ABSTRACT 
 
 
 
 
Title of Document: LATCHING 
MICROELECTROMECHANICAL SHOCK 
SENSOR SYSTEMS: DESIGN, MODELING, 
AND EXPERIMENTS 
  
 Luke J. Currano, Ph.D., 2010 
Directed By: Professor Balakumar Balachandran, Department 
of Mechanical Engineering 
Assistant Professor Miao Yu, Department of 
Mechanical Engineering 
 
 
Latching shock sensors are acceleration threshold sensors that trigger when the 
acceleration level exceeds the designed acceleration threshold. The latching 
mechanism provides a mechanical memory, which keeps the sensor in a triggered, or 
latched, state until the sensor is reset. The attractive feature of this type of sensor is 
that it does not require  power during monitoring; power is only needed to query and 
reset the sensor. Several devices have been presented in the literature, but with  
limited experimental data and models that provide little to no insight into the 
dynamics of the latching event. The aim of this work is to further the understanding of 
the physics and design of micromechanical latching shock sensors by conducting a 
combination of careful experiments and development of original reduced-order 
  
models. These efforts enable one to obtain a detailed picture of the latching dynamics 
for the first time. 
 Latching shock sensors have been designed, fabricated, and experimentally 
evaluated in this work. The model predictions have been compared to the 
experimental results to verify the validity, including a quantitative comparison of the 
position of the shock sensor during a latching event captured via high-speed 
videography. This is the first time a latching event has been imaged in this class of 
sensors, and the first time, the model predictions of position versus time histories 
have been validated through experiments. The models have also been used to conduct 
detailed numerical studies of the shock sensor, amongst other things to predict a  latch 
?bounce? phenomenon during an acceleration event. To understand more thoroughly 
how the various design parameters affect the latching threshold of the sensor, various 
parametric and optimization studies have also been conducted with the reduced-order 
models to guide designs of future latching acceleration threshold sensors. 
 
 
 
 
 
 
 
 
 
 
 
  
 
 
 
 
 
 
LATCHING MICROELECTROMECHANICAL SHOCK SENSOR SYSTEMS: 
DESIGN, MODELING, AND EXPERIMENTS 
 
 
 
By 
 
 
Luke J. Currano 
 
 
 
 
 
Dissertation submitted to the Faculty of the Graduate School of the  
University of Maryland, College Park, in partial fulfillment 
of the requirements for the degree of 
Doctor of Philosophy 
2010 
 
 
 
 
 
 
 
Advisory Committee: 
Professor Balakumar Balachandran, Chair and Advisor 
Assistant Professor Miao Yu, Co-Advisor 
Professor Shapour Azarm 
Associate Professor Elisabeth Smela 
Professor Inderjit Chopra, Aerospace Engineering, Dean?s Representative 
Dr. Madan Dubey, U.S. Army Research Laboratory 
 
 
 
 
 
 
 
  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
? Copyright by 
Luke Joseph Currano 
2010 
 
 
 
 
 
 
 
 
 ii 
 
Dedication 
 
This dissertation is dedicated to my wife, Kim, for her steady patience and support 
throughout the last four years. 
 
 iii 
 
Acknowledgements 
I would first like to thank my advisors, Dr. Miao Yu and Dr. Balachandran, and all 
the members of my committee, who have all helped focus my research and improve 
this dissertation in numerous ways. Many thanks to Danny Gee for help with 
fabricating and testing sensors at various points, and to Brian Isaacson for fabrication 
support. And thanks to the cleanroom support staff at ARL for keeping the cleanroom 
running smoothly and the equipment in good working order. I am deeply indebted to 
Romeo del Rosario and Kwok Tom for providing applications and transition 
pathways for this research. Thanks also to Madan Dubey for encouraging me to 
undertake this PhD, and to Brett Piekarski, Paul Amirtharaj, John Pellegrino, and all 
the management of ARL for their support and flexibility in accommodating my 
course schedule and research. 
 
 iv 
 
Table of Contents 
DEDICATION.............................................................................................................. ii 
ACKNOWLEDGEMENTS.........................................................................................iii 
TABLE OF CONTENTS............................................................................................. iv 
LIST OF TABLES....................................................................................................... vi 
LIST OF FIGURES .................................................................................................... vii 
1. INTRODUCTION ................................................................................................ 1 
1.1 Background and Motivation ......................................................................... 1 
1.2 Acceleration Threshold Sensor Literature Review....................................... 4 
1.3 Friction Measurements for MEMS Literature Review ............................... 10 
1.4 Scope of this Work...................................................................................... 11 
1.5 Organization................................................................................................ 12 
2. DESIGN AND FABRICATION ........................................................................ 14 
2.1 Sensor Design ............................................................................................. 14 
2.1.1 Sensor Limitations and Failure Modes ............................................... 17 
2.2 Rotational Actuator Design......................................................................... 20 
2.3 Device Fabrication...................................................................................... 23 
2.4 Wafer Level Packaging Process.................................................................. 27 
2.5 Summary and Author?s Contributions........................................................ 31 
3. EXPERIMENTAL STUDIES AND RESULTS ................................................ 32 
3.1 Sensor Testing............................................................................................. 32 
3.1.1 Shock Characterization ....................................................................... 33 
3.1.2 High-Speed Video of Sensor Latching ............................................... 41 
3.1.3 Characterization of Electrical Performance of Latch.......................... 50 
3.1.4 Post-Latch Shock Testing ................................................................... 53 
3.1.5 Harmonic Excitation ........................................................................... 55 
3.2 Thermal Actuator Testing........................................................................... 69 
3.2.1 V-Beam Actuator................................................................................ 69 
3.2.2 Rotational Thermal Actuator Testing ................................................. 73 
3.3 Friction Testing........................................................................................... 81 
3.4 Summary and Contributions ....................................................................... 89 
4. MODELING AND NUMERICAL RESULTS .................................................. 90 
4.1 Latching Progression .................................................................................. 90 
4.2 Reduced-Order Latching Sensor Model ..................................................... 91 
4.2.1 One Degree-of-Freedom Model.......................................................... 92 
4.2.2 Two Degree-of-Freedom Model......................................................... 97 
4.3 Equivalent Stiffness for Suspension Springs and Latch ........................... 108 
4.4 Comparison of 1DOF and 2DOF Model Results...................................... 109 
4.5 Sensor Parametric Studies......................................................................... 112 
4.6 Sensor Optimization Studies and Robust Design ..................................... 120 
4.7 V-Beam Actuator Model........................................................................... 130 
4.8 Offset-beam Rotational Thermal Actuator Model.................................... 136 
4.8.1 Parametric Studies of Rotational Thermal Actuator......................... 145 
4.9 Summary and Contributions ..................................................................... 151 
5. RESULTS AND DISCUSSION....................................................................... 154 
 
 v 
 
5.1 Shock Testing  Model Fit.......................................................................... 154 
5.2 Comparison of High-Speed Video Images with Model Predictions......... 156 
5.3 Experimental Confirmation of Latch Bounce Effect Observed in Model 160 
5.4 Rotational Thermal Actuator Results........................................................ 162 
5.5 Summary and Contributions ..................................................................... 165 
6. CONCLUDING REMARKS............................................................................ 167 
6.1 Summary................................................................................................... 167 
6.2 Contributions............................................................................................. 168 
6.3 Possible Improvements and Future Directions ......................................... 172 
A. APPENDIX....................................................................................................... 175 
A.1 Fabrication Details ......................................................................................... 175 
A.1.1 Detailed Fabrication Process Flow ......................................................... 175 
A.1.2. Process Recipe Details........................................................................... 178 
A.2 MATLAB SCRIPTS.......................................................................................... 180 
A.2.1 Rotational Actuator MATLAB Model.................................................... 180 
A.2.2. 1DOF Shock Sensor MATLAB Model ................................................. 181 
A.2.3. 2DOF Shock Sensor MATLAB Model ................................................. 184 
A.2.4. Shock Sensor Parametric Study Code.................................................... 189 
A.2.5. Genetic Algorithm Codes....................................................................... 195 
A.2.6. Fmincon Optimization Routine.............................................................. 202 
A.3. ANSYS Scripts ............................................................................................. 204 
A.3.1. Shape Factor Code ................................................................................. 204 
A.3.2. Latch Contact Force/Deflection Code.................................................... 206 
BIBLIOGRAPHY..................................................................................................... 208 
 
 
 vi 
 
List of Tables 
Table 3.1. Design parameters for the two fabricated versions of the shock sensor.... 34 
Table 3.2. Comparison of sensor design values and those calculated from harmonic 
measurements.............................................................................................................. 62 
Table 3.3. Comparison of latch design values and those calculated from harmonic 
measurements.............................................................................................................. 69 
Table 3.4. V-beam actuator stiffness values calculated from force testing data......... 72 
Table 4.1. Design parameters for shock sensor relevant to the lumped parameter 
model (corresponding physical dimensions are given in Table 3.1). ....................... 103 
Table 4.2. Baseline parameters used for parametric studies (taken from Design 2 
sensor)....................................................................................................................... 113 
Table 4.3. Design parameters used for optimization studies. All other parameters are 
the same as that for Design 2. ................................................................................... 123 
Table 4.4. Optimized Design for low sensitivity to friction-coefficient changes..... 129 
Table 4.5. Parameters used in FEA and numerical calculations............................... 135 
Table 4.6. Electro-thermo-mechanical model parameter definition ......................... 138 
Table 5.1. Parameter values used in simulations of  reduced-order model. ............. 154 
 
 
 vii 
 
List of Figures 
Figure 1.1. The first MEMS acceleration threshold switch by Frobenius et al. [6]. .... 6 
Figure 1.2. Out-of-plane intermittent acceleration threshold sensor design with 
compliant contact by Yang et al. [12]........................................................................... 7 
Figure 1.3. Intermittent acceleration threshold sensor with adjustable threshold and 
electrostatic hold-down by Jia et al. [13]...................................................................... 7 
Figure 1.4. Bistable acceleration threshold sensor design based on prestressed bilayer 
beams by Go et al. [14]................................................................................................. 8 
Figure 1.5. The first latching MEMS accelerometer by Ciarlo et al. [18].................... 9 
Figure 1.6. Latching acceleration threshold sensor design with multiple suspension 
springs by Jean [20]. ................................................................................................... 10 
Figure 2.1. Shock sensor (shown in latched position). The mass is in the center, the 
springs are at the top and bottom, and the reset actuators are at the far left and far 
right............................................................................................................................. 14 
Figure 2.2. Close-up of the latch and release mechanism........................................... 15 
Figure 2.3. Rotational actuator test structure.............................................................. 21 
Figure 2.4. Rotational actuator conceptual drawing and parameter definition........... 22 
Figure 2.5. Deformed shape of actuator...................................................................... 22 
Figure 2.6. Shock sensor fabrication process flow. Individual steps are a) deposition 
of pads for wirebonding on SOI wafer, b) deposition of AuSn solder rings for wafer 
bonding, c) DRIE of device structure, d) deposition of patterned metal contacts on 
latch sidewall, e) DRIE of cleave assist lines, and f) final device release by oxide 
undercut....................................................................................................................... 24 
Figure 2.7. Spinning photoresist over topography: a) thick photoresist completely 
covers corners of mesas, while b) thin photoresist can leave corners uncoated. After 
development, c) there may be some residue left in the corner, but most of the sidewall 
is cleared for coating with metal................................................................................. 26 
Figure 2.8. Sidewall contact metallization.................................................................. 27 
Figure 2.9. Wafer-level packaging process: a) deposition and patterning of oxide 
insulting ring and AuSn bond rings on  cap wafer, b) 375 ?m backside etch in cap 
wafer for cleaving lines and wide part of through hole, c) frontside etch of  cap wafer 
to finish narrow part of through hole, d) bond cap wafer to device wafer using AuSn 
eutectic bond, and e) wirebond to electronic package. ............................................... 28 
Figure 2.10. Electronic package: a) unpackaged die, b) wafer-bonded die, and c) 
wafer-bonded and wire-bonded shock sensor die. Each die has two sensors designed 
to latch at different acceleration threshold levels. ...................................................... 29 
Figure 2.11. Damaged bond pad after wafer bonding, showing bubbles due to HF 
penetration through micropores in the bondpad and subsequent undercut of the 
chromium adhesion layer............................................................................................ 30 
Figure 3.1. Shock table test setup for basic shock experiments.................................. 35 
Figure 3.2. Typical acceleration time history produced by shock table, low-pass 
filtered at 400 Hz (acceleration units are g, or multiples of gravitational acceleration).
..................................................................................................................................... 35 
Figure 3.3. Acceleration to latch over first ten cycles for thirteen devices (Design 1). 
Error bars cover minimum to maximum acceleration levels...................................... 36 
 
 viii 
 
Figure 3.4. Acceleration to latch plotted for thirteen individual devices over first 10 
cycles (Design 1)......................................................................................................... 37 
Figure 3.5. SEMs showing wear at top edge of latch contact after 10 cycles. Region of 
wear is outlined in red................................................................................................. 38 
Figure 3.6.  Repeated cycle testing performed first in positive direction, then, in 
negative direction. Data reported is averaged over 13 devices that successfully latched 
every time in the positive direction............................................................................. 39 
Figure 3.7. Long term repetitive shock cycling results for 36 Design 1 sensors on one 
wafer. .......................................................................................................................... 40 
Figure 3.8. Acceleration threshold to latch for sixteen Design 2 devices over 7 cycles. 
The error bars indicate the maximum and minimum acceleration to latch for any one 
individual device......................................................................................................... 41 
Figure 3.9. Shock table test setup for high-speed video capture. (a) Schematic of the 
entire test setup, and (b) photograph of the shock table with a device mounted under 
the microscope for testing........................................................................................... 42 
Figure 3.10. Analysis of high-speed video frame. (a) Frame from high-speed video 
showing parallel traces. (b) Analysis of the image - two adjacent local maxima 
indicate position of mass............................................................................................. 44 
Figure 3.11. High-speed video of Design 1 sensor showing latching progression. The 
time between frames is 0.235 ms, and the first frame corresponds to the onset of the 
acceleration pulse........................................................................................................ 46 
Figure 3.12. High-speed video of Design 2 sensor showing latching progression. The 
time between frames is 0.235 ms, and the first frame corresponds to the onset of the 
acceleration pulse........................................................................................................ 47 
Figure 3.13. Position data extracted from Design 1 high-speed video. The error bars 
indicate uncertainty in the position measurements taken from the video frames due to 
blurring........................................................................................................................ 48 
Figure 3.14. Position data extracted from Design 2 high-speed video, along with the 
frame synchronization signal from the camera and electrical monitoring of the latch 
closure......................................................................................................................... 49 
Figure 3.15. Electrical sensing of latching event. Positive voltage indicates the device 
is latched. .................................................................................................................... 50 
Figure 3.16. Contact resistance of latched sensor over 100 cycles............................. 52 
Figure 3.17. Latch contact metallization: a) before use and b) after 100 cycles. Metal 
shows significant degradation after cycling, but retains good contact resistance....... 53 
Figure 3.18. Electrical monitoring of latched shock sensor subjected to secondary 
shock of 53.7 g............................................................................................................ 54 
Figure 3.19. Electrical monitoring of latched shock sensor subjected to secondary 
shock of 168.7 g.......................................................................................................... 54 
Figure 3.20. Experimental setup for harmonic excitation........................................... 56 
Figure 3.21. Laser query site for harmonic excitation of sensor. ............................... 57 
Figure 3.22. Frequency response of vibration table with 2V pseudorandom driving 
signal: a) magnitude and b) phase............................................................................... 59 
Figure 3.23. Frequency response for Design 1: a) magnitude and b) phase............... 60 
Figure 3.24. Frequency response for Design 2: a) magnitude and b) phase............... 61 
Figure 3.25. Frequency-response function: a) Design 1 and b) Design 2................... 64 
 
