amplification mechanism experience the same contact force.

So they are acting as two springs connected in series;

F= KN (δNB – δNT) (5.1)

Where, KN is acupuncture needle stiffness, δNB and δNT are displacement in the horizontal

plane of needle base and tip respectively.

Choice of an acupuncture needle having same stiffness as that of the displacement

amplification mechanism to be tested is very important. If soft acupuncture needle is used,

compliance of needle will govern the compliance of the system (Lai et al., 2004).

Acupuncture needle is made from stainless steel having stiffness calculated using equation

5.2.

KN = 𝟑𝟑𝑬𝑬𝟐𝟐𝑳𝑳𝟑𝟑

(5.2)

89

Where E represents Young’s modulus having value of 190 GPa, I is the moment of inertia,

L is the length and D is diameter of the needle.

Comparing needle tip displacement with respect to needle base displacement we can easily

find acupuncture needle displacement (Lai et al., 2004). Needle base displacement is

measured using the calibrated knobs on the micro-force probe. Needle tip displacement is

measured by taking a series of high-resolution pictures of the needle tip through the high

resolution cameras connected with the microscope. The displacements were measured in

pixels by the image processing software.

The prototypes are suspended 2 μm above the polysilicon substrate. It is essential to make

sure the needle-tip is not in contact with the polysilicon substrate. During experimentation

measures forces magnitudes can go significantly higher if the acupuncture needle is not

handled properly and it starts dragging on the substrate. Dragging can be detected by

moving the probe forwards and back: if the tip drags, the forward and backward

measurements will not overlap (Lai et al., 2004).

Stress analysis of the mechanism is not carried out. However, nodal displacements and

force analysis can be used for calculation stresses and finding factor of safety in future.


A uniaxial input force of magnitude 106.1 μN, applied using acupuncture needle produces

maximum deflection of 4 μm which is then amplified 7.6 times for configuration 1. Input

displacement vs. input force plot is shown in Figure 5-9.

90

Figure 5-9 Deflection vs. input force plot of configuration 1


A 4.25 μm deflection produced by application of 165.7 μN force being amplified 16 times

gives deflection of 68 μm. Input displacement vs. input force plot is shown in Figure 5-10.


91


A force 69.5 μN producing 4.2 μm deflection is amplified 16.14 times gives deflection of

67.79 μm. Input displacement vs. input force plot is shown in Figure 5-11.


5.5 Natural Frequency Calculation

Natural frequency of the mechanism is found using equation 5.3.

𝒇𝒇 = 𝟏𝟏𝟐𝟐𝟐𝟐 �

𝑲𝑲𝒎𝒎𝒆𝒆𝒇𝒇𝒇𝒇

(5.3)

Where f is natural frequency, K is the stiffness and meff is effective mass of the mechanism.

Stiffness of the mechanism is calculated using equation 5.4. Output force was measured

92

experimentally at the amplified output end using acupuncture needles, with known

stiffness. Displacements were measured in pixels, by taking high resolution pictures, and

converted to micrometers by image processing software. Stiffness of three configuration

are shown in Table 5-1.

k = 𝑭𝑭𝒙𝒙

(5.4)

Effective mass of the mechanism has been found out by Rayleigh's method that equates

kinetic and potential energy as shown below.

𝑲𝑲. 𝑬𝑬𝒎𝒎𝒊𝒊𝒙𝒙 = 𝑷𝑷. 𝑬𝑬𝒎𝒎𝒊𝒊𝒙𝒙

Potential energy is found from equation 5.5.

PEmax = ∫ 𝐤𝐤𝐲𝐲𝐲𝐲(𝐥𝐥)𝐦𝐦𝐦𝐦𝐦𝐦

𝟎𝟎 𝐲𝐲𝐲𝐲𝐲𝐲 = 𝟏𝟏𝟐𝟐

𝐤𝐤𝐲𝐲 𝐲𝐲(𝐥𝐥)𝐦𝐦𝐦𝐦𝐦𝐦𝟐𝟐 (5.5)

Here, l, w and t are length, width and thickness respectively. Ky is the spring constant.

The maximum kinetic energy is found from equation 5.6.

KEmax = ∫ 𝟏𝟏𝟐𝟐

𝒍𝒍𝟎𝟎 𝒑𝒑𝒑𝒑𝑬𝑬𝒑𝒑(𝒙𝒙)𝟐𝟐𝒅𝒅𝒙𝒙 = ∫ 𝟏𝟏

𝟐𝟐𝒍𝒍

𝟎𝟎 𝒑𝒑𝒑𝒑𝑬𝑬𝝎𝝎𝟐𝟐𝒚𝒚(𝒙𝒙)𝟐𝟐𝒅𝒅𝒙𝒙 (5.6)

93

In general, kinetic energy is given by equation 5.7

KEmax = 𝟏𝟏𝟐𝟐

𝒑𝒑𝒑𝒑𝑬𝑬𝝎𝝎𝟐𝟐 ∫ 𝒚𝒚(𝒙𝒙)𝟐𝟐𝒍𝒍𝟎𝟎 𝒅𝒅𝒙𝒙 (5.7)

Applying Rayleigh's method,

𝟏𝟏𝟐𝟐

𝒌𝒌𝒚𝒚 𝒚𝒚(𝒍𝒍)𝒎𝒎𝒊𝒊𝒙𝒙𝟐𝟐 = 𝟏𝟏

𝟐𝟐𝒑𝒑𝒑𝒑𝑬𝑬𝝎𝝎𝟐𝟐 ∫ 𝒚𝒚(𝒙𝒙)𝟐𝟐𝒍𝒍

𝟎𝟎 𝒅𝒅𝒙𝒙 (5.8)

Resonant frequency is given in equation 5.9.

𝝎𝝎𝒄𝒄𝟐𝟐 = 𝒌𝒌𝒚𝒚 𝒚𝒚(𝒍𝒍)𝒎𝒎𝒊𝒊𝒙𝒙

𝟐𝟐

𝒑𝒑𝒑𝒑𝑬𝑬𝝎𝝎𝟐𝟐 ∫ 𝒚𝒚(𝒙𝒙)𝟐𝟐𝒍𝒍𝟎𝟎 𝒅𝒅𝒙𝒙

= 𝒌𝒌𝒚𝒚

𝒎𝒎𝒆𝒆𝒇𝒇𝒇𝒇 (5.9)

The effective mass can be found from equation 5.10.