 ix 
 
Figure 3.26. Laser query site for harmonic excitation of latch................................... 65 
Figure 3.27. Velocity response of the latch as a function of frequency: a) magnitude 
and b) phase. ............................................................................................................... 66 
Figure 3.28. Velocity response of vibration table over frequency range used to test 
latch: a) magnitude and b) phase. ............................................................................... 67 
Figure 3.29. Frequency-response function of the latch: a) magnitude and b) phase. . 68 
Figure 3.30. V-beam force-deflection test structure. .................................................. 70 
Figure 3.31. Single V-beam actuator response with no resisting spring, with different 
initial design angles..................................................................................................... 71 
Figure 3.32. Response of a single 3 degree V-beam actuator beam with different 
resisting springs. ......................................................................................................... 71 
Figure 3.33. Actuator loadline for single beam, 3-degree V-beam device, using a 
driving current of 1mA. .............................................................................................. 72 
Figure 3.34. Experimental free deflection data for actuators for L = 400 ?m and 
different actuator beam widths. Wider beams result in less free deflection, but should 
be capable of providing higher forces......................................................................... 74 
Figure 3.35. Experimental free deflection data for actuators for w = 7 ?m and 
different actuator beam lengths. Longer actuators generally provide higher free 
deflection but generate less force................................................................................ 75 
Figure 3.36. Moment-rotation angle relationships for 400 ?m long rotational actuators 
of varying width.......................................................................................................... 76 
Figure 3.37. Force-deflection relationships for 400 ?m long rotational actuators and 
different widths. .......................................................................................................... 77 
Figure 3.38. Comparison of free deflection for V-beam, U-beam, and rotational 
thermal actuators......................................................................................................... 78 
Figure 3.39. Comparison of actuator free deflection under vacuum and atmospheric 
conditions.................................................................................................................... 79 
Figure 3.40. Frequency response of offset beam actuators in air for L = 400 ?m, 
various widths. ............................................................................................................ 80 
Figure 3.41. Friction test structure design. ................................................................. 81 
Figure 3.42. Friction test structure contact pad closeup. ............................................ 82 
Figure 3.43. White-light optical profilometer operational principle. Camera is scanned 
to find position of constructive interference (where sample distance equals reference 
distance) for each pixel. .............................................................................................. 83 
Figure 3.44. Force developed by actuators at 3 ?m travel.......................................... 84 
Figure 3.45. Plot of friction coefficient versus cycle number showing decrease over 
the first several cycles before settling to final value................................................... 86 
Figure 3.46. Scanning electron microscope image of plastic deformation along area of 
sliding contact. Right half of image does not contact friction pad, while the left half 
does. ............................................................................................................................ 87 
Figure 3.47. Forward-looking infrared (FLIR) measurement of temperature profile of 
friction device. The active area corresponding to the actuator is hot, while the contact 
pad is at the ambient temperature. .............................................................................. 88 
Figure 4.1. Shock sensor phases of latching: i) pre-contact, ii) during contact, iii) 
latched but still moving, and iv) latched and resting. ................................................. 91 
 
 x 
 
Figure 4.2. Mass and latch interaction for single DOF model: a) before contact and b) 
during contact.............................................................................................................. 93 
Figure 4.3. Geometry of mass-latch interaction for two DOF model: a) before contact, 
and b) during contact (with exaggerated contact deformation for clarity). ................ 98 
Figure 4.4. Finite element model of latch contact. ................................................... 101 
Figure 4.5. Comparison of contact forces predicted by Eq. (4.28) and finite element 
model over the relevant range of applied force. ....................................................... 102 
Figure 4.6. Design 2 sensor results obtained  from two DOF model for displacement 
of mass (y
m/c
) and displacement of latch (x) when ?=0.1, and acceleration pulse is 120 
g for 5.9 ms. .............................................................................................................. 104 
Figure 4.7. Alternating contact points which cause bouncing of mass after latching.
................................................................................................................................... 104 
Figure 4.8. Time history of contact force for a Design 2 sensor as predicted by 2 DOF 
model when ?=0.1, and acceleration pulse is 120 g for 5.9 ms................................ 105 
Figure 4.9. Position of latch during initial phase of contact, displaying bounce of up 
to 0.4 ?m (blowup of the contact section in the latch position graph from Figure 4.6). 
Point A is where initial contact is made, points B are where contact is lost during a 
bounce, and points C are where contact is reestablished after a bounce. ................. 106 
Figure 4.10.  Kinetic energy of mass and latch during latching event for Design 2 
sensor, when ?=0.1 and acceleration pulse is 120 g for 5.9ms................................. 107 
Figure 4.11. Comparison of single degree-of-freedom results from [9] to two degree-
of-freedom model results for Design 2 sensor, ? = 0.1, acceleration = 120 g for 
5.9 ms........................................................................................................................ 109 
Figure 4.12. Comparison of normal forces associated with 1DOF and 2DOF models.
................................................................................................................................... 110 
Figure 4.13. Comparison of predicted threshold acceleration to latch using one 
degree-of-freedom and two degree-of-freedom models for Design 1 and Design 2 
devices with various friction coefficients. Applied acceleration is a half-sine pulse 
with duration of 8 ms for Design 1 and 5.9 ms for Design 2.................................... 111 
Figure 4.14. Effect of vertical mass dimension on acceleration to latch. All other 
parameters are the same as those for the Design 2 sensor. ....................................... 114 
Figure 4.15. Effect of suspension spring length on acceleration to latch. All other 
parameters are the same as those for the Design 2 sensor. ....................................... 114 
Figure 4.16. Effect of flexure width on acceleration to latch. All other parameters are 
the same as those for the Design 2 sensor................................................................. 115 
Figure 4.17. Effect of latch length on acceleration to latch. All other parameters are 
the same as those for the Design 2 sensor................................................................. 116 
Figure 4.18. Effect of latch width on acceleration to latch. All other parameters are 
the same as those for the Design 2 sensor................................................................. 116 
Figure 4.19. Effect of friction coefficient on latching level for Design 2 sensor. .... 117 
Figure 4.20. Effect of varying latch width and effective mass of latch on magnitude of 
bounce for Design 2 sensor. Applied acceleration pulse is 140 g for 5.9 ms. .......... 118 
Figure 4.21. Modeled response of shock sensor with various damping factors. ...... 120 
Figure 4.22. Algorithm to find the acceleration threshold required to latch any 
individual design....................................................................................................... 122 
Figure 4.23. Fmincon optimization run using Design 2 as the initial point.............. 125 
 
 xi 
 
Figure 4.24. Best value of objective function (friction sensitivity) from first 7 
generations of genetic algorithm by using 100 points in population. The first 
generation had no feasible points.............................................................................. 127 
Figure 4.25. Fmincon optimization run using best point found from genetic algorithm 
as starting point......................................................................................................... 128 
Figure 4.26. Comparison between the sensitivity of Design 2 and optimized design to 
changes in the friction coefficient.  Note that optimized design has almost no change 
in threshold acceleration over a wide range of friction coefficients......................... 130 
Figure 4.27. Notation and geometry used for V-beam actuator stiffness model. ..... 131 
Figure 4.28. Comparison of analytical and finite element spring constants for V-beam 
actuator...................................................................................................................... 136 
Figure 4.29. Offset beam rotational thermal actuator parameter definition. ............ 137 
Figure 4.30. Error in shape factor approximation with respect to finite-element 
calculations. .............................................................................................................. 139 
Figure 4.31. Temperature dependence of resistance for an offset beam actuator..... 140 
Figure 4.32. Spring constant model for rotational actuator. ..................................... 143 
Figure 4.33. Vacuum and atmosphere temperature profiles for an actuator with 
L = 400 ?m, w = 5 ?m, and an applied current of 5 mA. ......................................... 145 
Figure 4.34. Optimization of r
1
 for maximum free rotation, with h = 20 ?m and 
L = 400 ?m................................................................................................................ 146 
Figure 4.35. Optimum value of  netrual axis offset r
1
 vs beam width w. ................. 146 
Figure 4.36. Effect of L and h on free deflection, at constant 50 mW power........... 147 
Figure 4.37. Optimization of L and h for maximum blocked moment, with r
1
 = 5 ?m 
and w = 5 ?m. ........................................................................................................... 148 
Figure 4.38. Optimum relationship between L and h for w = 5 ?m, r
1
 = 5 ?m........ 148 
Figure 4.39. Rotational actuator limited by buckling. .............................................. 149 
Figure 4.40. Plastically deformed rotational actuator............................................... 149 
Figure 4.41. Maximum deflection achievable from a rotational actuator as a function 
of L and w, with r
1
 = 5 ?m and h = 20 ?m. .............................................................. 150 
Figure 4.42. Maximum deflection achievable from a rotational actuator as a function 
of r
1
 and w, with L = 500 ?m and h = 20 ?m. .......................................................... 151 
Figure 5.1. Friction coefficients found from model fit. ............................................ 155 
Figure 5.2. Time-stamped images generated from model results and from high-speed 
video for Design 1 sensor. ........................................................................................ 158 
Figure 5.3. Comparison of model with high-speed video for Design 1 sensor. ....... 159 
Figure 5.4. Comparison of model with high-speed video for Design 2 sensor. ....... 160 
Figure 5.5. Electrical monitoring of contact of latch for a Design 1 sensor: a) full time 
history, and b) expanded view of contact period. Positive spikes in voltage indicated 
the latch is in contact with the mass, zero voltage indicates no contact. Arrows 
indicate contact is reestablished after a bounce. ....................................................... 162 
Figure 5.6. Comparison of model free deflection predictions with experimental data: 
Actuators with L = 400 ?m and  varying actuator beam widths............................... 163 
Figure 5.7. Comparison of model free deflection predictions with experimental data 
for actuators with w = 7 ?m and  varying actuator beam lengths............................. 163 
Figure 5.8. Predicted versus measured torsional stiffness for the rotational offset-
beam thermal actuators. ............................................................................................ 164 
 
 1 
 
1. Introduction 
In this chapter the background and motivation for this work is provided, along 
with a review of the relevant prior work in the literature, the scope of the work and 
the organization of the dissertation. 
1.1 Background and Motivation 
Every synthetic system must ultimately fail. Buildings may withstand the 
ravages of the natural world for hundreds of years; a laptop might function for a few 
years; a tungsten filament light bulb might last a few hundred hours. Three very 
different systems, with very different fabrication techniques and orders of magnitude 
difference in expected lifetime, but each will eventually succumb. The causes of 
failure might include fatigue, flaws in the materials or fabrication, unexpected loading 
beyond the design limit, among others. The failure might be a gradual decay in 
performance or a sudden catastrophic cease of function. The consequences may be 
mild, such as the annoyance of the light going out; or severe, such as hydraulic failure 
on an airliner during flight. The inherent risk in a system can be evaluated according 
to the probability of failure and the severity of the consequences. 
Devices with a high probability of failure and severe consequences are high 
risk, while low probability and low severity equate to low risk. In high risk systems it 
is desirable to know when and how failure is likely to occur, so steps can be taken to 
repair the system and avoid the severe consequences associated with unexpected 
failure. The function of predicting failure before it occurs is performed by a health 
monitoring system. The benefits of health monitoring go beyond avoiding the 
unexpected catastrophic failure, however. In a perfect health monitoring system, 
 
 2 
 
routine scheduled maintenance can be eliminated completely in favor of condition-
based maintenance. This saves time, money, and reduces waste due to replacing 
components before the end of their useful life. 
Health monitoring is becoming more cost effective due to the widespread 
availability of inexpensive, miniature, low-power sensors [1,2]. Two fundamental 
approaches to carry out health monitoring are as follows: i) monitoring of the system 
performance and ii) monitoring of the environment in which the system is located. 
Each of these approaches has its own benefits. On one hand, implementation of 
environmental monitoring is typically simple, with a handful of sensors and simple 
allowable thresholds. Performance monitoring generally requires many more sensors 
and sophisticated data analysis algorithms. On the other hand, performance 
monitoring is more directly linked to system health. Historically, specification sheets 
list allowable environmental conditions, and environmental monitoring can be used to 
determine if the allowed specifications are exceeded. Due to the ease of 
implementation and the available specifications, environmental monitoring is 
currently the more commonly used approach.  
The more challenging applications of health monitoring are those with limited 
available power and space. Automobile companies have been very successful in 
integrating sensors in their vehicles because power and size are not major constraints. 
The result is that the average car today contains dozens of sensors, including 
accelerometers, gyroscopes, pressure sensors, temperature sensors, light sensors, and 
in some cases humidity sensors, rain sensors, and cameras [3]. When power and size 
are major factors, implementing health monitoring systems becomes more difficult 
 
 3 
 
and tradeoffs are necessary between the amount and precision of the data collected 
and the power consumed.  
The present work is motivated by a need to monitor acceleration in 
environmental monitoring systems with a minimal amount of power. Power available 
for environmental monitoring systems is often severely limited, especially for small, 
long-life systems. Low-power sensors are critical to enabling embedded health 
monitoring where power is not readily available. For example, a piece of ammunition 
may be stored for ten years or more on a pallet in a warehouse before being required 
to perform its function with near-perfect reliability. For many (if not most) health 
monitoring applications, temperature, humidity, and acceleration are core 
environmental parameters that must be monitored. By way of example, the Military 
Standard Design Requirements for Standard Electronic Modules lists specific ranges 
for these parameters (along with a few other specialty environments rarely 
encountered in commercial applications) [4]. There are many other sensors needed for 
specific applications, but these three are needed in most cases, and may be sufficient 
by themselves in many cases. The sensor targeted by this work is the accelerometer, 
because acceleration alone of these three core parameters of interest is generally a 
quickly-changing quantity. For temperature and humidity sensing, power 
management is not a major factor because the sensor can have very low duty cycles, 
waking up for a millisecond every minute or hour to take a reading. Because the 
temperature and humidity environment generally change over much longer 
timescales, a good representation of the environmental history of each can be 
obtained by interpolating between measurements widely spaced in time.  
 
 4 
 
Conversely, acceleration must be sampled at high frequencies because a 
damaging drop or impact may last only a few milliseconds. The acceleration sensor 
must therefore be constantly powered or a critical transient impulse might be missed. 
In addition, supporting circuitry such as signal conditioning electronics and analog to 
digital converters must be powered on to convert the sensor output into a form that 
the system can use. Together this can pose a significant drain on the power supply. 
An acceleration sensor that draws no power and requires no supporting electronics 
during monitoring is therefore ideal. The mechanical latching acceleration threshold 
sensor is one approach to meet this need at the expense of resolution - the information 
provided by each sensor is only whether the acceleration exceeded the threshold. If 
several sensors with different threshold levels are provided, the acceleration seen by 
the system can be classified into one of several bins, but the resolution will still be 
orders of magnitude lower than using a true accelerometer. 
1.2 Acceleration Threshold Sensor Literature Review 
Microelectromechanical system (MEMS) accelerometer technology is mature, 
with many different applications and hundreds of millions of dollars in annual sales 
[5]. MEMS acceleration threshold sensors have not enjoyed nearly as much interest 
as accelerometers because the potential applications are far more limited.  In many 
cases, an exact acceleration level or profile is needed, and a threshold sensor alone 
will not give this information.  Threshold accelerometers do have a niche in 
applications that either require ultra-low-power monitoring of acceleration or those 
that do not have a need for the high precision offered by a conventional 
accelerometer.  No power monitoring of acceleration might be necessary in systems 
 
 5 
 
with a lifetime of many years or where the power source is very limited.  The low-
precision realm includes primarily systems where a simple go/no go acceleration 
level can be defined, such as automobile airbags, projectile launch systems, or simple 
health monitoring systems that monitor whether allowable ranges have been 
exceeded. 
A handful of other acceleration threshold sensors have been studied by other 
researchers. They are all mechanical switches of one sort or another, where the switch 
changes state when the acceleration is exceeded. Some researchers use the name 
?inertial switch? or ?g-switch,? but the function is the same. 
There are two primary categories of acceleration threshold switch designs: 
contact between surfaces is broken following an acceleration (normally closed), or 
contact between surfaces is made following an acceleration (normally open).  In 
either category, the response to the acceleration can be classified as ?intermittent? 
(wherein the device reverts to its original state after the acceleration event is over) or 
?persistent? (wherein the device remains in the altered state after the acceleration 
event is over). An intermittent design is sufficient if the device is monitored 
constantly, such that the trigger event can be recorded as it occurs.  A persistent 
design is generally more complex, but also has the flexibility of not requiring the 
constant power of monitoring electronics. The persistent nature of the device serves 
as a mechanical memory and is suitable for system architectures in which the 
electronics are powered up to query the sensors at regular intervals. This is the typical 
architecture for monitoring of temperature and humidity, and a persistent threshold 
accelerometer enables the same mode for acceleration monitoring. 
 
 6 
 
Prior work on acceleration threshold sensors is mostly concentrated on 
intermittent, normally-open devices [6-13].  These devices generally consist of a 
spring, a mass, and a contact surface separated from the mass by a gap.  Some use 
out-of-plane motion of the mass [6, 7, 10-12] and others use in-plane motion [8, 9], 
but each relies on the intermittent contact between the mass and another surface.  
Several of these previous designs array the switches such that many different 
acceleration levels can be detected. The very first micromechanical acceleration 
contact switch was reported in 1972 by Frobenius, Zeitman White, O?Sullivan and 
Hamel [6], and consisted of gold cantilevers above a gold stationary electrode, with 
only the mass of the cantilever providing inertial force (Figure 1.1). More recently, 
Yang et al. [12] introduced multiple springs and thick proof masses for low 
acceleration thresholds, as well as a compliant electrode to decrease bounce and 
protect against shock damage (Figure 1.2).  Jia et al. [13] incorporated an electrode to 
 
Figure 1.1. The first MEMS acceleration threshold switch by Frobenius et al. [6]. 
 
 7 
 
provide for electrostatic tuning of the threshold level and to hold the switch in the 
closed state (Figure 1.3). While this blurs the line between intermittent and persistent 
designs, the author considers this an intermittent device because electrical energy is 
required to hold the device in the closed state. 
 
Figure 1.2. Out-of-plane intermittent acceleration threshold sensor design with 
compliant contact by Yang et al. [12]. 
 
Figure 1.3. Intermittent acceleration threshold sensor with adjustable threshold 
and electrostatic hold-down by Jia et al. [13]. 
 
 
 8 
 
 
Figure 1.4. Bistable acceleration threshold sensor design based on prestressed 
bilayer beams by Go et al. [14]. 
 
A few persistent, normally open designs have been studied as well.  Persistent 
contact has been achieved using bistable mechanisms [14-17] as well as physical 
locks or latches [18-24]. The first report of a bistable acceleration threshold switch 
was by Go, Cho, Kwak, and Park in 1996 [14], and it used a prestressed bimorph 
beam with out-of-plane contacts (see Figure 1.4). More recent work makes use of 
bistable geometries that can be easily defined lithographically [15-17]. 
The shock sensor that is the subject of this dissertation is a physically latching 
acceleration switch. This requires sliding contact between two surfaces, generally 
considered to be the most challenging of the four classes of MEMS devices (no 
moving parts, moving but no contact, contact but no sliding, and contact with sliding) 
from a design perspective because there are more potential failure modes [25]. Partly 
due to this complexity, and partly because of the wide availability of standard MEMS 
accelerometers, very few physically latching acceleration switches have been 
reported. However, there is a fundamental advantage; the physical latch allows the 
 
 9 
 
device and all associated circuitry to be completely powered off most of the time to 
save power, with the system waking up to query the sensor either at pre-programmed 
intervals or when a user instructs it to query by pressing a button, for instance. 
The earliest acceleration threshold switch with a physical latching mechanism 
was reported by Ciarlo in 1992 [18], and consisted of two cantilevers perpendicular to 
each other with shaped ends to provide a locking mechanism (Figure 1.5). More 
recently, multiple suspension springs, large proof masses, and electro-thermal reset 
actuators have been included by Whitley, Kranz, Kesmodel, and Burgett [19], and 
Jean et al, [20-22] (Figure 1.6). Guo et al. [23, 24] presented a device that physically 
separates the mass from the latching mechanism and electrical contacts to reduce the 
chances of secondary shocks or vibration causing loss of contact after latching. All of 
the reports of physically latching acceleration switches have very limited 
experimental data (at most test results from one or two devices are reported). Each 
also relies on simplistic (often static) models to predict the threshold acceleration 
and/or response time that in every case neglect the interaction of the sensor with the 
latch. 
 
Figure 1.5. The first latching MEMS accelerometer by Ciarlo et al. [18]. 
 
 10 
 
 
Figure 1.6. Latching acceleration threshold sensor design with multiple 
suspension springs by Jean [20]. 
 
 
1.3 Friction Measurements for MEMS Literature Review 
From a design perspective, the most challenging MEMS are those that require 
surfaces with sliding contact between them. Accounting for friction and wear in the 
contacting surfaces introduces an added level of complexity in modeling and 
uncertainties in material properties. Although friction and wear are well understood 
and extensively characterized on the macroscale, the same is not true at the 
microscale. 
The characterization of friction and determination of friction coefficients on 
the microscale has been carried out by using various techniques. Perhaps the most 
popular method is to scrape the tip of an atomic force microscope or a scanning probe 
microscope across the surface of a film and measure the normal and transverse forces 
by using the tip itself [26, 27]. This method can not be easily adapted to sliding 
contacts where the two surfaces are vertical rather than horizontal, as is the case in 
many MEMS. In addition, the friction properties may depend on the surface 
preparation conditions, with published friction coefficient values ranging from 0.01 to 
0.8 for silicon/silicon nitride contact [28]. For these reasons, many MEMS have been 
used for friction measurements on both horizontal surfaces [29-31] and vertical 
 
 11 
 
surfaces [32-37]. All of these devices have relied on electrostatic actuation to 
generate the needed normal and tangential forces. The electrostatic actuators include 
parallel-plate and comb-drive type actuators. In this work, a design for making 
friction measurements by using MEM thermal actuators is detailed and presented 
along with the obtained experimental results. The advantages of using thermal 
actuators include far lower operating voltages, a much smaller footprint, and typically 
a much longer range of motion. Furthermore, in this design, one can create a wide 
range of forces by changing a single parameter, the offset angle of the V-beam 
actuator. 
 
1.4 Scope of this Work 
The focus of this work is to enable the development of an acceleration 
threshold sensor system with the following features: i) uni-axis, bi-directional 
acceleration threshold sensing (i.e. +/- accelerations in the y-axis must be detected 
and differentiated from each other); ii) a physical latching mechanism to serve as a 
mechanical memory of a threshold event; iii) electrical sensing of the latch closure; 
and iv) a reset function to use the sensor again after it has been triggered. The 
dissertation effort encompasses the design, modeling, fabrication and experimental 
study of this sensor system. The experimental study is used to learn more about the 
dynamics of this type of sensor and verify that the model is sufficiently representative 
of the sensor operation. The model is intended to serve as a framework for design of 
the sensors. As such, parametric studies and optimization codes are also included in 
the scope of the work. 
 
 12 
 
Various parts of this dissertation have been adapted from journal and 
conference papers written through the course of this dissertation research. These are 
noted at the top of each chapter as applicable. Although some of these papers have 
several co-authors, the co-authors can be grouped into one of the following three 
categories: i) the author?s advisors, who primarily guided the research directions, ii) 
collaborators who provided fabrication or testing support, wherein all of the 
underlying procedures have been developed by the author of this dissertation, and iii) 
collaborators on subjects related to the dissertation topic but not included in the 
dissertation, such as the development of a current ramp circuit to power the reset 
actuators discussed in reference [37].   This latter material, which is not a part of the 
author?s contributions, is not included in this dissertation. 
1.5 Organization 
This dissertation is organized into six chapters, an appendix, and a 
bibliography section. In the first chapter, the background and motivation for this work 
is provided along with a review of prior work on MEMS shock sensors. In Chapter 2, 
the shock sensor design under consideration is presented and the fabrication process 
flow developed to realize the design is discussed. In Chapter 3, the experimental 
studies carried out with the sensor, reset actuators, and friction test structures are 
discussed. In Chapter 4, the author has discussed the development of models for the 
shock sensor and the reset actuators, as well as parametric and optimization studies 
undertaken with the models. In Chapter 5, comparisons between the model 
predictions and experimental results are made for the purpose of verification. In 
Chapter 6, the conclusions that can be drawn from the dissertation work are presented 
 
 13 
 
along with the author?s contributions and thoughts on future research directions.  
Appendix A is included to provide fabrication details and information related to the 
numerical studies.   
 
 14 
 
2. Design and Fabrication 
In this chapter, an overview of the shock sensor as a system is presented. The 
design and working principle of the shock sensor under consideration are introduced 
along with a novel rotational actuator developed during this work. The author has also 
presented the fabrication processes employed to make the devices found in the 
experimental portion of this work. Portions of this chapter have been adapted from 
the author?s journal papers on the latching shock sensor [38] and the rotational 
thermal actuator [39]. 
2.1 Sensor Design 
The sensor consists of a series of springs attached to the substrate at one end 
and the mass at the other, as shown in Figure 2.1.  The pictured device makes use of a  
3.1mm
2.55mm
 
Figure 2.1. Shock sensor (shown in latched position). The mass is in the center, 
the springs are at the top and bottom, and the reset actuators are at the far left 
and far right. 
 
 15 
 
two-fold symmetric design with four springs to reduce the sensitivity to off-axis and 
rotational accelerations compared with single and dual-spring configurations. When 
the system undergoes acceleration along the y-axis, the motion of the mass forces two 
of the springs into compression and the other two springs into extension. The sensor 
incorporates latches anchored to the substrate, which engage the mating pieces 
attached to the mass after the sensor has traveled the designed setback distance 
(Figure 2.2). The latches are cantilevered so that they are stiff in the y-direction and 
less stiff in the x-direction. This allows them to move out of the way as the mass 
pushes past them. Four latches are used in this device with two of them meant for 
positive acceleration and the other two meant for negative acceleration along the 
same axis. Once the mass pushes past the latches, it may continue to move but will be 
prevented by the latches from returning to its resting state. As the acceleration 
 
Figure 2.2. Close-up of the latch and release mechanism. 
 
 16 
 
dissipates, the mass comes to rest against the flat back surface of the latches. These 
flat surfaces are coated with metal, and when the mass touches both metal contacts, 
an electrical circuit is closed between the two latches. This serves as the shock-
detection mechanism. 
The sensor is reset following the detection and recording of the shock event 
by using thermal actuators (Figure 2.2).  Here, V-beam style thermal actuators (also 
called bent-beam actuators) are used, since it is difficult to get sufficient force with 
U-beam style actuators (also called hot-beam/cold-beam actuators). The bent-beam 
actuators have been described in previous studies [6-8], and here, for clarity, a brief 
description of their operation is provided. The baseline actuator is a conductive beam 
divided into two segments that are rotated in-plane slightly by design to give the full 
beam a ?V? shape. When current is driven through the beam, Joule-heating causes 
thermal expansion of each segment. Since the beam structures are fixed at the ends, 
the two segments push against each other and generate an in-plane motion in the 
direction they are pointed towards. Due to the shallow angle and slender beam profile 
required for thermal strain amplification of the V-beam structure, most of the 
generated force is not along the actuation direction. The force component that is not 
along the actuation direction axially compresses the V-beam structure. If this axial 
force component is large enough to cause buckling of the structure, the output force 
from the actuator can drop dramatically.  However, multiple beam structures can be 
used in parallel to overcome this limitation.  
In the present work, the required actuation force to reset the shock sensor 
necessitated the use of five parallel beam structures. For the chosen V-beam 
 
 17 
 
dimensions (5 ?m width and 600 ?m length), a minimum of five beams is required to 
reliably unlatch the device while avoiding buckling of the individual V-beams. The 
actuators here push against the latches to disengage them from the mass. It is worth 
noting that the reset actuators are not connected to the latches; there is a gap of 3 ?m 
in between them and this gap can be seen in Figure 2.2. 
2.1.1 Sensor Limitations and Failure Modes 
The sensor design investigated here is limited to sensing of a single 
acceleration threshold along one axis. By having an array of multiple devices and 
rotating the orientation, two axes of sensing and multiple threshold levels can be 
incorporated onto a single chip. To provide a third sensing axis, a second chip would 
have to be made with a sensing axis oriented in a perpendicular direction to the first 
axis. The sensor also cannot be used to sense a second acceleration event after it has 
been latched and before it is reset ? therefore, two (or more) events may be 
interpreted as a single event. There is also a finite delay in the sensor response, as the 
sensor must travel past the latch, and return to rest against the latch as the imposed 
acceleration dies out. For the designs discussed here, the delay is on the order of 
1.5 to 3 ms. With design modifications that separate the electrical contacts from the 
latch, the dissipation of the acceleration is not required before electrical contact is 
established [19, 23]. These same design modifications prevent the loss of electrical 
contact due to vibration or secondary shocks occurring after the sensor is latched.   
In the MEMS industry, there is a widely accepted characterization of devices 
into four classes based on the complexity of design and the number of potential 
failure modes [25]. Class 1 consists of devices with no moving parts, and this class is 
 
 18 
 
considered the simplest to design since it has the fewest failure modes. Class 2 
consists of devices with moving parts, but no contact. Class 3 consists of devices with 
moving parts that make contact with other parts of the device or substrate, but no 
sliding is involved in the contact. Class 4 consists of devices that have moving parts 
that engage in sliding contact with other parts of the device or substrate.  The shock 
sensor belongs to Class 4, as the mass slides past the latch during an acceleration 
event, and the latch slides out of the way of the mass during the reset operation. In 
addition to the failures modes for devices in Classes 1-3 (particle contamination, 
charging, electrostatic discharge, stiction, fatigue, creep, impact-induced cracking or 
fracture), the general failure modes for devices in Class 4 also includes the unique 
failure modes associated with friction-induced heating and wear. 
For the specific shock sensor designs described in this work, the author has 
observed failure by stiction and shock-induced fracture of the latches, suspension 
springs, and reset actuators. Wear is observed in the contacting surfaces, including 
both the rounded side of the latch and the contact metallization on the flat side of the 
latch, but this has not been observed to cause failures when the devices have been 
tested up to 100 cycles. 
Shock-induced fracture of the thin members usually occurs when the wafer is 
cleaved to separate individual devices. At least one thin member fractures during 
cleaving in approximately one out of every three or four devices. This failure mode 
has been observed during shock testing as well, but is far less common. 
Stiction is the biggest problem with this design and this is by far the most 
common failure mode, due to the very large mass (2.5 mm x 3.1 mm), relatively 
 
 19 
 
compliant suspension springs, and small (2-4 ?m) gap between the mass and the 
substrate. Stiction failures occur in fabrication if the released device is exposed to 
liquid, but this is relatively easy to avoid with a gas phase release process. They also 
occur in storage if the device is stored in a non-conductive plastic carrier, because the 
carrier collects charge from the air and attracts or repels the mass, eventually causing 
it to contact the substrate and stick in place. Stiction failures also occur in operation 
of the device, and the author hypothesizes that this is due to rubbing of the mass 
against the substrate, resulting in exchange of electrons and a buildup of charge 
similar to that what occurs during rubbing of a balloon against a wool sweater. 
While not technically device failure, during shock testing, the mass does 
sometimes engage one latch but not both, resulting in a situation where the device 
needs to be reset but the circuit is not closed. This may be caused by slight 
asymmetries in the fabrication process, where one latch is stiffer than the other, for 
instance, or from an acceleration that is not applied exactly parallel to the sensing 
axis. This effect could be alleviated by designing a sensor with only one latch, by 
using a spring anchor as the second terminal for sensing the  change in resistance 
when the switch closes. 
The electrothermal reset actuators are also subject to overheating if the 
applied current is too large. Overheating can result in plastic deformation or fracture 
of the actuator beams. This can be avoided by using a reset signal with a defined 
pulsewidth. If the actuator beams are too narrow, too long, or the bend angle is too 
shallow, the actuators may also buckle before the device is reset. This can be 
addressed in the actuator design, ensuring that the output force generated is sufficient 
 
 20 
 
to reset the latch and the axial load on each beam does not exceed the Euler buckling 
load [40]. 
2.2 Rotational Actuator Design 
Thermal actuators are often not given full consideration in MEMS because of 
their high power consumption relative to many electrostatic and piezoelectric 
actuators, but they do have certain advantages. They are useful in some MEMS 
devices because they can simultaneously provide large forces and large 
displacements. They also require relatively low voltage inputs (often less than 10 V 
[41-46]), especially when compared to electrostatic actuators. The focus of most 
research efforts on thermal actuators has primarily been on two types of actuators, 
namely, bent-beam or V-beam actuators (so called because of their shape) and hot-
arm/cold-arm or u-beam actuators. Bent-beam actuators supply very large forces 
(typically hundreds of micronewtons to a few millinewtons) with translational 
deflections extending to 30 ?m [41, 42]. Hot-arm/cold-arm actuators are generally 
limited to small forces (less than 10 ?N) but can supply relatively large free 
displacement along an arc (up to 50 ?m) [44]. 
The author therefore designed an offset-beam rotational thermal actuator, 
shown in Figure 2.3, to overcome the low force limitation of u-beam actuators while 
reducing the required power compared to V-beam actuators [39]. The offset beam 
actuator therefore provides the best of both worlds: free displacements approaching 
those of similarly sized U-beam actuators, and maximum output forces approaching 
those of similarly sized bent-beam actuators, with power consumption on the order of 
u-beam actuators. Output forces of up to 1.44 mN along with displacements greater 
 
 21 
 
than 20 ?m were achieved, indicating that these actuators provide better force-
displacement performance than hot arm/cold arm style actuators. Although a direct 
comparison to bent beam actuators is difficult because the displacement profiles are 
different (rotational vs. translational), similar free deflections can be obtained from 
the offset-beam rotational actuators with about 40% less current and 40% less voltage 
(i.e., 64% less power). 
The actuator under consideration (Figure 2.3) is fundamentally a rotational 
actuator, although it is not difficult to transform the output to near-linear translation 
over small angular stroke lengths. The actuator consists of two flexible beams 
connected to a central displacement amplification beam with an offset between their 
respective axes, as shown in Figure 2.4. As in other MEMS thermal actuators, current 
passing through the flexible beams generates joule heating and this causes each beam 
to expand. Due to the offset between the two actuator beams, the axial expansion is 
converted into a torque on the amplification beam about the point P, shown in Figure 
2.4. The central beam then rotates according to the amount of torque applied and the 
 
Figure 2.3. Rotational actuator test structure. 
 