𝒎𝒎𝒆𝒆𝒇𝒇𝒇𝒇 = 𝒑𝒑𝒑𝒑𝑬𝑬 ∫ � 𝒚𝒚(𝒙𝒙)𝒚𝒚(𝒍𝒍)𝒎𝒎𝒊𝒊𝒙𝒙

�𝟐𝟐

𝒅𝒅𝒙𝒙 𝒍𝒍𝟎𝟎 (5.10)

For the displacement amplification mechanism under consideration, effective mass can be

found using equation 5.11.

94

𝒎𝒎𝒆𝒆𝒇𝒇𝒇𝒇 = ∫ 𝒑𝒑(𝒙𝒙)𝟐𝟐(𝒙𝒙) � 𝒚𝒚(𝒙𝒙)𝒚𝒚(𝒍𝒍)𝒎𝒎𝒊𝒊𝒙𝒙

�𝟐𝟐

𝒅𝒅𝒙𝒙 𝒍𝒍𝟎𝟎 (5.11)

Here A(x) is the cross-sectional area.

Effective mass of the configuration 2 of the mechanism is calculated below.

𝒎𝒎𝒆𝒆𝒇𝒇𝒇𝒇= 𝟒𝟒𝒑𝒑𝒑𝒑𝑭𝑭𝑭𝑭𝟐𝟐𝑬𝑬 ∫ �𝒚𝒚𝑭𝑭𝑭𝑭𝟐𝟐(𝒙𝒙)𝒚𝒚𝒎𝒎𝒊𝒊𝒙𝒙

�𝟐𝟐

𝒅𝒅𝒙𝒙 𝒍𝒍𝟎𝟎 + 𝟐𝟐𝒑𝒑𝒑𝒑𝑭𝑭𝑭𝑭𝑬𝑬 ∫ �𝒚𝒚𝑭𝑭𝑭𝑭(𝒙𝒙)

𝒚𝒚𝒎𝒎𝒊𝒊𝒙𝒙�

𝟐𝟐𝒅𝒅𝒙𝒙 𝒍𝒍

𝟎𝟎 +

𝒎𝒎𝒄𝒄 � 𝒚𝒚𝒄𝒄𝒚𝒚𝒎𝒎𝒊𝒊𝒙𝒙

�𝟐𝟐

(5.12)

For the guided-end flexure connected to the anchor and the plate is given by equations 5.13

and 5.14 respectively.

𝒚𝒚𝑭𝑭𝑭𝑭𝟐𝟐(𝒙𝒙) = 𝑭𝑭/𝟒𝟒𝑬𝑬𝟐𝟐

�− 𝒙𝒙𝟑𝟑

𝟔𝟔+ 𝒍𝒍𝒙𝒙𝟐𝟐

𝟒𝟒� (5.13)

𝒚𝒚𝑭𝑭𝑭𝑭(𝒙𝒙) = 𝑭𝑭/𝟒𝟒𝑬𝑬𝟐𝟐

�− 𝒙𝒙𝟑𝟑

𝟔𝟔+ 𝒍𝒍𝒙𝒙𝟐𝟐

𝟒𝟒� + 𝟏𝟏

𝟐𝟐𝒚𝒚𝒎𝒎𝒊𝒊𝒙𝒙 (5.14)

95

Considering the structure as a rigid body, the maximum displacement of the shuttle is given

below:

𝒚𝒚𝒎𝒎𝒊𝒊𝒙𝒙 = 𝑭𝑭𝒍𝒍𝟑𝟑

𝟐𝟐𝟒𝟒𝑬𝑬𝟐𝟐 (5.15)

Substituting equation 5.13 to 5.15 gives

𝒚𝒚𝑭𝑭𝑭𝑭(𝒙𝒙)𝒚𝒚𝒎𝒎𝒊𝒊𝒙𝒙

= − �𝒙𝒙𝒍𝒍�

𝟑𝟑+ 𝟑𝟑

𝟐𝟐�𝒙𝒙

𝒍𝒍�

𝟐𝟐 (5.16)

𝒚𝒚𝑭𝑭𝑭𝑭𝟐𝟐(𝒙𝒙)𝒚𝒚𝒎𝒎𝒊𝒊𝒙𝒙

= − �𝒙𝒙𝒍𝒍�

𝟑𝟑+ 𝟑𝟑

𝟐𝟐�𝒙𝒙

𝒍𝒍�

𝟐𝟐+ 𝟏𝟏

𝟐𝟐 (5.17)

After integration and some tedious algebra

𝒎𝒎𝒆𝒆𝒇𝒇𝒇𝒇= 𝒎𝒎𝑭𝑭𝑭𝑭𝟐𝟐 �𝟏𝟏𝟑𝟑𝟑𝟑𝟓𝟓

�+ 𝒎𝒎𝑭𝑭𝑭𝑭 �𝟖𝟖𝟑𝟑𝟗𝟗𝟎𝟎

�+ 𝒎𝒎𝒄𝒄 (5.18)

Calculating the masses gives

𝒎𝒎𝑭𝑭𝑭𝑭𝟐𝟐= 2.33 × 10-12 kg

𝒎𝒎𝑭𝑭𝑭𝑭= 2.79 × 10-12 kg

𝒎𝒎𝒄𝒄= 6.33 × 10-12 kg

Substituting the above calculated masses in equation 5.18, gives an effective mass of

10.22×10-12 kg for configuration 2 of the mechanism. Following the same procedure,

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effective masses of configuration 1 and 3 are also calculated as shown in the Table 5-1.

Natural frequencies and stiffness’s of three configurations are calculated using equation

5.3 and 5.4 respectively. Comparison of natural frequencies of three configurations of

displacement amplification mechanism achieved experimentally and from FEM using

IntelliSuite® is shown in the Table 5-2. Single degree of freedom system is taken into

consideration so only 1st mode is compared. The error percentage between experimental

and FEM results may be because of difference in boundary conditions, fabrication defects

and calculation errors.

Table 5-1 Natural frequencies of three configurations of the displacement amplification mechanisms

Effective Mass (meff) Stiffness (k) Natural Frequency (f)

Configuration 1 16.55 × 10-12 kg 3.12 N/m 69.13 kHz

Configuration 2 10.22 × 10-12 kg 3.6 N/m 94.50 kHz

Configuration 3 16.55 × 10-12 kg 3.0 N/m 67.7kHz

97

Table 5-2 Comparison of natural frequencies of three configurations of displacement amplification mechanism

Natural Frequency

(Intellisuite®)

Natural Frequency

(Experimental)

Error %

Configuration 1 55.79 69.13 kHz 19

Configuration 2 86.66 94.50 kHz 8.2

Configuration 3 54.84 67.7kHz 19.05

5.6 Comparison of Amplification Factors

A comparative study is carried out between results of kinematic equations, direct stiffness

method, FEA using IntelliSuite® and experimentation as shown in the Table 5-3.