 22 
 
bending stiffness of each of the actuator beams. The length of the yoke or moment 
arm r
2
 defines the amount of translation that can be derived from the resulting 
rotation. The deformed shape of the actuator is similar to the second bending 
modeshape of the complete system, as shown in Figure 2.5. 
 
2.55
mm
P
r
2
r
1
x
y
L
 
Figure 2.4. Rotational actuator conceptual drawing and parameter definition. 
 
 
2.55
mm
vernier scale
resisting 
spring
 
Figure 2.5. Deformed shape of actuator. 
 
 23 
 
 The intent in designing this actuator was to replace the V-beam reset 
actuators with these rotational actuators to achieve reset with lower power levels. In 
practice, the reset operation required stabilization of the actuator to prevent buckling 
of the slender beams, and this negated the expected performance improvements. 
However, the rotational actuator design, modeling, and testing is included here 
because it is likely to be a useful contribution for other applications. Also, 
independent from this work, Heo and Kim have also developed MEMS thermal 
actuators with the same operation principle around the same time [47]. 
2.3 Device Fabrication 
The mass, spring, latches, and reset actuator of the sensor are made of low-
resistivity silicon (1 to 3 m?-cm) to lower the voltage required by the actuators and 
the resistance of the sensor in the latched-state. The sensor is fabricated on a silicon-
on-insulator (SOI) wafer with a 20 ?m thick device layer, a 2?m buried oxide layer, 
and a 500?m thick, 1-10?-cm handle wafer. The fabrication process flow is shown in 
Figure 2.6, with further details on specific recipes and device settings given in 
Appendix A. 200/2000 ? of chrome gold (Cr/Au) is deposited first on the anchors of 
each of the springs, latches, and actuators via ebeam evaporation and patterned by 
liftoff (step a). These pads serve as wirebonding sites for connection to an electronic 
package. Gold-tin (AuSn 80/20 weight %) rings are then deposited via ebeam 
evaporation and patterned by liftoff around each of the anchors to provide for wafer-
to-wafer bonding with a cap wafer later in the process (step b). Next, the complete 
device profile, including springs, mass, latches, and actuators, is patterned with 
contact photolithography and deep reactive ion etching (DRIE) is used to transfer the 
 
 24 
 
profile to the silicon device layer in a single step (step c). The wafer is re-patterned 
with a thick negative photoresist that allows patterning over features up to about 
30?m and sputtered Cr/Au or aluminum is deposited on the mating sidewalls of the 
latches to lower the contact resistance (step d). The last patterning step on the device 
wafer is a backside DRIE of trenches about 375 ?m deep and 100 ?m wide in 
between the individual die to provide cleaving lines for die separation (step e). 
 
Figure 2.6. Shock sensor fabrication process flow. Individual steps are a) 
deposition of pads for wirebonding on SOI wafer, b) deposition of AuSn solder 
rings for wafer bonding, c) DRIE of device structure, d) deposition of patterned 
metal contacts on latch sidewall, e) DRIE of cleave assist lines, and f) final device 
release by oxide undercut.   
 
 25 
 
Attempts to release the device in a standard liquid HF/supercritical drying 
process were found to be unreliable. Many of the devices adhered to the wafer during 
release due to the large mass and relatively compliant springs. Therefore, a vapor-
phase HF release process was used (step f), performed in a Primaxx MEMS-CET 
etcher that provided good results with no stiction or residue; this etcher is also less 
aggressive in attacking the Cr adhesion layer under the bond pads. 
Coating sidewalls with a lithographically-defined pattern (step d) is a 
challenging process step that merits additional detailed information. Spin-coating 
over deep topography causes significant thickness variations,  with thicker photoresist 
layers forming inside wells and thinner layers forming on the mesas, and especially 
thin layers at the corner of the mesas (see Figure 2.7). The author developed this 
process by using a negative photoresist because the various thicknesses would cause 
features on the top of mesas to be over-exposed and the sidewalls to be under-
exposed. With negative photoresist, the exposure and subsequent bake cause the resist 
to cross-link, which prevents exposed features from developing. Even if the resist is 
not fully exposed all the way through the thickness, the top surface will be cross-
linked, impeding development underneath any exposed areas. 
The photoresist used was Futurex NR5-8000, spun at 500rpm for 5 seconds, 
then 1000rpm for 40 seconds to yield a nominally 15 ?m thick layer. The viscosity 
and slow spin speed allow the resist to fully coat the corners of mesas up to 20?m 
high (see Figure 2.7). Thinner resist or faster spin speeds can cause gaps in the resist 
coating at these corners, which will result in metal adhering during the liftoff process 
in undesired locations. 
 
 26 
 
a)
b)
c)
 
Figure 2.7. Spinning photoresist over topography: a) thick photoresist 
completely covers corners of mesas, while b) thin photoresist can leave corners 
uncoated. After development, c) there may be some residue left in the corner, 
but most of the sidewall is cleared for coating with metal. 
 
Once a complete coating is achieved, an exposure dose about 15 percent higher 
than usual worked well with these devices. On a Karl Suss MA6 contact aligner, with 
dose of 1000mJ/cm
2
, the best results were achieved with an exposure time of 7.8 
seconds (on a silicon wafer with no topography, the exposure time is 6.8 seconds). 
The usual postbake procedure at 100 ?C for 120 seconds was used, and the resist was 
developed in Futurex RD6. Since it is difficult with standard microscopy equipment 
to view and monitor development on the sidewalls themselves,  development was 
stopped when inspection revealed full development at both the top and bottom of the 
mesa edge. Some resist remains undeveloped where the resist pools at the bottom 
corner of the mesa (Figure 2.7 c), but the important part for this process is that part of 
 
 27 
 
the sidewall is now free of resist, so an electrical connection can be made from the 
top of the mesa to the sidewall. The end result of the metal deposition and liftoff is 
shown in Figure 2.8. 
 
2.4 Wafer Level Packaging Process 
Vacuum packaging reduces the power required for the reset actuators 
dramatically, since the dominant loss mechanism is heat transfer through the air into 
the underlying substrate. This has been observed before for thermal actuators [6]. 
Vacuum probe station tests on the fabricated devices have shown that a sensor that 
resets at 15V/108mA in air at atmospheric pressure will reset at 7V/50mA at an air 
pressure of 140mT. Further characterization is needed to determine the pressure 
levels up to which the benefit of the low surrounding pressure can be utilized.  
contact metal
sidewall
curved latch 
surface
10?m
 
Figure 2.8. Sidewall contact metallization. 
 
 28 
 
The author developed a AuSn wafer-level vacuum packaging process to lower 
the reset actuator current and voltage (see Figure 2.9). The procedure for this process 
is as follows. First, a silicon dioxide layer 0.5?m thick is  deposited on a second 
double-side-polished silicon wafer (standard, not SOI). Subsequently, AuSn rings 
matching those on the device wafer are deposited on top of the oxide layer via e-beam 
evaporation. The oxide layer is removed in the area inside the ring with an RIE etch. 
sensor
2-step via
wirebond
sealed cavity
a)
b)
c)
d)
e)
cleave lines
Si cap wafer
SiO
2
AuSn
Au
SOI device 
layer
SOI handle 
layer
 
Figure 2.9. Wafer-level packaging process: a) deposition and patterning of oxide 
insulting ring and AuSn bond rings on  cap wafer, b) 375 ?m backside etch in 
cap wafer for cleaving lines and wide part of through hole, c) frontside etch of  
cap wafer to finish narrow part of through hole, d) bond cap wafer to device 
wafer using AuSn eutectic bond, and e) wirebond to electronic package. 
 
 29 
 
On the opposite side of the cap wafer, cleaving lines and through holes are etched 
about 375?m deep by using DRIE. The cleaving lines match those in the device 
wafer. This first part of the through holes is substantially wider than the bond pads to 
accommodate a wire bonding capillary. These holes are then finished from the front 
side of the cap wafer with another DRIE; this part of the hole is smaller than the 
diameter of the AuSn ring. The cap wafer is then aligned and bonded to the device 
wafer with an AuSn eutectic bond, at a temperature of 300?C in a 50T H
2
N
2
 
atmosphere by using 1 psi bond pressure. The result of this packaging process is that 
the device bond pads are exposed to the environment so that electrical contact can be 
made, while the sensor is contained in a sealed chamber (see Figure 2.9). Electrical 
connection between the sensor and the bond pads is accomplished through the low-
resistivity silicon device layer of the SOI wafer. 
The wafer is either cleaved apart or diced, and the individual sensors are 
placed in an electronic package and wire bonded (Figure 2.10). It is worth noting that 
 
Figure 2.10. Electronic package: a) unpackaged die, b) wafer-bonded die, and c) 
wafer-bonded and wire-bonded shock sensor die. Each die has two sensors 
designed to latch at different acceleration threshold levels. 
 
 30 
 
each of the springs is wire bonded to a pin-out on the package for diagnostic use. The 
resistance between each of the springs can be checked to determine if the spring is 
intact or broken. The typical spring-spring resistance in the fabricated devices is 
about 5 k?. 
One issue encountered with the bonding process was that the Cr/Au bond pads 
sometimes showed discoloration and bubbles after bonding (see Figure 2.11). The 
author believes this to be due to pinholes in the bond pads, which allow the 
underlying chromium to be attacked by the vapor HF. The bond pads appeared 
undamaged after the release process, but when the wafer is heated above 275?C, a 
reaction apparently takes place between the gold and the silicon substrate causing the 
bubbles and discoloration. Attempts to wire bond to the damaged bond pads failed, 
since the wire bond would not stick to the pad. By increasing the thickness of the gold 
 
 
Figure 2.11. Damaged bond pad after wafer bonding, showing bubbles due to 
HF penetration through micropores in the bondpad and subsequent undercut of 
the chromium adhesion layer. 
 
 31 
 
layer from 2000 ? to 4000 ?, it is found that the pinholes could be eliminated and the 
bond pads remained intact after bonding.   
2.5 Summary and Author?s Contributions 
An overview of the sensor and reset actuator design and fabrication process 
has been presented in this chapter. While the details of the sensor design are unique, 
the device is functionally very similar to those presented in [19] and [16, 20]. The 
primary novel aspect of the sensor design is patterned coating of only the flat contact 
surfaces with metal, which eliminates false positive readings when the sensor is in 
contact with the latch but not yet fully latched. The rotational thermal actuator design 
presented in Section 2.2 is a novel actuator concept, although as mentioned above, 
Heo and Kim independently developed a very similar actuator around the same time 
[47]. In the area of fabrication, the author?s main contributions are the process for 
patterning and liftoff of metal on the sidewalls, and the wafer-level packaging 
scheme, which eliminates the need for filled vias. 
 
 32 
 
3. Experimental Studies and Results 
This chapter contains a description of the experimental portion of this work. 
This includes shock testing of the sensor to determine the minimum acceleration 
required to latch, shock testing of already-latched sensors to investigate possible loss 
of contact, and high-speed imaging of the latching process to examine the physics and 
to compare with models developed in Chapter 3.4. Harmonic excitation of the shock 
sensors is carried out to obtain an indirect measurement of the stiffness of the springs 
and latches. Testing of both the V-beam style and rotational offset-beam thermal 
actuators is also presented, as is friction testing conducted by using the V-beam 
thermal actuators. Portions of this chapter have been adapted from the author?s two 
journal papers [38, 48] and two conference papers on the shock sensor [49, 50], as 
well as his journal paper on the rotational thermal actuator [39] and conference paper 
on the friction test device [51]. 
3.1 Sensor Testing 
Two different versions of the shock sensor have been fabricated and tested. 
They are denoted throughout the text as Design 1 and Design 2, and correspond to 
nominal acceleration thresholds of 50 g?s and 100 g?s. The dimensions for each 
design are summarized in Table 3.1. This work is limited to the study of two different 
designs because the detailed experimental characterization undertaken here takes 
considerable time, and an in-depth study of more designs would not be practical. Two 
devices is also a minimum number needed to begin a validation of the model 
prediction of the device response. Once a basic understanding of the fundamental 
physics of the device performance has been established through experiments and 
 
 33 
 
simulations of these two designs has been carried out, the reduced-order models can 
be used to explore the parametric design space and identify new designs to provide a 
more targeted study of trends and identified phenomena.  
3.1.1 Shock Characterization 
As the sensor is intended to be a shock sensor, the fundamental measure of 
performance is how the sensor responds to a sudden acceleration event. To 
characterize the magnitude of the acceleration required to latch the sensor, shock 
testing was performed. In this section, the author describes the shock testing 
methodology and the acceleration threshold results obtained for each of the two 
designs. 
The shock table testing was performed on a GHI Systems Linear Shock 
Machine (LSM-100). This machine is a horizontal shock machine, so the initial 
velocity is produced purely through a compression of two springs. The shock itself 
occurs when the table hits a programmer and comes to rest. The LSM-100 produces 
pulse durations between 0.1 and 30 ms, with velocity change limits of 63.5cm/sec to 
635cm/sec depending on the amount of spring compression (setback) and the 
hardness of the programmer stop. The programmer that produces the longest duration 
acceleration pulses was used for all testing except the high-speed video tests. The 
acceleration pulse was measured by using a single-axis accelerometer screwed into a 
threaded mount in the back of the table and captured by using computerized data 
acquisition software. The accelerometer data were filtered at 400Hz to eliminate high-
frequency noise in the data.  
 
 34 
 
The author?s test setup is shown in Figure 3.1, with a typical acceleration 
profile captured during the testing shown in Figure 3.2. The shock sensor wafer was 
attached to the shock table with thin, low-profile, double-sided tape. The setback was 
set to a level known not to cause latching, and the device was shocked. The wafer was 
inspected under a microscope to determine if any devices latched. If only one of the 
two latches engaged on any device, it was considered to be a non-latching event and 
the device was manually reset by pushing the latch out of the way by using a fine 
wire held in a pair of tweezers. If both latches engaged on any device, the device was 
considered to have latched. The setback level was increased slightly (in increments of 
0.1-0.2cm) and the test was repeated until all functioning devices on the wafer had 
latched. This set of events constitutes a single cycle. After each cycle, the sensors 
were all reset by applying an out-of-plane shock sufficient to pull the masses up and 
over the latches to reset. The peak acceleration of the latching event as measured 
from the filtered accelerometer data were recorded for each device in each cycle.  
 
Table 3.1. Design parameters for the two fabricated versions of the shock sensor.  
Parameter Description Design 1 Design 2 
H
m
 (mm) height of mass 2.025 2.025
W
m
 (mm) width of mass 4.0 3.1
t (?m) sensor thickness 3.42e-7 2.65e-7
L
f
  (mm) length of suspension spring 1.945 1495
W
f
  (mm) width of suspension spring 12 12
w
L
 (?m) width of latch cantilever 8 8
l
L
  (?m) length of latch cantilever 460 460
r (?m) latch radius 40 40
d
i
 (?m) horizontal offset 15 15
y
0
 (?m) initial sensor travel to latch 150 150
crit
y&&  (g) nominal threshold acceleration for latching  50 100
 
 35 
 
Twenty-nine devices on the same wafer were tested repeatedly on the shock 
table by using above described the full-wafer test procedure. Of these, latching 
thresholds were obtained in every cycle for 13 devices. Nine of the remaining devices 
developed stiction after several cycles, evident from manually pushing the mass with 
a small wire under the microscope and observing that it did not return to the designed 
 
 
Figure 3.1. Shock table test setup for basic shock experiments. 
 
-10
0
10
20
30
40
50
1020304050
Time (ms)
Accel
e
r
a
tio
n
 (g
)
 
Figure 3.2. Typical acceleration time history produced by shock table, low-pass 
filtered at 400 Hz (acceleration units are g, or multiples of gravitational 
acceleration). 
 
 36 
 
equilibrium position. This indicates some force countering the restoring force of the 
springs, which is likely due to electrostatic charge buildup from the mass rubbing 
against the wafer during the shock events. The mass is low resistivity silicon, but 
silicon is known to develop a so-called ?native oxide? layer when exposed to air for 
more than a few hours that is insulating, and the author believes that this native oxide 
is developing an electrostatic charge that causes the mass to stick lightly to the 
substrate. The mass can be moved, but there is a discernible stiction force, and hence,  
data from devices that exhibited this behavior were not included in the analysis. 
Seven other devices would not latch in the first (or first few) cycles, then started to 
latch thereafter. The data from these devices were also not included in the analysis. 
The average acceleration required to latch the thirteen ?good? devices are 
shown over the first ten cycles in Figure 3.3. From the minimum and maximum 
accelerations to latch shown for a device in each cycle (plotted as error bars), it is 
30
35
40
45
50
55
60
024681012
Cycle Number
A
c
cel
e
r
a
t
i
on t
o
 Lat
ch 
(
g
)
 
Figure 3.3. Acceleration to latch over first ten cycles for thirteen devices (Design 
1). Error bars cover minimum to maximum acceleration levels. 
 