Comparison of FEM and Experimental amplification factors of configuration 1, 2 & 3 is

shown in the Table 5-4. Reason of difference in amplification factor achieved

experimentally and that of IntelliSuite® is because of fabrication defects and difference in

boundary conditions. While the variance in amplification factor achieved using

IntelliSuite® and that of DSM is because of difference in FEM techniques, that is, DSM is

using matrix approach and IntelliSuite® is using XFEM. Comparison of amplification

factor achieved with kinematic and experimental analysis shows a difference because of

computation errors and assumptions made in kinematic analysis in which displacement of

flexures is achieved with geometric approach.

98

Table 5-3 Comparison of displacement amplification factors

Amplification Method % Error

Kinematic analysis 18.75 -

FEM (DSM) 15.17 18.71

FEM(IntelliSuite®) 17.09 8.85

Experimental 16 14.6

Table 5-4 Comparison of FEM and experimental amplification factors of configuration 1, 2 & 3

Ampification Factor Configuration 1 Configuration 2 Configuration 3

FEM 9.17 17.09 17.04

EXPERIMENTAL 7.6 16 16.14

Comparison plots of force analysis for configurations 1, 2 and 3 are shown in the Figures

5-12, 5-13and 5-14 below. Difference in results of FEM simulations and experimental data

may be because of fabrication errors, human error and lab uncertainties.

99

Figure 5-12 Comparison of FEM and experimental results of force analysis of configuration 1


100


5.7 Conclusions

Testing of prototypes developed with PolyMUMPs is discussed in detail. Experimentation

methodology is first discussed before presenting empirical results of the designed

amplification mechanism. Amplification factors of 7.6, 16 and 16.14 are achieved with

three configurations. These amplification factors are compared with FEM and analytical

results. Comparison of the three shows a close coordination. Reason of difference in

amplification factor achieved experimentally and that of IntelliSuite® is because of

fabrication defects and difference in boundary conditions.

Sensitivity can be increased by creating larger displacements or by employing larger comb

array sensors. Large displacements can be achieved via displacement amplification or via

softer spring suspensions. Displacement amplification is a better choice as it does not alter

the noise floor of the readout circuitry. Adding large combs add extra mass and need

101

supporting flexures for keeping combs in-plane. Adding extra mass will alter the natural

frequency and more high forces will be needed to displace proof mass. Softer suspension

can increase the displacement but integrity of the mechanism is compromised. Moreover,

softer suspension will also produce undesired out of plane motions that will ultimately

lower the amplification factor. Displacement amplification mechanism is a good option as

they can be easily incorporated with the transducer structure, don’t consume high power

and can be fabricated using commercial fabrication processes that is economical.

102

CHAPTER 6. VARIANT OF DISPLACEMENT

AMPLIFICATION MECHANISM

6.1 Introduction

In previous chapter, three configurations of proposed displacement amplification

mechanism have been tested. Results depict that configuration 2 and configuration 3 can

produce amplification factor of the order of 16 and 16.14 respectively. All configurations

produce amplification in the direction perpendicular to the direction of applied

displacement. So, keeping into consideration that a combination of different configurations

can give even higher amplification factor, configuration 2 and configuration 3 of the

designed amplification mechanism have been combined. Primary goal of the new design

of displacement amplifier presented in this chapter, is to investigate the amplification factor

produced by combination of different configurations of already designed mechanism and

secondary to get amplification in parallel direction instead of perpendicular. A mechanism

consisting of configuration 2 and configuration 3 is designed as shown in Figure 6-1.

Mathematical model of the mechanism is developed using kinematic approach followed

by parametric analysis to optimize the design and finalize the design parameters. FEA is

performed using commercial software IntelliSuite®. Prototype has been fabricated using

PolyMUMPs process and tested at MEMS Lab, Queen’s University, Kingston, Canada.

Comparison of amplification factors is presented at the end of the chapter. The mechanism

is fixed at the two ends shown by point 1 in the Figure 6-1.

103

Figure 6-1 Schematic diagram of displacement amplification mechanism

6.2 Design Description

The proposed amplification mechanism shown in Figure 6-1 is combination of two

configuration no. 2 (marked as 1 and 2 in Figure 6-1) and one configuration no. 3 (marked

as 3 in Figure 6-1) amplification mechanisms. The mechanism is fixed at both ends marked

as 1. Input displacement, U1y, applied to side masses m1 to m4 produces deflection of micro

flexures in compliant mechanisms 1 and 2; that is amplified at output, U4x, in perpendicular

direction. This amplified output act as an input for compliant mechanism 3 that is again

amplified to give output displacement, U8y, parallel to the displacement U1y.

104

The amplification factor was expected to be more than the amplification factor achieved

during experimentation in previous chapter but it is small than the one achieved earlier.

This is because of increased stiffness value of the whole amplification mechanism that has

become three times as compared to the individual compliant mechanism. Parametric

analysis has been carried out to finalize the dimensions of the mechanism. Polysilicon has

been used to fabricate prototype using PolyMUMPs process. The improved actuation can

be utilized in applications, like micro-pump and micro-gyroscopes, where large

displacement motion of the actuator is needed (Shakoor et al., 2005). The displacement

amplification mechanism can be used on actuation side for micro-pumps. However, this

mechanism can be incorporated in micro gyros on sensing side to increase the overlap area

of electrostatic comb drives, thus increasing the differential capacitive signal.

Figure 6-2 Spring model of displacement amplification mechanism

105

6.3 Analytical Modelling

6.3.1 Mathematical Model

The kinematic equations of the amplification mechanism are derived using geometric

approach shown below:

U4x = L1 (sin cos-1 �𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝟏𝟏 − 𝑼𝑼𝟏𝟏𝒚𝒚

𝑳𝑳𝟏𝟏� – sinα1) (6.1)

U8y = L2 (sin cos-1 �𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝟐𝟐 − 𝑳𝑳𝟏𝟏 (𝒄𝒄𝒊𝒊𝒊𝒊𝒄𝒄𝒄𝒄𝒄𝒄−𝟏𝟏 (𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝟏𝟏− 𝑼𝑼𝟏𝟏𝒚𝒚/𝑳𝑳𝟏𝟏) – 𝒄𝒄𝒊𝒊𝒊𝒊𝒄𝒄𝟏𝟏) 𝑳𝑳𝟐𝟐

� –

sinα2) (6.2)

Here U1y is input displacement, U4x and U8y are output displacements of compliant

mechanism 1 and 3 respectively, L1 is length of flexure of compliant mechanism 1 and 2,

α1 is the angle made by the flexure of compliant mechanism 1 and 2. L2 is length of micro-

flexure of compliant mechanism 3, α2 is the angle made by the micro-flexure of compliant

mechanism 3.

The force matrix for the mechanism is given below:

�𝑭𝑭𝒙𝒙 𝒄𝒄𝒖𝒖𝑬𝑬𝑭𝑭𝒚𝒚 𝒄𝒄𝒖𝒖𝑬𝑬𝑴𝑴𝒄𝒄𝒖𝒖𝑬𝑬

� =

⎣⎢⎢⎢⎡

𝑬𝑬𝑬𝑬𝒑𝒑𝑳𝑳

𝟎𝟎 𝟎𝟎

𝟎𝟎 𝑮𝑮𝑬𝑬𝑬𝑬𝒑𝒑𝟑𝟑

(𝟒𝟒𝑮𝑮𝑳𝑳𝟑𝟑 + 𝒄𝒄𝒊𝒊 𝑬𝑬𝑳𝑳𝒑𝒑𝟐𝟐 ) 𝑬𝑬𝑬𝑬𝒑𝒑𝟑𝟑

𝟔𝟔𝑳𝑳𝟐𝟐

𝟎𝟎 𝑬𝑬𝑬𝑬𝒑𝒑𝟑𝟑

𝟔𝟔𝑳𝑳𝟐𝟐 𝑬𝑬𝑬𝑬𝒑𝒑𝟑𝟑

𝟏𝟏𝟐𝟐𝑳𝑳 ⎦⎥⎥⎥⎤

�𝒖𝒖𝒊𝒊𝒊𝒊𝒖𝒖𝒊𝒊𝒊𝒊𝜽𝜽𝒊𝒊𝒊𝒊

� (6.3)

106

Here Fxout and Fyout are forces in x and y direction respectively, Mout being the moment

about z-axis, ux and uy are input displacements along x-axis and y-axis respectively, αs is

shear coefficient and θin is the angular displacement.

6.4 Numerical Analysis

The mechanism is capable of amplifying input displacements in parallel direction

depending on the micro-flexure length, angle made by micro-flexure and input

displacement. A code has been written to simulate the kinematics of the amplification

mechanism. Different parameters of the mechanism especially amplification factor and

output displacements at different critical locations have been analyzed for input

displacement varying from 0.1 μm to 2 μm. Three amplification factors (AF) have been

defined as given in the equations:

AF1 = U4X / U1Y (6.4)

AF2 = U8Y / U4X (6.5)

AF3 = U8Y / U1Y (6.6)

Input displacement U1y has been plotted against AF1 and U4x as shown in Figure 6-3 and

Figure 6-4 respectively. Plots show that with increasing U1y, U4x increases but AF1

decreases which depicts that the mechanism amplifies more for less displacements and is

efficient for small displacement amplification. For example, for 1 μm displacement the AF1

is 10.22. The amplification factor can be controlled by changing the flexure length and

angle.

107

Figure 6-3 Input displacement U1y vs. AF1

Figure 6-4 Input displacement U1y vs. output displacement U4x

U4x has been plotted against AF2 and U8y as shown in Figures 6-5 & Figure 6-6

respectively. Plots show that with increasing U4x, U8y increases and AF2 decreases. The

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 29

9.5

10

10.5

11

11.5

X: 1Y: 10.22

KINEMATICS

INPUT DISPLACEMENT U1y(micrometer)

AM

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0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

5

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X: 1Y: 10.22

KINEMATICS


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108

mechanism amplifies more for small values of U4x, however with increasing U4x output

displacement U8y increases. For example, for 10.22 μm displacement the AF2 is 0.644.

Figure 6-5 Input displacement U4x vs. AF2

Figure 6-6 Input displacement U4x vs. output displacement U8y

0 2 4 6 8 10 12 14 16 18 200.5

0.6

0.7

0.8

0.9

1

1.1

1.2

X: 10.22Y: 0.6444

AM

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2KINEMATICS

INPUT DISPLACEMENT U4x(micrometer)

0 2 4 6 8 10 12 14 16 18 201

2

3

4

5

6

7

8

9

10

X: 10.22Y: 6.586

KINEMATICS

INPUT DISPLACEMENT U4x(micrometer)

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8y

(mic

rom

ete

r)

109

AF3 and U8y have been plotted against input displacement U1y as shown in Figures 6-7 and

Figure 6-8 respectively. Graphs show that with increasing U1y, U8y increases and AF3 first

decreases more and then decreases in a linear way.


Figure 6-8 Input displacement U1y vs. Output displacement U8y

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 24

5

6

7

8

9

10

11

12

X: 1Y: 6.586

KINEMATICS


AM

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0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 21

2

3

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6

7

8

9

10

X: 1Y: 6.586

KINEMATICS


OU

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(mic

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110

6.5 Parametric Analysis

Parametric analysis has been performed to finalize the dimensions of the displacement

amplification mechanism. This analysis enlightened the effect of change in length and

angle on amplification factor. Having knowledge of these dimensional parameters on

amplification factor, optimization becomes possible and more efficient design can be made.

The mechanism, shown in Figure 6-1, is fixed at link 1 and input displacement varying

from 0.1 μm to 2 μm is applied at masses m1 to m4. Minimum feature size and thickness in

PolyMUMPs process is 2 μm, so width is chosen to be 2 μm and thickness is as per

PolyMUMPs fabrication process rule. Effects of change in length of micro-flexure, from

50 µm to 950 µm and angle of micro-flexure, from 1o to 10 o is discussed below.

The plot shown in Figure 6-9 depicts that with increasing angle amplification decreases. It

can also be seen ∆AF1 decreases with increasing angle and become almost constant for

higher angles. It means that increasing angle doesn’t affect AF1 very much which restricts

the range of controlling amplification factors to small flexure angles.