 37 
 
readily apparent that there is a spread in the data. Nevertheless, it appears that the 
average latching acceleration level decreases somewhat over the first several cycles. 
If the latching threshold over the same number of cycles is plotted for each individual 
device (Figure 3.4), there is a clear downward trend visible for almost every device 
studied. In fact, for every individual device the acceleration to latch is lower in Cycle 
10 than in Cycle 1. 
The author attributes this effect to wear in the contact surfaces, smoothing 
asperities, and a decrease in the value of the apparent coefficient of friction. This is 
consistent with the findings of sidewall friction measurements on SOI devices 
presented in Section 3.3, and the SEMs of these cycled devices show evidence of 
25
30
35
40
45
50
55
60
024681012
Cycle Number
A
cc
e
lerat
i
on t
o
 Lat
ch (g
)
 
Figure 3.4. Acceleration to latch plotted for thirteen individual devices over first 
10 cycles (Design 1). 
 
 38 
 
wear at the top edge of the latch sidewall (Figure 3.5). Contact (and therefore wear) 
only occurs at the top edge due to a slight taper in the sidewall from top to bottom. 
To confirm that the downward trend in the latching acceleration level is due to 
contact wear and not softening of the springs or changes in the mass itself, the wafer 
was turned around and tested in the negative direction for 5 cycles. If any changes in 
the mass or springs were the cause, the first cycle would be expected to exhibit the 
lower threshold from Cycle 10 in the positive direction since the same mass and 
springs are used for both directions. However, the latches for the negative direction 
are completely separate and not affected by latching events in the positive direction, 
so changes in the latches would be expected to result in a return to the original (high) 
latching threshold in the first negative cycle. 
 
Figure 3.5. SEMs showing wear at top edge of latch contact after 10 cycles. 
Region of wear is outlined in red. 
 
 39 
 
The resulting average peak acceleration to latch is shown in Figure 3.6, with 
the average values for the positive cycling shown for clarity. The same decreasing 
trend was observed in the negative direction cycling, with the average acceleration to 
latch jumping back up from the 40.2g level measured in the last cycle of the positive 
direction testing to 46.4g. This is not quite as high as the 49.1g average peak 
acceleration seen in the very first positive direction test. This discrepancy can be 
attributed to the fact that during the positive direction testing a few devices were 
inadvertently latched in the negative direction while attempting to free half-engaged 
devices. This would tend to lower the friction coefficient for a few of the devices, 
bringing the overall average down. 
The longer term trend is shown in Figure 3.7 for a different wafer over fifty 
cycles of repeated shock testing. For the cycles in between the plotted data points, the 
acceleration was set to a sufficient level to latch all of the devices. Sixty-eight devices 
Number of Cycles
Av
er
ag
e  Acc
e
l
e
r
a
ti
o
n
 
to 
L
a
tc
h (
g
)
38
40
42
44
46
48
50
024681012
positive direction
negative direction
 
Figure 3.6.  Repeated cycle testing performed first in positive direction, then, in 
negative direction. Data reported is averaged over 13 devices that successfully 
latched every time in the positive direction. 
 
 40 
 
were tested on this wafer, of which 36 devices were latched at all of the cycles for 
which data was recorded (1, 2, 5, 10, 20 and 50 cycles). The figure shows the average 
acceleration to latch for these 36 devices. As seen in the previous case, a decrease in 
the threshold acceleration level is seen over the first ten cycles. After twenty and fifty 
cycles, the threshold acceleration required to latch increases again. The author 
attributes this phenomenon to a new resisting force arising from electrostatic stiction, 
which arises over time as the mass collect opposite electrostatic charges due to 
scraping together. Over many cycles, this begins to pull the mass down until it 
eventually scrapes against the substrate. 
Design 2 devices from the same wafer used in Figure 3.3 were also tested over 
seven cycles. Out of twenty-five functional devices tested, 16 successfully latched for 
every cycle. The results for these 16 devices are shown in Figure 3.8. There is no 
40.00
45.00
50.00
55.00
60.00
65.00
70.00
0 102030405060
Number of Cycles
A
c
c
e
l
e
ra
ti
o
n
 t
o
 L
a
tc
h
 (g
)
 
Figure 3.7. Long term repetitive shock cycling results for 36 Design 1 sensors on 
one wafer. 
 
 41 
 
meaningful decrease in the acceleration to latch observed over these first seven 
cycles. The author hypothesizes that this is because the Design 2 springs are stiffer 
compared to those of Design 1 (1.32 N/m as compared to 0.60 N/m for Design 1). 
The friction force is therefore expected to constitute a smaller percentage of the 
overall force resisting the motion of the mass, so slight changes in the friction 
coefficient would not be expected to have as large an effect on the acceleration 
required to latch the device. 
 
3.1.2 High-Speed Video of Sensor Latching 
To provide insight into the latch dynamics and experimental verification for 
the models, the latching of the shock sensor was also observed on high-speed video to 
follow the progression of the mass from resting, to contact, to the latched state. The 
60
70
80
90
100
110
120
130
140
012345678
Test Number
A
v
e
r
age
 A
c
c
e
l to 
Latc
h
 
(g)
 
Figure 3.8. Acceleration threshold to latch for sixteen Design 2 devices over 7 
cycles. The error bars indicate the maximum and minimum acceleration to latch 
for any one individual device. 
 
 42 
 
experimental setup for high-speed video is shown in Figure 3.9. The acceleration 
applied to the shock sensor was monitored with the Dytran 3200B6T accelerometer 
threaded directly into the shock table. For these tests, the shock sensor was 
wirebonded to a dual inline package for electrical monitoring of the sensor and 
electrical reset in between experiments. For electrical monitoring, a voltage divider 
circuit was constructed. One latch of the sensor was connected to a DC power supply 
set to output 5 V, the other latch was connected to a 1 M? resistor. The opposite end 
of the resistor was connected to ground. When the sensor latches, the circuit is closed 
and a voltage is detected across the 1 M? resistor. A computer with a data acquisition 
system was set up to power the accelerometer, capture the acceleration data and the 
voltage sensed through the sensor. 
HS C
a
m
e
r
a
Microscope
Shock table
Table motion
Oscilloscope
for triggering/
frame sync
Trigger
Frame 
sync
Accelerometer
PC for data Acquisition
Laptop for 
HS Video
Device under 
test
DC Power supply 
for device reset
(a) (b)
Microscope
Device 
under test
Shock table
 
Figure 3.9. Shock table test setup for high-speed video capture. (a) Schematic of 
the entire test setup, and (b) photograph of the shock table with a device 
mounted under the microscope for testing. 
 
 43 
 
The microscope setup used a 5x objective and 10x multiplier (for a total 
magnification of 50x) were used to image the sensors. The accelerometer output was 
used to trigger an oscilloscope which in turn triggered the high-speed camera. The 
oscilloscope captured the accelerometer output, the trigger signal sent to the camera, 
and the frame sync output from the high-speed camera. Because the oscilloscope used 
only has three channels, the voltage divider signal could not be captured on the 
oscilloscope. However, by matching the accelerometer trace from the data acquisition 
system and the oscilloscope, the trigger and frame sync can be synchronized with the 
voltage divider signal. 
The individual frames from the high-speed video were analyzed using a 
simple MATLAB routine (Figure 3.10). The routine reads the image file into an 
array, resizes the image to exclude irrelevant parts of the frame based on user input, 
and sums the image brightness values (ranging from 0-255) over each column. The 
gold traces on the mass are significantly brighter than the background and are easy to 
pick out when the column sum is plotted. Because the background brightness varies 
with position (it is brightest in the center of the frame) and the sensor moves through 
the frame during the latch event, a global maximum does not necessarily suffice. 
Instead the frame is searched for a pair of local maximum representing the two 
parallel gold traces across the mass. 
 
 44 
 
Similarly, the position of the chip can be tracked by assuming there is no 
significant deflection of the second (inactive) latch in the direction parallel to the 
sensor axis. By inactive, the author refers to the latch for negative deflection when the 
direction of the applied acceleration is such that the mass travels in the positive 
direction. The mass never contacts this latch and the ratio of stiffness to mass is very 
high, so the assumption of no deflection should be valid. Another local minima can be 
Parallel traces 
on mass
Active Latch
Inactive Latch
(a)
(b)
Intens
it
y
 (
A
rb. Units)
1
1.5
2
2.5
Horizontal Position (pixels)
0 200 400 800 1000
3
1200
0.5
Local Maxima 
corresponding to gold 
traces on mass
600
 
Figure 3.10. Analysis of high-speed video frame. (a) Frame from high-speed 
video showing parallel traces. (b) Analysis of the image - two adjacent local 
maxima indicate position of mass. 
 
 45 
 
found, and this minima corresponds to the gold coating on top of the latch. By 
subtracting the motion of this local minima (and therefore the motion of the shock 
table under the camera) from the motion of the mass, the relative motion of the mass 
with respect to the translating reference frame of the chip can be found. This relative 
motion is the parameter tracked by the model and also indicates whether the sensor is 
latched. If the relative motion exceeds 150 ?m, the mass has moved completely past 
the latch and will remain latched until the sensor is reset. 
The full interaction of the shock sensor and latch is shown in Figure 3.11 for 
Design 1 and in Figure 3.12 for Design 2. The frame rate for both videos was 4261 
frames per second, the highest frame rate available with a reasonable resolution on 
the high-speed camera used. Fewer frames are captured for Design 2 because the 
overall time to latch is lower. The position data extracted from each video are shown 
in Figure 3.13 for Design 1 and Figure 3.14 for Design 2. For the Design 1 video, 
time zero was estimated by matching the model presented in 4.2.2 to the position data 
from the video. To eliminate this uncertainty for the Design 2 video, a frame sync 
was set up from the high-speed video camera to pinpoint the exact time for each 
frame as well as a voltage divider to electrically monitor the contact. These are all 
plotted together for Design 2 in Figure 3.14. 
 
 46 
 
 
Figure 3.11. High-speed video of Design 1 sensor showing latching progression. 
The time between frames is 0.235 ms, and the first frame corresponds to the 
onset of the acceleration pulse. 
 
 47 
 
 
 
 
Figure 3.12. High-speed video of Design 2 sensor showing latching progression. 
The time between frames is 0.235 ms, and the first frame corresponds to the 
onset of the acceleration pulse. 
 
 48 
 
Examining the position versus time data, the timing of the latch event can be 
estimated. For both sensors, the mass must travel 92 ?m before contact is made with 
the latch, and 150 ?m before the device is fully latched. For Design 1 (Figure 3.13), 
contact is made after about 1.8 ms and the device is latched after about 2.6 ms. The 
acceleration pulse lasted approximately 8 ms in this test, so the device is latched well 
before the peak acceleration is even achieved. There was no electrical monitoring 
 
-25
0
25
50
75
100
125
150
175
200
Mass Position
Disp
l
a
c
e
me
n
t
 (
?
m)
A
c
c
e
le
ra
t
i
on
 (g
)
-30
-25
-20
-15
-10
-5
0
012345678
Time (ms)
Shock Table
Acceleration (meas)
 
 
Figure 3.13. Position data extracted from Design 1 high-speed video. The error 
bars indicate uncertainty in the position measurements taken from the video 
frames due to blurring. 
 
 49 
 
during this test, so the exact time when the electrical circuit closes is unknown. For 
the Design 2 test (Figure 3.14), a harder programmer was used to reduce blurring, so 
the acceleration pulse duration was approximately 2.2 ms. First contact and latching 
both occur sometime between 0.75 and 1 ms, and electrical closure of the switch 
occurs around 1.3 ms. It is also evident from the contact voltage output trace that the 
switch bounces several times after latching. The electrical circuit finally closes for 
good at 2.9 ms (outside the scale of the graph). 
D
i
s
pl
a
c
em
e
n
t (
?
m)
Accel
e
r
a
tio
n
 (g)
-120
-100
-80
-60
-40
-20
0
20
00.511.522.5
Table Acceleration
(meas)
Time (ms)
Vol
t
ag
e (
V
)
-50
0
50
100
150
200
-6
-5
-4
-3
-2
-1
0
1
2
3
4
Mass Displacement
Contact Voltage
Frame Sync
 
Figure 3.14. Position data extracted from Design 2 high-speed video, along with 
the frame synchronization signal from the camera and electrical monitoring of 
the latch closure. 
 
 50 
 
 
3.1.3 Characterization of Electrical Performance of Latch 
Because the sensor is intended to be electrically sensed, the author 
characterized the electrical performance of the sensor. The sensor was connected in 
series to a 1M? resistor during shock testing to create a switched voltage divider. 
One latch was connected to the positive terminal of a DC power supply set for 5V 
output, and the other latch was connected to one terminal of the resistor. The second 
terminal of the resistor was connected to the power supply ground. The voltage across 
the resistor was monitored during the test on one channel of the data acquisition 
system. The output from an accelerometer mounted on the shock table was monitored 
on a separate channel. The result of one such test is shown in Figure 3.15. The 
voltage across the resistor jumps to 4.97V after the shock sensor comes to rest against 
the latches and closes the circuit. Since the latch contact metallization is 
-20
0
20
40
60
80
100
120
0 1020304050
Time (ms)
A
c
c
e
l
e
ra
ti
o
n
 (g
)
-1
0
1
2
3
4
5
6
O
ut
pu
t
 V
o
l
t
ag
e 
(
V
)
Acceleration Profile
Output Voltage
 
Figure 3.15. Electrical sensing of latching event. Positive voltage indicates the 
device is latched. 
 
 51 
 
lithographically defined and only present on the flat back surface of the latches, the 
resistance during the sliding phase of contact is orders of magnitude higher (typically 
> 20M?), and hence, there is no false trigger before the mass pushes all the way past 
the latches. 
The time delay between the acceleration event and the voltage output seen in 
Figure 3.15 is a result of the mass continuing to travel past the latch before coming 
back and resting against the metalized contact surface. The length of the time delay 
primarily depends on the time taken by the mass to move past the latch and the 
magnitude and duration of the applied acceleration. 
In order to investigate potential degradation of the latch metallization due to 
repeated latch/reset events, the authors cycled the shock sensor over 100 times on the 
shock table, resetting the device with the integrated electro-thermal reset actuators 
after each cycle. During reset, as the contact metallization on the latches and mass 
scrape together, there is some concern that the subsequent wear might cause the 
contact resistance to increase dramatically after a small number of cycles. The results 
of the repeated cycling testing are shown in Figure 3.16. In creating this figure, data 
for a 6 unsuccessful latch attempts were omitted along with data for cycles one 
through six, which were used for the electrical confirmation of latching experiments 
reported above, because the contact resistance was not measured directly in these 
tests. The cycling was completed over the course of about an hour, eliminating the 
possibility of environmental degradation. It is evident from the data that the contact 
resistance does not change even over 100 cycles. It is also notable that the contact 
resistance is remarkably consistent from event to event. Even the outlier data points 
 
 52 
 
differ from the mean by less than 4%. The standard deviation over all 93 cycles is 
1.5?, less than 0.5% of the mean value. 
The contact metallization on the latch sidewalls was examined in the SEM 
before and after the 100 latch-reset cycles (Figure 3.17). Despite the very good 
contact resistance over all 100 cycles, significant wear and flaking off of the gold 
contact is observed. This is most likely due to the scraping of the contact surfaces 
against each other during the reset actuator operation. During reset, the actuators push 
the latches out of the path of the mass to allow the device to reset. The excellent 
repeatability of the contact resistance over many cycles is even more remarkable in 
light of the gross degradation of the contact metallization evident in the SEM images 
(Figure 3.17). It is expected that this degradation will eventually lead to poor 
electrical contact and a higher contact resistance. However, this type of device is 
intended for monitoring potential damage-inducing shocks, and 100 cycles is 
probably much more than what would be tolerated by the system the sensor is being 
300
302
304
306
308
310
312
0 2040608010
Cycle
C
o
n
t
act
 R
e
si
st
a
n
ce
 (
o
h
m
)
 
Figure 3.16. Contact resistance of latched sensor over 100 cycles. 
 
 53 
 
used to monitor. The contact wear is not expected to affect the useful life of the 
sensor in most applications. 
3.1.4 Post-Latch Shock Testing 
After the sensor has latched, it will remain latched until it is reset. However, 
the electrical contact between the mass and latch can be momentarily lost due to a 
subsequent acceleration in the latching direction, if the acceleration is large enough. 
To demonstrate this effect and understand the limitations of the sensor, an electrical 
monitoring signal was applied to a latched Design 1 shock sensor, and the sensor was 
shocked with progressively higher pulses until contact was lost. A constant current of 
50 ?A was applied across the latches, and the voltage across the latches was 
monitored. When the latches lose contact, a sudden increase in voltage is detected.  
For the device tested, the lowest shock level observed to result in loss of 
contact was 53.7 g (Figure 3.18). With progressively higher acceleration pulses, the 
contact is lost at a similar acceleration level and remains open until the acceleration 
decrease below this level, with a delay of up to 300 ?s at the onset and 800 ?s as the 
contact metal
sidewall
curved latch 
surface
10?ma) 12?mb)
 
Figure 3.17. Latch contact metallization: a) before use and b) after 100 cycles. 
Metal shows significant degradation after cycling, but retains good contact 
resistance. 
 