111


Input displacement U1y has been plotted against U4x as shown in Figure 6-10. The plot

shows increase in output displacement U4x with increasing U1y. The magnitude of output

is more for small angles and becomes constant for higher angles which illustrates that for

higher angles output displacement U4x will not increase significantly which restricts the

magnitude of output displacement. The output displacement U4x is acting as an input for

middle compliant mechanism. The displacement, U4x, is orthogonal to input displacement

U1y; however, the mechanism amplifies the displacement in parallel direction.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 25

10

15

20

25

30

35

40

45

50

X: 1Y: 14.85

SENSTIVITY ANALYSIS(ANGLE)

INPUT DISPLACEMENT U1y(micrometers)

AM

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1

1 Degree2 Degree3 Degree4 Degree5 Degree6 Degree7 Degree8 Degree9 Degree10 Degree

112


AF2 has been plotted against U4x.The plot shown in Figure 6-11 shows that with increasing

angle amplification factor decreases. It can also be seen that ∆AF2 almost remains the same

with increasing angle initially and converges for higher angles. It illustrates that increasing

angle doesn’t affect AF2 very much which restricts the range of controlling amplification

factors to small flexure angles.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

5

10

15

20

25

30

35

40SENSTIVITY ANALYSIS(ANGLE)


OU

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(mic

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ete

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113


The displacement U4x has been plotted against U8y as shown in Figure 6-12. The plot shows

an increase in output displacement U8y with increasing U4x. The magnitude of output is

more for small angles and decreases for higher angles which show that for higher angles

output displacement U4x will not increase significantly which restricts the magnitude of

output displacement.

0 5 10 15 20 25 30 35 400.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4SENSTIVITY ANALYSIS(ANGLE)

INPUT DISPLACEMENT U4x(micrometers)

AM

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2


114


AF3 has been plotted against U1y as shown in Figure 6-13. The plot shows that with

increasing angle amplification factor decreases. It can also be seen ∆AF3 almost remain

same with increasing angle initially and converges for higher angles. It means that

increasing angle doesn’t affect AF3 very much which restricts amplification factors.

Moreover, the trends show nonlinear behavior for small angles and linear for higher angles.

0 5 10 15 20 25 30 35 400

2

4

6

8

10

12

14

16



OU

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icro

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115


The displacement U1y has been plotted against U8y as shown in Figure 6-14. The plot

shows an increase in output displacement U8y with increasing U1y. The magnitude of output

is more for small angles and decreases for higher angles which show that for higher angles

output displacement U8y will not increase significantly which restricts the magnitude of

output displacement.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

10

20

30

40

50



AM

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116

Figure 6-14 Input displacement U1y vs. output displacement U8y

Amplification factor can be increased or decreased with the help of length variation

keeping all other parameters constant. The plot shows that with increasing length

amplification factor increases. It can also be seen ∆AF1 decreases with increasing length

and become almost constant for higher flexure lengths. It means that increasing length of

micro-flexure doesn’t affect AF1 for higher length values. AF1 is plotted against input

displacement U1y for different flexure lengths as shown in Figure 6-15.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

2

4

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8

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12

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16



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icro

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117


Input displacement U1y has been plotted against U4x as shown in Figure 6-16. The plot

shows an increase in output displacement U4x with increasing U1y. The magnitude of output

becomes constant for higher lengths which show that for higher values of micro-length

output displacement U4x will not increase significantly which restricts the magnitude of

output displacement. With increasing micro-flexure lengths, ∆AF1 decreases and

ultimately become constant. U4x is also acting as an input for 3rd compliant mechanism

which again gives output in perpendicular direction U8y making the mechanism a parallel

displacement amplification mechanism.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 25

10

15

20SENSTIVITY ANALYSIS(LENGTH)


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50

150

250

350

450

550

650

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950

118


AF2 has been plotted against U4x as shown in Figure 6-17. The plot shows that with

increasing micro-flexure length amplification factor increases. It can also be seen ∆AF2

decreases with increasing length and become constant for higher values of micro-flexure

length. It means that increasing micro-flexure length doesn’t affect AF2 very much which

restricts the range of controlling amplification factors to optimize micro-flexure lengths.

The displacement U4x has been plotted against U8y as shown in Figure 6-18. The plot shows

an increase in output displacement U8y with increasing U4x. The magnitude of output is

more for large flexure lengths.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

5

10

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20

25



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(mic

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119



AF3 has been plotted against U1y as shown in Figure 6-19. The plot shows that with

increasing flexure length amplification increases. It can also be seen ∆AF3 decreases with

0 5 10 15 20 25 300.2

0.4

0.6

0.8

1

1.2

1.4

1.6SENSTIVITY ANALYSIS(LENGTH)


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120

increasing flexure length and become consistent for large flexure lengths. It means that

increasing flexure length doesn’t affect AF3 very much which restricts amplification

factors. The displacement U1y has been plotted against U8y as shown in Figure 6-20. The

plot shows an increase in output displacement U8y with increasing U1y. The magnitude of

output is more for large micro-flexure lengths but ∆U8y decreases which restricts the

magnitude of output displacement.


0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

5

10

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25



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Figure 6-20 Input displacement U1y vs. output displacement U8y

Based on the parametric analysis, overall size of the mechanism and PolyMUMPs process

following parameters are finalized as given in the Table 6-1. Same parameters has been

used for FEA simulations and fabrication.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

2

4

6

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10

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122

Table 6-1 Physical properties and dimensions of device

Parameter Value

Length of micro-flexure (μm) 250

Angle made by micro-flexure (Degrees) 3o

Width of micro-flexure (μm) 2

Thickness of micro-flexure (μm) 2

Overall size of mechanism 910x665x2

6.6 Finite Element Analysis

IntelliSuite® has been used to carryout FEA simulations. Static analysis of the mechanism

is presented here. The mechanism is constrained at the two extreme ends, same has been

shown in figure 6-1 at point 1 marked at both ends. 1 µm in-plane displacement applied to

four side masses applied along y-axis, output displacement of 7.04 µm is achieved along

y-axis, giving an amplification factor of 7.04 as shown in Figures 6-21(a & b).

123

(a)

(b)

Figure 6-21 Displacement produced along (a) X-axis and (b) Y-axis by the proposed

mechanism with input displacement of 1 µm applied to four side masses

124

Application of 1 µm in-plane displacement of four side masses will produce reactions

forces of magnitude 1590 µN along X-axis and 2320 µN along Y-axis as shown in Figures

6-22 (a & b).

(a)

125

(b)

Figure 6-22 Reaction forces produced by the proposed mechanism with input displacement of 1 µm applied to four side masses along (a) X-axis and (b) Y-axis

6.7 Experimentation

Prototypes of the mechanism are fabricated using PolyMUMPs and experimentation

carried out at MEMS Lab, Queen’s University, Canada. Detailed methodology for testing

is followed as discussed in chapter 5.To test the amplification mechanism four side masses

are to be given input displacement and output displacement sensed. Thermal chevron were

designed to provide input displacement to side masses and electrostatic combs to measure

the output displacement in terms of change in capacitance. However, during testing thermal

chevrons did not produced enough displacement that can be sensed by electrostatic combs.

126

So, design and functionality of thermal chevrons and electrostatic comb is not discussed

here. To restrict out of plane motion, supporting flexures were incorporated attached to the

ends of combs.

Probe heads equipped with tungsten needles have been used to provide input displacement

and image processing software has been used to calculate the output displacement

produced. Individual images, having resolution 2594x1944, of the mechanism were taken

at different displacements before and after application of displacement. The displacements

were measured in pixels and converted to micrometers by image processing software.

6.7.1 Kinematic Analysis

Figure 6-23 shows the fabricated prototype of amplification mechanism. An input

displacement of 1.6 μm has been applied to the side masses using tungsten micro-needles

giving amplified output displacement of 9.2 μm. Hence the amplification factor is 5.8.

Figures 6-24 (a & b) shows initial and final position of the mechanism after application of

force.

127

Figure 6-23 Microscopic image of proposed micro displacement amplification mechanism

128

(a)

(b)


129

6.7.2 Force Analysis

An input force of magnitude 154.21 μN, applied inward individually to masses m1, m2, m3

and m4 using four probe heads equipped with four acupuncture micro-needles, produces

inward displacement of 7.57 μm, that is amplified 5.8 times producing output displacement

of 43.91 μm on both outputs ends. Force vs. displacement plots depicts a linear relationship

between the two entities as shown in Figure 6-25.

Figure 6-25 Displacement vs. input force plot of micro displacement amplification mechanism

6.7.3 Comparison of Results

A comparative study is carried out between analytical, FEM and experimental results as

shown in the Table 6-2. Results are compared and are found in close agreement. Reason

for error between experimental results and that got from FEM and analytical model is

because of difference in boundary conditions and fabrication defects.

0

20

40

60

80

100

120

140

160

180

200

0 1 2 3 4 5 6 7 8

Inpu

t For

ce(µ

m)

Input Displacement(µm)

130

Table 6-2 Comparison of analytical, FEM and Experimental results

ANALYTICAL FEM EXPERIMENTAL

AMPLIFICATION

FACTOR

6.58 7.06 5.8

% Error _ 7.2% 15.18%

6.8 Conclusions

Design and analysis of a displacement amplification mechanism consisting of

configuration 2 and configuration 3 of previously developed amplification mechanism,

having an amplification factor of 5.8 (experimental) is presented in this chapter. Parametric

analysis shows that with increasing length of micro-flexures amplification factor increases

and decrease with increase in micro flexure angle, from 1o to 10o. Prototypes are fabricated

from polysilicon using PolyMUMPs process. The mechanism is experimented and results

compared with analytical and FEA results. Comparison of results show a close

coordination.

131

CHAPTER 7. MICRO DISPLACEMENT AMPLIFICATION

MECHANISM INCORPORATED WITH CHEVRON SHAPED

THERMAL ACTUATOR

7.1 Introduction

We envisage that the proposed mechanism can be incorporated for sensing and actuating

applications in the prescribed manner. The design presents a thought process that how the

amplification mechanism can be applied as a sensor or actuator. It’s a nontraditional way

out to achieve the best possible output. From simulation point of view, we have shown the

viability of this structure of the amplifier in such configuration for use as sensor and

actuator. This chapter consists of static and thermal simulations carried out using

IntelliSuite® software of configuration 3 designed with thermal chevrons and electrostatic

comb drives.

7.2 Design Description

The proposed model consisting of chevron shaped thermal actuators, an amplification

mechanism and an electrostatic comb drives for sensing displacements is shown in Figure

7-1. The device is designed keeping in view the parametric studies conducted in chapter 3

and PolyMUMPs process.

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Figure 7-1 Schematic diagram of thermally actuated displacement amplification mechanism

A chevron-type electro-thermal actuator has been utilized for actuating the side masses,

whereas electrostatic interdigitated comb-drives have been used for sensing the amplified

displacement. Chevron actuators are operated via Joule heating. When an input voltage is

applied to the two electrical pads connecting the two terminals of the actuation resistor, a

current will flow through the resistor generating heat due to the Joule effect. The generated

heat will propagate inside the structure and will cause the thermal expansion of the layers.

Chevron-type electro-thermal actuators consist of an array of silicon beams mirrored along

the central axis to generate unidirectional stroke. Beam pairs are designed with a pre-

bending angle β, when heated expand and buckle to produce in-plane movement. These

thermal actuators consume low excitation voltages and produce greater force and

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displacements. Their low damping coefficients ensures high quality factor (Hickey et al.,

2003).

The displacement amplification mechanism is designed such that the two side masses, m1

and m2, move together under same driving force given by thermal chevrons defining 1-

DOF drive-mode. The comb drive is free to move in the orthogonal sense direction under

the influence of translational induced single DOF normal force. The suspension connecting

side masses m1 and m2 with the output via anchor consists of four micro-flexures of same

dimensions, two side masses, one end constraint and another end free to move. The

unidirectional displacement produced by thermal actuators is amplified in the direction

perpendicular to the direction of applied displacement.

When voltage is applied to thermal chevrons, displacement is produced which is then

amplified 20.18 times. Steady state static thermal electrical analysis is performed under

variable resistivity and voltage bias of 2V producing reaction forces of magnitude 194.20

µN and 150.91 µN along X-axis and Y-axis respectively, thus producing displacement of

0.11 µm and 2.22 µm along X-axis and Y-axis respectively. Time domain simulations of

device are carried with constant electrical resistivity, variable voltage and convective

boundary conditions. Modal analysis of the mechanism is carried out to predict the natural

frequencies and associated mode shapes of mechanism during free vibrations. The desired

mode is at frequency of 286.160 KHz. Simulation results prove the viability of the

mechanism as an amplification device with enhanced voltage-stroke ratio.

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7.3 Mathematical Modeling

Neglecting small angle change, chevron actuator can be modeled as fixed-fixed beam.

Stiffness can be calculated by using equation 7.1 (Roark, Young and Plunkett, 1989).