 54 
 
acceleration dissipates. The highest acceleration pulse applied was 168.7g, shown in 
Figure 3.19. 
The implications of this loss of contact depend on how the shock sensor is 
integrated into a larger electronic system. The shock sensor can be used in one of 
three ways: the system can query the sensor at set intervals and record whether the 
sensor has latched in the time between intervals; it can be used as a wakeup/trigger 
sensor for a standard accelerometer; or it can be constantly monitored for latch 
-10
0
10
20
30
40
50
60
0 5 10 15 20
Time (ms)
A
ccel
e
r
a
t
i
on
 (
g
)
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
V
ol
t
age
 (
V
)Acceleration Profile
Output Voltage
 
Figure 3.18. Electrical monitoring of latched shock sensor subjected to 
secondary shock of 53.7 g. 
 
-20
0
20
40
60
80
100
120
140
160
180
0 5 10 15 20
Time (ms)
A
c
c
e
l
e
ra
t
i
o
n
 (g
)
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
V
o
l
t
ag
e (
V
)
Acceleration Profile
Output Voltage
 
Figure 3.19. Electrical monitoring of latched shock sensor subjected to 
secondary shock of 168.7 g. 
 
 55 
 
contact. In the first two scenarios, the event will generally be interpreted by the 
system as a single event. In the last scenario, the system interpretation depends on 
software. In the simplest case, the software is written to log every new contact as a 
shock event, and both the original latching event and post-latch secondary shock will 
be recorded as separate events. If the software is written such that after latching, the 
system does not look for events until the sensor has been reset, only the original event 
will be recorded. The acceleration required for a latched device to lose contact will 
generally be smaller than the acceleration required to latch, because there is no 
contact force between the latch and mass to overcome. Therefore it may not be 
desired to record the post-latch secondary shock. This should be considered carefully 
in the implementation of this type of sensor. 
3.1.5 Harmonic Excitation 
Harmonic excitation of the shock sensor was carried out to determine the first 
resonance frequency of the shock sensor to verify the mass and spring constant 
calculated from the device dimensions. The harmonic excitation of the shock sensor 
was realized by using a small vibration table. The frequency response was measured 
using a laser Doppler vibrometer (LDV) mounted on a microscope. The microscope 
was in turn mounted on a boom stand and suspended above the vibration table. A 
laser is sent through the microscope and the LDV system is used to measure the 
Doppler shift in the return signal. The frequency shift is related to the velocity 
component parallel to the direction of the laser. This means that the measurement is 
well-suited for out-of-plane measurements. Since the shock sensor mass moves in-
plane, the chip must be mounted perpendicular to the microscope field of view. 
 
 56 
 
However, in order to provide a clear path for the laser beam to hit the mass and 
return, the chip was mounted perpendicular to the vibration table and the table was 
tilted approximately 10 degrees off of the axis of the microscope. The motion of the 
vibration table is therefore correctly coupled to the sensor mass, and the sensor mass 
moves nearly (but not exactly) parallel to the direction of the laser. The test setup is 
shown in Figure 3.20. 
 
Figure 3.20. Experimental setup for harmonic excitation. 
 
 57 
 
The laser Doppler vibrometer used was a Polytec MSV-300 system, with a 
detachable head that can mount on any microscope. The bandwidth for the 
measurement was set to 1 kHz, with 1600 discrete frequencies scanned. The driving 
signal for the vibration table was a pseudorandom (broadband) signal, with a 2 Volts 
rms magnitude for Design 1 and 3 Volts for Design 2. 
The measurements were carried out at a single point on the device. The laser 
was focused on the sidewall of the beam extending from the side of the mass towards 
the latches because this is the point with the widest uninterrupted view of the sidewall 
of the mass. The test site is shown in Figure 3.21. The latches were manually broken 
off to prevent them from interfering with the motion of the mass. The LDV was set to 
perform a fast fourier transform to output the frequency response. Multiple frequency 
spectra (between 8 and 21) were averaged by the LDV system for each device and the 
resulting magnitude and phase components of the velocity, displacement, and 
acceleration were saved. Since the velocity measurement is a direct one, the 
excitation
direction
laser spot
 
Figure 3.21. Laser query site for harmonic excitation of sensor. 
 
 58 
 
acceleration and displacement are calculated from the velocity measurements. 
Since the frequency-response of the vibration table itself is not flat, a 
spectrum was also taken off of the surface of the block on which the chip was 
mounted. The resulting frequency-response reference data are shown in Figure 3.22. 
Representative frequency responses for Designs 1 and 2 are shown in Figure 3.23 and 
Figure 3.24, respectively. In each case, the velocity data has been plotted since these 
are the actual measurements (displacement requires a numerical differentiation which 
can introduces an additional source of uncertainty). The rolloff between 100Hz and 
200Hz in each of these device response plots is due to the frequency-dependent 
response of the vibration table itself.  
The first resonance frequency of Design 1 was 174.8 +/- 3 Hz over 5 devices 
tested from one wafer. The first resonance frequency of Design 2 averages 294.8 +/- 
5 Hz over 5 devices tested from one wafer. The devices were chosen in rows from the 
center to the edge of the wafer, since the deep reactive ion etch process which defines 
the features has some variation from center to edge. The resonance frequency was 
nearly identical in all designs, with a variation from the maximum to the minimum of 
less than 6 Hz for Design 1 and less than 9 Hz for Design 2. 
 
 
 59 
 
a)
b)
-200
-150
-100
-50
0
50
100
150
200
0 200 400 600 800 1000
Frequency (Hz)
P
has
e (
degr
ees
)
0
0.0001
0.0002
0.0003
0.0004
0.0005
0.0006
0.0007
0 200 400 600 800 1000
Frequency (Hz)
Ve
l
o
c
i
t
y
 (
m
/s
)
 
Figure 3.22. Frequency response of vibration table with 2V pseudorandom 
driving signal: a) magnitude and b) phase. 
 
 
 60 
 
a)
b)
-0.0005
0
0.0005
0.001
0.0015
0.002
0 200 400 600 800 1000
Frequency (Hz)
V
el
o
c
i
ty
 (
m
/
s
)
-200
-150
-100
-50
0
50
100
150
200
0 100 200 300 400 500 600 700 800 900 1000
Frequency (Hz)
P
h
a
s
e (
d
eg
r
e
e
s
)
 
Figure 3.23. Frequency response for Design 1: a) magnitude and b) phase. 
 
 
 61 
 
a)
b)
0
0.001
0.002
0.003
0.004
0.005
0.006
0.007
0 200 400 600 800 1000
Frequency (Hz)
Ve
l
o
c
i
t
y
 (
m
/
s
-200
-150
-100
-50
0
50
100
150
200
0 200 400 600 800 1000
Frequency (Hz)
P
h
a
s
e
 (d
e
g
re
e
s
)
 
Figure 3.24. Frequency response for Design 2: a) magnitude and b) phase. 
 
 
 
 62 
 
The quality factor and damping ratio can be estimated from the frequency- 
response data as well. The quality factor is computed as the resonance frequency 
divided by the bandwidth between the half-power points, or the points at which the 
magnitude of the response is equal to the maximum response magnitude divided the 
square root of two. 
 
)(
LoHi
R
FF
F
Q
?
=  (3.1) 
For small values of damping, the damping ratio can be estimated as [52]  
 .
2
1
Q
=?  (3.2) 
The parameters extracted from the harmonic measurements along with the 
designed values are shown in Table 3.2. The measured first resonance frequency 
differs significantly from the values calculated by using the designed dimensions of 
the spring and mass, however. For Design 1, the design dimensions indicate a first 
natural frequency of 211 Hz and 356 Hz for Designs 1 and 2, respectively. The mass 
calculation is likely to be very close to accurate assuming the density value of 
2.33 g/cm
3
 is correct, because the dimensions of the mass are very large and small 
Table 3.2. Comparison of sensor design values and those calculated from 
harmonic measurements. 
 Design 1 
(design 
value) 
Design 1 
(from harmonic 
measurements) 
Design 2 
(design 
value) 
Design 2 
(from harmonic 
measurements) 
Resonance Frequency 
(Hz) 
211 174.8 356 294.8 
Mass (?g) 342 -- 265 -- 
Spring Constant (N/m) 0.601 0.412 1.324 0.907 
Spring Width ( ?m) 12 10.6 12 10.6 
Quality Factor Q -- 4.9 -- 11.2 
Damping Ratio ? 
-- 0.102 -- 0.045 
 
 63 
 
variations in the patterning and etching process will have minimal effect on the 
overall mass. The springs, however, are designed to be just 12 ?m wide, and small 
variations in the width can cause large changes in the stiffness and correspondingly 
large changes in the resonance frequencies. With this in mind, the width of the spring 
was varied until the calculated first natural frequency matched the measured first 
resonance frequency for each design. The effective spring width calculated in this 
manner (using the average first resonance frequencies reported above) was 10.6 ?m 
for both cases. 
The frequency-response function (FRF) for each design was also obtained 
from the harmonic test data. The FRF is a way to decouple the uneven response of the 
vibration table from that of the device. The frequency-response function is 
determined using 
 
?
?
?
?
?
?
?
?
=
input
output
log20FRF(dB)
10
 (3.3) 
where, in this case, the output is the measured device response magnitude and the 
input is the measured vibration table response measurement. The resulting FRF for 
both designs is shown in Figure 3.25. There is quite a bit of noise introduced in the 
FRF plots because of the low signal-to-noise ratio in the reference measurement of 
the vibration table. 
 
 64 
 
a)
b)
0
10
20
30
40
50
60
70
80
90
0 200 400 600 800 1000
Frequency (Hz)
Ga
i
n
 (
d
B
)
-20
-15
-10
-5
0
5
10
15
20
25
0 200 400 600 800 1000
Frequency (Hz)
Ga
i
n
 (
d
B
)
 
Figure 3.25. Frequency-response function: a) Design 1 and b) Design 2. 
 
 
 
 
 
 65 
 
excitation 
direction
Laser 
test site
 
Figure 3.26. Laser query site for harmonic excitation of latch. 
 
The harmonic responses of the latches were also characterized to verify the 
stiffness coefficients used in the model. The test site for the latch characterization is 
shown in Figure 3.26. The velocity plot of the latch response is shown in Figure 3.27. 
All of the resonance peaks except the largest peak (just below 19kHz) are due to the 
response of the vibration table, as seen in the vibration table response shown in 
Figure 3.28. The frequency-response function derived by comparing the latch 
response to the table response as described above is shown in Figure 3.29.  The 
design parameters for the latch and the parameters extracted from the harmonic 
measurements are summarized in Table 3.3. Note that the effective width of the latch 
calculated from the harmonic measurements is only 5.5 ?m, as compared to the 
designed latch width of 8 ?m. 
 
 
 
 
 66 
 
 
0.00E+00
1.00E-04
2.00E-04
3.00E-04
4.00E-04
5.00E-04
6.00E-04
0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000
V
e
l
o
c
i
ty
 (
m
/s
)
-180
-120
-60
0
60
120
180
0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000
Frequency (Hz)
P
has
e
 (
degr
ees
)
a)
b)
 
Figure 3.27. Velocity response of the latch as a function of frequency: a) 
magnitude and b) phase. 
 
 67 
 
P
hase (
degr
ees
)
0.00E+00
1.00E-05
2.00E-05
3.00E-05
4.00E-05
5.00E-05
6.00E-05
7.00E-05
0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000
V
e
l
o
c
i
ty
 (m
/s
)
a)
b)
-180
-120
-60
0
60
120
180
0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000
Frequency (Hz)
 
Figure 3.28. Velocity response of vibration table over frequency range used to 
test latch: a) magnitude and b) phase. 
 
 68 
 
 
Frequency (Hz)
-300
-200
-100
0
100
200
300
0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000
P
ha
s
e d
i
f
f
er
en
c
e
 (
d
eg
r
e
es
)
-20
-10
0
10
20
30
40
50
0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000
Ga
i
n
 (
d
B
)
a)
b)
 
Figure 3.29. Frequency-response function of the latch: a) magnitude and b) 
phase. 
 
 69 
 
 
3.2 Thermal Actuator Testing 
3.2.1 V-Beam Actuator 
In the friction studies detailed later in this dissertation, thermal actuators are 
used to apply the normal and transverse forces. In order to gauge the magnitudes of 
these forces, the actuator stiffness is required. If the actuator stiffness is constant over 
the deflection range, it allows extrapolation of the force exerted by the actuator with a 
resisting load from the measured free deflection of the actuator via the actuator 
loadline, or force-deflection response. The actuator stiffness is the slope of the 
loadline. In order to construct the actuator loadline, V-beam test structures based on 
the friction test structures were constructed on the same wafer. The test devices 
(Figure 3.30) consist of an array of V-beams connected to a resisting spring. Devices  
with the same size and number of drive beams with various stiffness resisting springs 
were used to measure multiple points on the actuator loadline. The deflection of the 
spring with a given drive current was measured by using an integrated vernier scale 
on the yoke connecting the actuator to the spring. The inherent assumption is that the 
Table 3.3. Comparison of latch design values and those calculated from 
harmonic measurements. 
 Design value Calculated from harmonic 
measurements 
Resonance Frequency 
(kHz) 
30.97 18.79 
Effective Mass (?g) 0.117 0.104 
Spring Constant (N/m) 4.44 1.44 
Latch Width ( ?m) 8 5.5 
Latch Length ( ?m) 460 460 
Latch thickness ( ?m) 20 20 
Quality Factor Q -- 332 
Damping Ratio ? 
-- 0.0015 
 
 70 
 
compression in the yoke is negligible. The exerted force was calculated by using the 
designed spring constant of the resisting spring. 
V-beam test structures with an initial tilt angle of two to four degrees and one 
to five parallel beams were fabricated and tested. The resisting springs coupled to the 
actuators had designed spring constants of 0 (no spring), 50, 100, 150, 300, 450, and 
600 N/m. A single actuator loadline is constructed by plotting measurements at a 
single applied current with all of the different spring constants on the same graph. 
The response of a single-beam actuator for each of the different angles is shown in 
 
Figure 3.30. V-beam force-deflection test structure. 
 
 71 
 
Figure 3.31. The response of the three degree, single actuator beam with each of the 
different resisting springs is shown in Figure 3.32. 
A representative actuator loadline for a single, 3-degree V-beam device is 
shown in Figure 3.33. A linear least-squares fit has been plotted along with the data, 
0
5
10
15
20
25
30
0 2 4 6 8 10121416182022242628
2 Degree
3 Degree
4 degree
Current (mA)
D
e
fl
ec
t
i
on (
?
m)
 
Figure 3.31. Single V-beam actuator response with no resisting spring, with 
different initial design angles. 
 
Current (mA)
D
e
f
l
ecti
on (
?m)
0
5
10
15
20
25
02468101214161820
k = 0 N/m
k = 50 N/m
k = 100 N/m
k = 150 N/m
k = 300 N/m
 
Figure 3.32. Response of a single 3 degree V-beam actuator beam with different 
resisting springs. 
 
 72 
 
and the equation for the fit is shown on the plot as well. The slope of the fit line is the 
negative of the stiffness of the actuator, and the y-intercept is the predicted blocked 
force (the force corresponding to zero displacement). The actuator stiffness values 
calculated from the loadlines are summarized in Table 3.4. 
Deflection (?m)
Fo
rce
 (?
N)
y = -324.13x + 1107.5
0
100
200
300
400
500
600
01234
 
Figure 3.33. Actuator loadline for single beam, 3-degree V-beam device, using a 
driving current of 1mA. 
 