Kchev=M�𝟏𝟏𝟗𝟗𝟐𝟐𝐄𝐄𝟏𝟏𝐋𝐋𝐂𝐂

𝟑𝟑 �=M 𝟏𝟏𝟔𝟔𝐄𝐄𝐄𝐄𝐰𝐰𝟑𝟑

𝐋𝐋𝐂𝐂𝟑𝟑 (7.1)

Here, E is the elastic modulus, I is second moment of inertia, L is length of beam, w is

width and t is thickness and M is beams quantity.

Chevron actuators are operated via Joule heating. The deflection of the chevron is given

by equation 7.2 (Sinclair, 2000).

Defchev = [L2 + 2L(∆L) – L cos2(α)]1/2 – L sin(α) (7.2)

Where L is the length of a single beam, ∆L is the elongation of the beam due to thermal

expansion, and α is the pre-bending angle.

When the temperature increases‚ mutually constrained thermal expansion of the two beams

pairs will result in a linear motion by the combined bending of the beams. Point A will be

forced to move in x direction. By applying the voltage difference, temperature of

combination of beams will increase, deformation occurs through bending in which free

expansion is impeded by two supports‚ which will move the point A along x-axis only.

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The beam is subject to a force, F, and deflection, ux, that can be found using Castigliano’s

displacement theorem and by considering that bending and axial deformations produce by

total strain energy. The equation is:

u1y=u0 = 𝐀𝐀𝛂𝛂𝐀𝐀𝐀𝐀 𝐥𝐥𝟑𝟑𝐬𝐬𝐬𝐬𝐬𝐬𝐬𝐬 𝟏𝟏𝟐𝟐𝟏𝟏𝐳𝐳 𝐜𝐜𝐜𝐜𝐬𝐬𝟐𝟐𝐬𝐬+𝐀𝐀𝐥𝐥𝟐𝟐 𝐬𝐬𝐬𝐬𝐬𝐬𝟐𝟐𝐬𝐬

(7.3)

Where 𝑙𝑙 is length of single thermal chevron, A is cross-sectional area and Iz is moment of

inertia about the z-axis‚ respectively. Where α is the material coefficient of linear thermal

expansion (measured in 1/°C) and ∆T is the temperature variation, β is pre-bending angle

made by beam (Lobontiu and Garcia, 2004).

Using classical chevron actuator, the vertical displacement is negligible due to the

mechanical constraints imposed by the use of constant width beams and pre-bending angle

α and there is no need for additional constraints to confine the motion within the horizontal

plane (Varona, Tecpoyotl-Torres and Hamoui, 2009b).

7.4 Modal Analysis

FEM has been used to do modal analysis and determine the natural mode shapes and

frequencies of the device during free vibration. The natural frequencies of the device are

tabulated in Table 7-1. The desired mode is at frequency of 286.160 kHz.

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Table 7-1 Natural frequencies of the proposed device

Mode Number Natural Frequency (kHz)

Mode 1 66.96

Mode 2 94.05

Mode 3 185.25

Mode 4 213.13

Mode 5 250

Mode 6 252.72

Mode 7 258.35

Mode 8 285.23

Mode 9 286.16

Mode 10 304.54

7.5 Steady State Static Thermal Electrical Analysis

A steady state static thermal electrical analysis is performed on the device. Material chosen

is polysilicon having material properties as shown in the Table 7-2. Resistivity is variable

depending on temperature variation as shown in the equation 7.4(Corporation, 2007).

ρ(T) = ρO (1+8.3x10-4T+5x10-7T2) (7.4)

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where ρ is resistivity, T is temperature and ρ0 of 2x10-3

ohm.cm. Convective heat transfer

coefficient and sink temperature are taken 1 watt/(cm2 oC) and 25oC.

Table 7-2 Parametric values of variables used in steady state static thermal electrical analysis (I. Corporation 2007)

Sr.no Quantity Numerical Value

1 Density 2.33 g/cm3

2 Coefficient of thermal expansion 23.31×10-7/C

3 Resistance 0.00204-0.00365 ohm.cm

4 Thermal conduction 0.25W/cm/C

5 Specific heat 0.678J/g/C

6 Young’s Modulus 140GPa

7 Poisson ratio 0.22

8 Dielectric 4.2

The device is operated via a voltage bias of 2V that is applied to two end points of chevrons.

Thermal deformations are produced and displacement of 0.11µm is observed in x direction.

This displacement is amplified at the output to 2.22 µm giving an amplification factor of

20.18 as shown in Figure 7-2.

138

(a)

(b)

Figure 7-2 Displacement produced along (a) X-axis (b) Y-axis when a biased voltage

of 2V is applied to thermal chevrons

139

In-plane displacements have a direct relationship with biased voltages as shown in Figure

7-3. Plots show that increasing voltage from 1V to 6V varies displacement along X-axis

from 0.039µm to 3.21 µm and along Y-axis from 0.85µm to 26.37µm.

(a)

(b)

Figure 7-3 Displacement produced along (a) X-axis (b) Y-axis when a biased voltage

applied to thermal chevrons is varied from 1V to 6V

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Displacement Amplification decreases from 21.79 to 8.21 with varying voltage from 1V

to 6V as shown in Figure 7-4. Increasing voltage of thermal chevrons provide actuation

higher in magnitude giving higher amplitude outputs but amplification factor don’t

increases with same ratio. Elastic deformations of flexures at high displacement amplitudes

cause non-linear phenomenon to occur. Amplification factor decreases with increasing

displacement making the mechanism more efficient to amplifying small actuations.

Secondly with increasing voltage, temperature of thermal chevrons increases incorporating

nonlinear effects into the joint operations and mechanism performance.

Displacement of thermal chevrons will produce in-plane displacements that are amplified

using displacement amplification mechanism. These displacements will be accompanied

by in-plane reaction forces. Application of voltage bias of 2V will produce 194.2 µN force

along X-axis and 150.91 µN along Y-axis as shown in Figures 7-5(a & b).

Figure 7-4 Plot of decreasing amplification factor with varying voltages from 1V to 6V

141

(a)

(b)

Figure 7-5 Reaction produced along (a) X-axis and (b) Y-axis by application of voltage bias of 2V to thermal chevrons

142

Increasing voltage from 1V to 6V increases reaction force along X-axis from 80.59 µN to

1519.6 µN and along Y-axis from 68.52 µN to 1545.68 µN as shown in Figures 6(a & b)

respectively.

(a)

(b)

Figure 7-6 Reaction forces produced along (a) X-axis (b) Y-axis when a biased voltage applied to thermal chevrons is varied from 1V to 6V

143

Applying a voltage bias of 2V, temperature distribution during device operation is shown

in Figure 8-7. Increasing bias voltage from 1V to 6V, temperature of device increases from

62.53oC to 1376.44oC as shown in Figure 7-8. Melting temperature of polysilicon is

1400oC, so further increase in temperature will melt the device.

Figure 7-7 Temperature distribution of proposed device at 2V bias voltage

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Figure 7-8 Plot shows an increase in temperature with increasing bias voltage

7.6 Conclusions

This chapter presents static and thermal simulations carried out using IntelliSuite®

software of configuration 3 designed with thermal chevrons and electrostatic comb drives.

A steady state static thermal electrical analysis under variable resistivity condition for

voltages varying from 1V to 6V shows a decrease in amplification factor from 20.14 to 8.

Increasing bias voltage from 1V to 6V, temperature of device increases from 62.53oC to

1376.44oC which limits the voltages below 5V since melting temperature of polysilicon is

1400oC. Modal analysis of the mechanism shows the desired mode at 286.160 KHz

frequency. The viability of the configuration 3 of the proposed mechanism for use as sensor

and actuator is presented only from simulation point of view.

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CHAPTER 8. CONCLUSIONS AND RECOMMENDATIONS

8.1 Conclusions

An efficient micro displacement amplification mechanism capable of amplifying

displacement with maximum amplification factor of 16 (approximately) has been

successfully experimentally determined. The proposed mechanism can be utilized in three

configurations giving different amplification factors. The design was analytically modeled

and simulated using MATLAB. Results were found in good agreement with that of the

simulated results using FEM based software IntelliSuite® and experimental results

substantiating the viability of this amplification mechanism. The parametric analysis

played a key role in addition to analytical modeling and numerical simulation in predicting

the behavior of mechanism. A variant of amplification mechanism consisting of two

configurations of basic amplification mechanism has been designed and analyzed. All

prototypes were fabricated using commercial fabrication process PolyMUMPs and tested

at MEMS Lab, Queens University, Kingston, ON, Canada.

Based on the research work presented in this dissertation, following conclusions are

enumerated below:

1. PolyMUMPs was used for the fabrication of all micro displacement amplification

mechanism prototypes. FEA of micro displacement amplification mechanism was

carried out using IntelliSuite®. Necessary masks required for PolyMUMPs were

generated in IntelliMask®.

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2. Design of proposed displacement amplification mechanism along with its three

configurations and different materials available for micro-fabrication have been

described. Analytical model of the mechanism is developed to solve kinematics.

3. Numerical results from analytical model of the mechanism have been investigated

and discussed. Kinematic analysis has been carried out using computer codes

written in MATLAB. Results show that length and angle of micro-flexure are two

major geometric parameters that control the performance of amplification

mechanism. The amplification factor is directly proportional to the length and

inversely proportional to the angle of the mechanism. Results also show that the

amplification factor of the mechanism is restricted to a specific range of length and

angle.

4. Finite element analysis of the designed mechanism was performed using direct

stiffness matrix method, which is an implementation of FEM. The mechanism was

disintegrated in to seven elements having eight nodes. Application of distributed

force of magnitude 2000 µN and appropriate boundary conditions showed that the

input displacement is amplified 15.17 times at the output.

5. Experimental analysis of the mechanism depicts that amplification factor of 7.6, 16

and 16.14 are achieved, for configuration 1, 2 and 3 respectively.

6. Development of prototypes using PolyMUMPs is discussed. Implementation of

fabless technology along with theory and design rules of PolyMUMPs is achieved.

Results of analytical modeling and FEM are compared with experimental results.

Comparison shows a close match between the results from both techniques.

147

7. Design and development of displacement amplification mechanism consisting of

two configurations of designed amplification mechanism having experimentally

achieved amplification factor of 5.8 is also presented. Static analysis results validate

the developed model. Parametric analysis shows that with increasing length of

flexure hinges, from 50 µm to 950 µm, amplification factor increases and become

constant at length of 550 µm. The amplification factor shows a decrease with

increase in micro flexure angle, from 1o to 10o. However after 5o, change in

amplification factor is not so significant, making mechanism almost insensitive to

input displacement above 5o angle.

8. The mechanism is simple, easy to fabricate and can be incorporated with micro

grippers to amplify the displacement at the jaw tip.

9. The proposed mechanism configuration 3 designed with thermal chevrons and

electrostatic comb drives can be incorporated for sensing and actuating

applications. The design presents a thought process for researchers who want to use

this amplification mechanism.

10. Successfully achieved the goal of design, fabrication and testing of two

displacement amplification mechanisms.

11. These mechanisms can be used for industrial application like microaccelerometer,

microgyroscope, microactuator, microgripper and micro energy harvester.

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8.2 Recommendations

(a) Incorporation of mechanism to develop an accelerometer

In future the designed mechanism may be incorporated with 3-axis accelerometer. Proof

mass displacement could be amplified 15-20 times at the output end where interdigitated

comb drives will sense change in capacitance. By incorporating the mechanism, effort may

be made to increase the performance and accuracy as the overlap area of the combs directly

effects the reliability of data acquisition. This change in capacitance could be calibrated for

displacements. This analog data can hence be digitalized and processed further with the

help of readout and operational circuitry developed for the designed accelerometer.

(b) Incorporation of mechanism to develop an energy harvester

The designed mechanism could also be incorporated as an energy harvester. Work on a

macro electromagnetic energy harvester is already underway in which permanent magnets

and insulated copper coils are planned to be used in the form of solenoid. Performance of

electromagnetic energy harvester rely primarily on change in magnetic flux within the coil.

When the magnetic flux changes, an electromotive force (e.m.f) is induced in the coil. If

the circuit is complete, the e.m.f causes current to flow through it. This e.m.f is directly

proportional to displacement permanent magnet covers so that more and more magnetic

lines are cut through the coil. Bi-stable mechanisms could be used to get output for a range

of frequencies instead of getting output only for one single mode frequency. After

successful implementation of the design concept at macro level, micro energy harvester,

could be fabricated and tested.

149

(c) Incorporation of mechanism to develop an actuator

The designed mechanism can also be used to increase the output of any conventional or

non-conventional actuator. In future two new actuators could be developed by

incorporating the designed mechanism with conventional actuators (e.g. capacitive and

thermal chevrons actuators).

(d) Incorporation of mechanism to develop a micro gripper

By incorporating the designed mechanism with a micro gripper, the combination could be

used to hold and transport micro-parts in micro-assembly, biological cells in microsurgery

and micro-particles in material science. Such micro gripper could be based on thermal or

electrostatic actuators and optical or capacitive force sensors.

150

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