Table 3.4. V-beam actuator stiffness values calculated from force testing data. 
Angle 
(Degrees) Beams 
Total Actuator 
Stiffness (N/m) 
Stiffness/Beam 
(N/m) 
2 1 138.9 138.9 
2 2 388.0 194.0 
2 3 648.5 216.2 
2 4 1014.9 253.7 
3 1 324.1 324.1 
3 2 736.1 368.0 
3 3 1046 348.7 
3 4 1355.4 338.9 
4 1 211.5 211.5 
4 2 313.6 156.8 
4 3 319.5 106.5 
4 4 790.3 197.6 
 
 
 73 
 
The stiffness per beam is expected to be constant for each designed tilt angle, 
but the calculated values do not reflect this. In addition, the stiffness per beam is also 
expected to increase with increasing tilt angle, and the calculations also do not reflect 
this expectation, as the four degree devices demonstrated a lower stiffness than the 
three degree devices. This is likely due to variations in the fabricated dimensions 
across the wafer. The resisting springs were either 10 ?m (for the nominal 0-300 N/m 
spring constants) or 15 ?m wide (for the 450-600 N/m spring constants). Each 
individual spring constant in these two ranges was designed by modifying the spring 
length. Variations of as little as 2 ?m in the as-fabricated spring widths can cause a 
50% change in the effective stiffness of the resisting spring. This would be enough to 
account for the observed discrepancies in the experimental data. For future iterations 
of this type of device, wider and longer springs would be less sensitive to small 
variations in the fabricated dimensions. 
 Since the trends from the measurements do not make sense, the author elected 
to use stiffness values derived from analytical and finite element models instead when 
calculating the forces exerted by actuators for the purpose of friction measurements. 
3.2.2 Rotational Thermal Actuator Testing 
Since thermal actuators are current-driven rather than voltage-driven devices, 
the rotational actuator free deflection was measured as a function of applied current 
(Figure 3.34 and Figure 3.35). The deflection was measured by using the angular 
vernier scale, which has gradations of 0.15 degrees. The error in the measurement is 
therefore estimated as +/- 0.075 degrees. Most of the tested devices had a small initial 
deflection before the current was applied, in the range of 0 to 0.15 degrees. This bias 
 
 74 
 
deflection was subtracted from the measurements before comparisons with the 
corresponding model predictions. The cause of the initial deflection is not yet known, 
but this could possibly be due to the compressively stressed buried thermal oxide film 
pushing in on the device anchors. This explanation is consistent with the fact that the 
initial deflection is always in the same direction as the deflection induced by thermal 
expansion. 
The moment/angular deflection characteristics of the actuators were also 
examined by using a series of test structures that included resisting cantilever springs 
with various designed spring constants. The angular deflection was measured as a 
function of current for each actuator/spring combination by using an angular vernier 
scale patterned on the wafer adjacent to the device. The actuator moment about point 
P was calculated by using the cantilever spring constant and the measured deflection. 
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
0 3 6 9 12 15 18 21 24
Current (mA)
R
o
t
a
t
i
on
 A
ng
l
e 
(
d
eg
)
5?m width
7?m width
10?m width
 
Figure 3.34. Experimental free deflection data for actuators for L = 400 ?m and 
different actuator beam widths. Wider beams result in less free deflection, but 
should be capable of providing higher forces. 
 
 75 
 
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
0 3 6 9 12 15 18
Current (mA)
R
ot
at
i
on A
n
gle (
deg
)
400?m length
500?m length
600?m length
 
Figure 3.35. Experimental free deflection data for actuators for w = 7 ?m and 
different actuator beam lengths. Longer actuators generally provide higher free 
deflection but generate less force. 
 
For a single actuator design, measurements at the same applied current can be 
combined to get a moment-rotation angle relationship. Figure 3.36 shows three such 
relationships, for actuators with L = 400 ?m and various widths. The applied current 
levels were chosen such that the free deflection of each of the three actuators was the 
same. Linear trendlines for each actuator design are plotted along with the data, and 
the equations describing them are also displayed on the graph. The slope of the linear 
trendline for each actuator beam width w is the torsion spring constant of the actuator, 
and the y-intercept corresponds to the actuator blocked moment. It can be seen that 
the actuator stiffness and blocked moment both increase with increasing beam width. 
Many applications for MEMS actuators use translational motion rather than 
rotation as the input. The central yoke allows for near-translational motion at the 
actuator output for small angular deflections. The translational deflection output is 
proportional to the yoke length r
2
, which for the considered actuators is kept constant 
at 485 ?m. As the yoke length is increased, the free deflection at a given current will 
 
 76 
 
increase linearly; however, the maximum output force will also decrease linearly 
because the available actuator moment about P remains the same. Stated another way, 
the area under the force deflection plot (which represents the feasible operation region 
of the actuator) remains constant, but the slope of the line can be changed simply by 
changing the yoke length. 
With the above caveat about yoke length, some design criteria for translational 
output and comparisons with existing purely translational actuators is desirable. A 
translation force-deflection plot is given for the rotational actuators in Figure 3.37. 
The translational measurements were conducted with the same test structures as those 
used for the torsion measurements. The force is obtained by the measured deflection 
of the resisting cantilever spring combined with the calculated cantilever spring 
constant.  
y = -8.59E-6x + 2.26E-7
y = -4.94E-6x + 1.29E-7
y = -3.03E-6x + 8.15E-8
0.00E+0
2.00E-8
4.00E-8
6.00E-8
8.00E-8
1.00E-7
1.20E-7
1.40E-7
0.008 0.013 0.018 0.023 0.028
Rotation (rad)
Mo
m
e
n
t
 (
N
m)
5um Width
7um Width
10um Width
 
Figure 3.36. Moment-rotation angle relationships for 400 ?m long rotational 
actuators of varying width. 
 
 77 
 
In Figure 3.37, a series of force-deflection profiles are shown for rotational 
actuators of constant length L = 400 ?m and varying widths. The deflection values 
shown are the displacement measurements at the actuator output projected onto the x-
axis. The slope of each linear trendline represents an approximate measure of the 
actuator linear stiffness. The actuator stiffness is observed to increase with the 
actuator width (the theoretical relationship is calculated in Section 4.8). The y-
intercept of the trendline is an approximate measure of the actuator blocked force 
(zero displacement force). It can be seen from the plot that when a wider actuator 
beam is used, higher forces are possible for the same displacement (yielding larger 
actuator work). The tradeoff is increased actuator current ? the voltage (and current 
density) remains nearly constant for a given displacement as the beam is widened. 
The free deflection of a rotational actuator is also compared with the 
commonly-used bent-beam and hot-arm/cold-arm style thermal actuators of similar 
y = -12.91x + 168.1
y = -21.01x + 266.7
y = -36.55x + 466.7
0
50
100
150
200
250
4 6 8 10 12 14
Displacement (?m)
F
o
rc
e
 (?
N
)
5?m Width
7?m Width
10?m Width
 
Figure 3.37. Force-deflection relationships for 400 ?m long rotational actuators 
and different widths. 
 
 78 
 
dimensions in Figure 3.38. For all of the actuators represented in this graph, 5 ?m 
wide hot beams are used, and they are all fabricated on SOI wafers with identical 1-3 
m?-cm resistivity device layers measuring 20 ?m in thickness. The hot-arm/cold-arm 
actuator is 1050 ?m long, the bent-beam actuator is 1200 ?m long, and the rotational 
actuator has a 1000 ?m span (L = 500 ?m in Figure 2.4) with an amplification beam 
of r
2
 = 485 ?m. The rotational actuator consumes slightly more power than the hot-
arm/cold-arm actuator but only about 36% as much as the bent-beam actuator for the 
same free deflection. 
The rotational actuator provides far more force than the hot-arm/cold-arm type 
of actuator. While force data are not available for this hot-arm/cold-arm actuator 
because there were no force test structures included, this type of actuator typically is 
limited to a few ?N of force before the actuator burns out or buckles [44-46]. The 
maximum measured force output from the bent-beam actuator design is 1.2mN at 
0
2
4
6
8
10
12
14
16
18
0 50 100 150 200 250 300 350
Power (mW)
Deflection (?m)
Rotational Actuator
Hot-arm/Cold-arm
Bent Beam
 
Figure 3.38. Comparison of free deflection for V-beam, U-beam, and rotational 
thermal actuators. 
 
 79 
 
24.7 mA/18.8 V drive, pushing against a spring with a stiffness of 50 N/m. As the 
current is increased from this point, the actuator beams start to buckle. The maximum 
force measured with the rotational actuator is 0.23 mN at 15 mA/12V, pushing 
against a spring with a stiffness of 69.5 N/m. Therefore, the rotational actuator is a 
good choice for applications that require large displacements and require more force 
than what a hot-arm/cold-arm actuator can provide, but not all of the force available 
with a bent-beam actuator. 
It has been shown in both simulations and testing for other types of thermal 
actuators that the dominant heat loss mechanism is conduction through the air to the 
substrate [45], followed by heat loss into the anchors. Containing these heat losses 
can greatly increase the actuator efficiency by increasing the equilibrium beam 
temperature for the same applied current. The heat loss through the air into the 
substrate is easily eliminated by operating the actuators under vacuum.  
0
0.5
1
1.5
2
2.5
3
3.5
0 102030405060
Power (mW)
D
ef
l
ect
i
o
n
 (
d
eg
)
6.2mT
Atmosphere
 
Figure 3.39. Comparison of actuator free deflection under vacuum and 
atmospheric conditions. 
 
 80 
 
An actuator was tested both in vacuum and atmospheric (ambient) conditions. 
The pressure during vacuum testing varied between 5.9 mT and 6.5 mT. The results 
are plotted in Figure 3.39. For the same free deflection, the actuator required 50% less 
current and 40% less voltage, consuming 70% less overall power. 
The frequency of operation for thermal actuators is generally limited by the 
thermal time constant of the system [41, 43, 53]. The rotational offset beam actuator 
is limited in the same way. The thermal time constant of the actuator depends 
primarily on the beam width, length, and thickness. Smaller devices have a lower 
thermal mass and are expected to have a larger cutoff frequency. Frequency response 
measurements were performed using a laser Doppler vibrometer while driving the 
device with a square wave input signal. The normalized frequency response in air is 
shown in Figure 3.40. The measurements shown correspond to actuators with L = 
0.01
0.1
1
10
10 100 1000 10000 100000
Frequency (Hz)
N
o
rm
. D
i
s
p
la
ce
me
n
t
5um width
7um width
10um width
 
Figure 3.40. Frequency response of offset beam actuators in air for L = 400 ?m, 
various widths. 
 
 81 
 
400 ?m. The cutoff frequencies extrapolated from this data are about 350, 285, and 
270 Hz for the 5 ?m wide, 7 ?m wide, and 10 ?m wide actuators, respectively.  
3.3 Friction Testing 
The friction coefficient of the silicon sidewalls must be determined for the 
shock sensor model. A novel friction test structure has been designed for this work to 
perform these measurements (reported in [51]). This friction test structure makes use 
of V-beam thermal actuators to separately apply normal and transverse forces to test 
blocks of various sizes (see Figure 3.41 and Figure 3.42). A stationary fixed block 
provides the other surface for contact. Both contacting surfaces are deep reactive ion 
etched silicon, the same as the sidewalls of the contacting pieces of the shock sensor 
system. 
 
Normal 
Actuator
Transverse 
Actuator
Stationary 
Contact Block
 
Figure 3.41. Friction test structure design. 
 
 
 82 
 
The test procedure used is to apply a fixed current to the normal actuator, 
pushing the test block against the stationary block. The transverse actuator current is 
then increased until the test block is observed to slip under a microscope. The 
actuator stiffness and the free deflection measured from an actuator test structure are 
used to calculate the forces applied by each actuator, by assuming a linear force-
displacement profile for an individual actuator: 
 )( xkF
freeact
?= ?  (3.4) 
where x is the measured displacement of the actuator during the friction test, k
act
 is the 
actuator stiffness and ?
free
 is the free displacement at the given current level. 
In the pictured device, each actuator has five parallel bent beams for the total 
force generation.  The number of beams can be changed to generate a wide range of 
forces. The beams used in this study have been constructed by using two 600?m long 
segments, with a 3 degree rotation and a 25?m wide yoke in the middle. The normal 
 
 
 
 
Figure 3.42. Friction test structure contact pad closeup. 
 
 83 
 
actuator yoke has been narrowed just beyond the actuator to be 5?m wide to provide 
low spring resistance for the friction measurement and additional thermal isolation 
between the actuator and the contact pad. 
The displacement x for these friction test structures is fixed at 3?m in all cases 
for both the normal and transverse actuators. The free deflections are measured as a 
function of applied current on freestanding actuators fabricated on the same wafer as 
the friction test devices. Deflection measurements have been obtained by using a 
white-light optical profilometer with a lateral resolution of 162nm (see Figure 3.43). 
This free-deflection data was used in conjunction with the FEA derived actuator 
stiffness to determine the force exerted by each actuator for a particular current level. 
The forces obtained depend primarily on the applied current and the number 
of parallel bent beams. In Figure 3.44, the force versus current profile results are 
reference 
mirror
white light source
y
camera
beam 
splitter
sample
 
 
Figure 3.43. White-light optical profilometer operational principle. Camera is 
scanned to find position of constructive interference (where sample distance 
equals reference distance) for each pixel. 
 
 84 
 
plotted for 2-beam and 5-beam actuator configurations. The forces developed by the 
thermal actuators are relatively high compared to many MEMS structures, and these 
forces easily extend into the mN range. 
The two constraints on the high end of the applied force are overheating that 
can melt the actuator and buckling of the individual actuator beams or the narrowed 
portion of the normal actuator. For the configurations used here, the narrowed portion 
of the normal actuator buckled at approximately 2.55 mN, which is a bit lower than 
the theoretical critical buckling load of 2.84 mN. Applying a larger force than this 
requires stiffening of the thin portion of the normal beam, and this can be easily 
accomplished if desired. The maximum current applied in Figure 3.44 was limited by 
overheating. 
It is expected that the force in Figure 3.44 should scale linearly with the 
number of beams in the actuator because the stiffness scales linearly, but this is 
0
2
4
6
8
10
12
14
16
18
6 8 10 12 14 16
Normalized Current (mA/beam)
For
c
e
 (
m
N
)
2-beam
5-beam
 
Figure 3.44. Force developed by actuators at 3 ?m travel. 
 
 85 
 
clearly not the case. In practice, the displacement of a 5-beam actuator is much larger 
than a 2-beam actuator with the same normalized current applied. The explanation for 
this is the added efficiency realized when more heated beams are placed alongside the 
original beams ? the heat losses per beam to the surrounding air decrease, so the same 
applied current per beam results in a larger temperature increase. This in turn leads to 
a larger free displacement at the same normalized current, and the larger free 
displacement results in a higher generated force. 
Friction test structures using 5-beam normal actuators and 2-beam transverse 
actuators have been studied to determine the friction coefficient of the DRIE silicon 
sidewalls. A steady current was applied to the normal actuator, and the current input 
to the transverse actuator was slowly ramped up while monitoring the contact pad 
under the microscope until slip was observed. The test was initially performed using a 
current in the normal actuator just above the level required to achieve contact; this 
was done to keep the normal force magnitude low. The required current in the 
transverse actuator to achieve slip is documented, then the current input into the 
normal actuator is increased and the test is repeated. 
Two wafers with friction test structures were fabricated with very different 
DRIE recipes; one of them provides fairly rough sidewalls (RMS roughness of 
~275 nm as measured using white light optical profilometry) and the other provides 
very smooth sidewalls (RMS roughness of ~60 nm). For the wafer with smooth 
sidewalls, the friction coefficient measurements ranged from 0.67 to 0.76, which 
agrees well with the value of ~0.7 obtained in macroscale friction measurements on 
silicon [28].  
 
 86 
 
Early tests on the rough sidewall wafer, however, appeared to indicate that the 
coefficient of friction was decreasing as the normal load was increased, to the point 
that a smaller transverse actuator current was needed to achieve slip after the normal 
actuator current was increased to a higher level than before.  To sort this result out, 
repetitive tests were performed with a previously uncycled test structure without 
increasing the normal actuator current between each test. The results showed that the 
coefficient of friction has a high initial value, before it drops with each subsequent 
cycle and plateaus out to a steady-state value (see Figure 3.45). The friction 
measurements were repeated after a period of 90 minutes, and again several weeks 
later on the same device with no change from the final steady-state value. It is 
supposed that as the surfaces rub against one another, some asperities are broken off 
and the surface roughness decreases each time until nearly all asperities above a 
0
0.5
1
1.5
2
2.5
3
0 5 10 15 20
Cycle Number
Fr
i
c
t
i
on C
oe
f
f
i
c
i
e
nt
Initial Measurement
After 1.5 hours
 
Figure 3.45. Plot of friction coefficient versus cycle number showing decrease 
over the first several cycles before settling to final value. 
 
 87 
 
critical size are gone. After this point the roughness stays relatively constant, and the 
friction coefficient does not change. The initial friction coefficient seen in these 
particular devices is several times larger than the final steady-state value (2.45 as 
compared to 0.70). This suggests that rough contact surfaces should be conditioned 
before use by putting the sliding surfaces through several cycles (for these devices, 15 
cycles was found to be enough). This ensures that the coefficient of friction will 
remain constant through the device?s lifetime. 
The contact surfaces on the rough wafer were examined after the experiments 
to see evidence of wear. There is a clear delineation between the area of the sidewall 
subjected to the sliding contact and the adjacent area not contacted (see Figure 3.46). 
Smoothing of the horizontal striations from the deep reactive ion etch is clearly 
visible in the left half of the image, which is in the contact region. The right half of 
the image shows the as-etched surface with the intact sharp striations, since this area 
 
Figure 3.46. Scanning electron microscope image of plastic deformation along 
area of sliding contact. Right half of image does not contact friction pad, while 
the left half does. 
 
 88 
 
was not in the contact region. 
To determine if heating of the contact surface occurs due to the thermal 
actuation technique, forward-looking infrared (FLIR) measurements were taken of the 
devices while the normal actuator was turned on and pressed against the contact 
surface. The measurements showed the contact area to be at room temperature even 
when the active region of the device is glowing red-hot (shown in Figure 3.47); this 
means that the friction measurements can be applied to room temperature contacts. 
Extension of the technique to allow for heated contact is possible, either by separate 
active heating of the contact pad or by altering the design to allow passive heating 
from the normal actuator. Active heating of the contact pad would be preferable, 
since the contact temperature could then be controlled independently from the applied 
force. 
 
contact 
pad
actuator
 
Figure 3.47. Forward-looking infrared (FLIR) measurement of temperature 
profile of friction device. The active area corresponding to the actuator is hot, 
while the contact pad is at the ambient temperature. 
 
 
 89 
 
 
3.4 Summary and Contributions 
The experimental techniques used to study the shock sensor and the reset 
actuators have been presented in this chapter. These techniques are intended to 
provide a framework for experimental study of this class of device, from 
identification of the lumped mass and spring constants, to measurement of the 
acceleration required to latch the device, to capturing and interpreting images of the 
actual latching event. A thorough experimental study of these latching shock sensors 
has never been reported before, and it is the author?s hope that the experiments 
described in this chapter will aid other research efforts on this type of device, and that 
other researchers will build on and improve the techniques reported here. In 
particular, further measurements of the relative position of the mass and latch during 
the latching process, including other design configurations and better temporal and 
spatial resolution, would significantly enhance the understanding of the dynamic 
behavior of latching acceleration switches. 
The experimental data collected have also been presented and analyzed in this 
chapter. Some key observations regarding latching acceleration threshold switches 
that are not described in previous literature include: the change in the friction 
coefficient over the first several cycles; the timing of the latching event ; that is, the 
time from the onset of the acceleration pulse to contact, to latching, and finally, to the 
sensor settling and closing the electrical circuit; the apparent onset of stiction due to 
electrostatic charge buildup in the sensor over tens to hundreds of cycles; and the 
surprising stability of the contact resistance over many cycles despite gross 
degradation in the metallization. 
 
 90 
 
4. Modeling and Numerical Results 
In this chapter, all of the modeling and numerical studies performed are discussed. 
These studies include the derivation of two distinct reduced-order models for the 
shock sensor operation, comparisons between the predictions of the two models, and 
various parametric and optimization studies intended to lend insights into the design 
of this class of sensor. In addition, a model is developed to calculate the force applied 
by a V-beam actuator (used for the friction characterization studies). A new thermal 
actuator model is also developed and applied to the rotational offset-beam actuator 
design. This new model takes into account the temperature-dependence of the thermal 
expansion coefficient of silicon, neglected in previous models of thermal actuators. 
Finally several parametric studies of the rotational actuator design are presented to 
gain insights into the design of these devices. Portions of this chapter have been 
adapted from the author?s journal articles on the latching shock sensor [38, 48] and 
the rotational actuator [39]. 
4.1 Latching Progression 
 The latching shock sensor goes through several distinct phases during an 
acceleration event. These are illustrated in Figure 4.1 for the sensor under 
consideration here. In the first phase, the acceleration has begun and the sensor is 
responding, but has not yet come into contact with the latch. In the second phase, the 
sensor makes contact with the latch and there are some interaction forces and some 
resulting motion of the latch. In the third phase, the sensor has continued past the 
latch, possibly overshooting so that contact is again broken and there are no 
interaction forces. In the fourth and final stage, the sensor comes to rest against the 
 
 91 
 
latch and the electrical circuit is closed. The following sections derive the governing 
equations for the sensor during the second phase, when the mass is pushing past the 
latch and the latch is moving. The governing equations for the first and third phase 
can be easily derived from these by setting the interaction forces to zero, and the 
governing equation for the last phase can be easily derived by setting the interaction 
forces to zero and adding the axial stiffness of the latch to the sensor spring constant. 
4.2  Reduced-Order Latching Sensor Model 
In this section, two separate models are developed to predict the response of 
the shock sensor. The first model developed makes use of a single degree-of-freedom 
 
Figure 4.1. Shock sensor phases of latching: i) pre-contact, ii) during contact, iii) 
latched but still moving, and iv) latched and resting. 
 
 92 
 
(the position of the mass), and this model is based on the assumptions that contact is 
maintained between the mass and latch during phase ii (Figure 4.1) and that the 
contact force can be estimated during this phase from the spring force of the latch. It 
eventually became clear that these assumptions were too limiting, since the true 
interaction forces could not be computed. Therefore, a two-degree-of-freedom model 
was developed in which one degree-of-freedom is used for the mass and another for 
the latch (the second latch is assumed to have an identical response to the first). This 
allows the interaction between the mass and latch to be studied in detail, including 
whether contact is indeed maintained in phase ii as well as the contact forces. 
The one degree-of-freedom model requires very little computation time, and it 
may therefore be useful for fast order-of-magnitude estimates in screening potential 
designs. The two-degree-of-freedom is far more computationally intensive, primarily 
because of the very small time steps needed for correctly capturing the onset of 
contact. This model is expected to provide more accurate predictions of the device 
response, as well as allow for a detailed study of the dynamics of the mass-latch 
interaction. 
4.2.1 One Degree-of-Freedom Model 
The simplest model of the shock sensor is a one degree-of-freedom lumped 
parameter representation. The suspension springs and mass are lumped into a single 
global mass and spring constant, and the interaction of the latch and mass is included 
as a nonlinear, position dependent resisting force. In order to solve for the interaction 
force, the assumption is made that the latch and mass remain in contact throughout 
the interaction period. This model may not be the most accurate, but it can be solved 
 
 93 
 
quickly using standard numerical techniques. It may also be useful for quickly 
evaluating various design options in the early stages of device design. The geometry 
used in developing this model is shown in Figure 4.2. 
We will use Lagrange?s Equations to derive the basic equation of motion for 
the one degree of freedom case [54]. The Lagrangian is defined as  
 ,VTL ?=  (4.1) 
where T is the kinetic energy of the system and V is the potential energy of the 
system. Since the mass of the latch is negligible compared to the sensor mass, the 
kinetic energy of the latch will be neglected. The kinetic and potential energy of the 
system can then be written as 
 ,)(
2
1
2
/ cmc
yyMT && +=  (4.2) 
x
d
i
y
0
(a)
r
(b)
?
F
f
N
y
m/c
Y
X
w
L
2
r
r + d
i
+ x
y
0 
-y
m/c
 
Figure 4.2. Mass and latch interaction for single DOF model: a) before contact 
and b) during contact. 
 
 94 
 
 ,
2
1
2
/ cm
kyV =  (4.3)  
where M is the effective mass of the sensor (including the effective mass of the 
suspension springs if desired), y
c
 is the displacement of the chip, y
m/c
 is the relative 
displacement of the mass with respect to the chip, and k is the total effective spring 
constant of the suspension springs. A dot above a parameter indicates the time 
derivative. The complete Lagrangian can therefore be written as 
 .
2
1
)(
2
1
2
/
2
/ cmcmc
kyyyML ?+= &&  (4.4) 
The Euler-Lagrange Equations can be used to derive the equations of motion 
for the system from the Lagrangian expression. There is one equation for each 
generalized coordinate. Since we have only one generalized coordinate (y
m/c
), there is 
only one equation of motion, 
 ,
/
//
cym
cmcm
Q
y
L
y
L
dt
d
=
?
?
?
?
?
?
?
?
?
?
?
?
?
&
 (4.5) 
where Q represents any non-conservative forces, which in this case are the interaction 
forces between the mass and latch. Referring to Figure 4.2, the interaction force 
consists of a normal and frictional force. Both of these vary in direction and 
magnitude according to position of the mass. 
 At this stage, it is assumed that the mass and latch remain in contact 
throughout the interaction period and that the contact is rigid (i.e., no deformation of 
the mass of latch occurs in the contact region due to the contact forces). The result of 
this assumption is a constraint on x, wherein the deflection of the latch can be related 
to the displacement of the mass by simple geometry, 
 
 95 
 
 )()(4
2
/0
2
icm
dryyrx +???=  (4.6) 
After using static considerations for simplicity, the normal force N can be derived 
from a force balance on the latch in the x-direction. 
 0)90cos(cos =???=
?
xkNNF
L
x
???  (4.7) 
Solving for N yields 
 
??? sincos ?
=
xk
N
L
 (4.8) 
Examining the geometry in Figure 4.2b, relations for sin ? and cos ? can be derived in 
terms of the generalized coordinate y
m/c
 as 
 
r
yy
cm
2
sin
/0
?
=?  (4.9) 
 
r
xdr
i
2
cos
++
=?
 (4.10) 
The frictional force in the y-direction which resists motion is therefore 
 
r
yyr
NysignNysignF
cm
cmcmfy
2
)(4
)(cos)(
2
/0
2
//
??
??=??= ??? &&  (4.11) 
 
()
()
2
/0
2
/0
2
/0
2
2
0
2
/
)(4*
)(4
)(4
)(
cm
cmcm
iL
cmfy
yyr
yyyyr
dryyrk
ysignF
??
????
????
??=
?
?
&
 (4.12) 
Taking the required derivatives and substituting into the Euler-Lagrange equation 
yields the equation of motion for the one degree-of-freedom approximation of the 
system dynamics. 
 
 96 
 
()
( )
()
,)(4
)(4
)(4
)(         
2
/0
2
/0
2
/0
2
2
0
2
/
/
//
cm
cmcm
iL
cm
cm
Lcmcmc
yyr
yyyyr
dryyrk
ysign
dy
dx
xkkyyyM
??
????
????
??
=+++
?
?&
&&&&
 (4.13) 
Substituting in for x and its derivative yields the final equation of motion as a 
second order nonlinear differential equation in y
m/c 
() []
( )
( )
()
,)(4
)(4
)(4
2)(                                   
)(4
2
)()(42
2
/0
2
/0
2
/0
2
2
0
2
/
2
/0
2
/02
/0
2
//
cm
cmcm
i
Lcm
cm
cm
icmLcmcmc
yyr
yyyyr
dryyr
kysign
yyr
yy
dryyrkkyyyM
??
????
????
??
=
??
?
+???+++
?
?&
&&&&
 (4.14) 
By introducing two states, y
1
 an y
2
, which represent the zeroth and first time 
derivative of y
m/c
, respectively, the author converts the second order ODE into two 
first order ODE?s. 
[]
()
( )
()
2
/0
2
/0
2
/0
2
2
0
2
/
2
/0
2
/02
/0
21
2
21
)(4
)(4
)(4
)(             
)(4
2
)()(4)(
)(
cm
cmcm
i
L
cm
cm
cm
icm
L
c
yyr
yyyyr
dryyr
m
k
ysign
yyr
yy
dryyr
m
k
m
ky
yy
dt
d
yy
dt
d
??
????
????
??
?
??
?
+??????=
=
?
?
&
&&
 (4.15) 
Once in this form, given an initial condition and the function for the 
acceleration of the chip, the system can be readily solved using numerical techniques. 
Unless otherwise specified, we used the ode45 solver available in MATLAB which 
makes use of an explicit Runge-Kutta formula for the numerical integration. 
The one degree-of-freedom model can be useful for quickly getting an order 
of magnitude estimate of the travel of the mass for a particular input acceleration 
function (
c
y&& ). The limitation is that this formulation does not include inertia of the 
latch, which may cause underestimation of the interaction forces. This also does not 
 
 97 
 
allow for study of the dynamics of the mass-latch interaction, so another model was 
created with two degrees of freedom. This model is presented next, and predictions 
from the two models are presented in Section 4.4. 
4.2.2 Two Degree-of-Freedom Model 
To avoid making the potentially erroneous assumption that the mass and latch 
remain in contact throughout the interaction period and to study the interaction in 
detail, one must include at minimum a second degree of freedom. This allows the 
mass and the latch to move independently, but significantly increases computation 
time and requires the use of very small time steps or more complex event detection 
during solution. The results are expected to provide more insights into the dynamics 
of the mass-latch interaction and to be more accurate than those obtained from the 
one degree of freedom model. 
The two degree-of-freedom model presented in this section is the 
recommended model to use for detailed device design. The model is validated in 
Section 5.2 using the results from the high speed video measurements of the sensor 
position time history presented in Section 3.1.2. 
The author will begin as with the one degree-of-freedom model by developing 
the Lagrangian and using the Euler-Lagrange equations to derive the equations of 
motion. For this model, however, the author will include the inertia of the latch in the 
kinetic energy expression and we will impose no constraint on the latch displacement 
x. The kinetic and potential energy for the system are  
 ,
2
1
2)(
2
1
22
/
xmyyMT
cmc
&&& ++=  (4.16) 
 
 98 
 
 ,
2
1
2
2
1
2
2
1
222
/
?
cLcm
kxkkyV ++=  (4.17)  
where m is the mass of the latch, ? is the apparent penetration depth of the contact 
region (see Figure 4.3), and k
c
 is an effective spring constant of the contact 
interaction. The factors of two in front of the second term in the kinetic energy and 
the second and third terms in the potential energy result from the fact that there are 
two latches in the system. The implicit assumption is that both latches have the same 
properties and move in unison. The Lagrangian can then be written according to Eq. 
(4.1) as 
 
                                                                     
.
2
1
2
2
1
2
2
1
2
1
2)(
2
1
222
/
22
/
?
cLcmcmc
kxkkyxmyyML ???++= &&&
 (4.18) 
 
Figure 4.3. Geometry of mass-latch interaction for two DOF model: a) before 
contact, and b) during contact (with exaggerated contact deformation for 
clarity). 
 
 99 
 
The equation of motion in the y-direction is found by substituting the 
Lagrangian into Eq. (4.6) and taking the required derivatives. There are two 
generalized coordinates (y
m/c
 and x), resulting in two second order ODE?s. With the 
addition of the second degree of freedom, the normal force is now conserved and the 
only nonconservative force Q is the friction force. The equations of motion are 
 () ???
?
? cos2)(sign2
/
/
// ccm
cm
ccmcmc
ky
y
kkyyyM &&&&& ?=
?
?
+++  (4.19) 
 ???
?
? sin2)(sign222
ccL
kx
x
kxkxm &&& ?=
?
?
++  (4.20) 
where sin? and cos?  can be written as 
 
?
?
?
++
=
r
xdr
i
2
cos  (4.21) 
 
?
?
?
?
=
r
yy
2
sin
0
 (4.22) 
The penetration depth ? is a function of the two generalized coordinates. By 
examination of the geometry in Figure 4.3, the penetration depth can be derived as 
()() ()()
()()?
?
?
?
?
>+++?
?+++?+++?
=
   rxdryy                                                                
rxdryy              xdryy-r
icm0
icm0icm0
2,0
2,2
22
/
22
/
22
/
? (4.23) 
The partial derivatives of ? can be obtained as 
 
 
22
/0
/0
/
)()(
)(
xdryy
yy
y
icm
cm
cm
+++?
?
=
?
??
 (4.24) 
 
22
/0
)()(
)(
xdryy
xdr
x
icm
i
+++?
++
?=
?
??
 (4.25) 
Substituting ? and its partial derivatives into Eq. 4.19 and 4.20 results in the final 
equation of motion used in our model. 
 
 100 
 
()
( )
()()
()
()()
?
???
?
???
?
?
?=
+++?
++
++
?
++
?=
+++?
?
+++
r
yy
kx
xdryy
xdr
kxkxm
r
xdr
ky
xdryy
yy
kkyyyM
c
icm
i
cL
i
c
icm
cm
ccmcmc
2
)(sign
2
2)(sign2
0
22
/0
22
/0
/0
//
&&&
&&&&&
(4.26) 
The value of the contact stiffness, k
c
, has not yet been addressed. For 
simplicity, a constant linear spring constant to represent the contact stiffness is 
attractive. In reference [55], Puttock and Thwaite derived expressions for the 
penetration depth as a function of applied force for various contact configurations 
with cylindrical and spherical bodies by using Hertzian contact theory. In Hertzian 
contact theory, the following assumptions are made: i) the contact surfaces are 
completely smooth, ii) the bodies are isotropic and linearly elastic, iii) the elastic 
limits are not exceeded, and iv) there are no friction forces in action. While these 
assumptions do not hold true for the silicon contact surfaces (the surfaces have 
roughness on the order of 60 nm, the silicon material has moderately anisotropic 
modulus and the sliding contact must necessarily include friction), simplifications are 
needed to derive a closed-form solution for the contact force-deformation 
relationship. According to Puttock and Thwaite, in general, the friction forces and 
varying elastic moduli lead to deviations of less than 10% from the ideal cases that 
they had studied.  On this basis, for the case of two parallel cylinders in contact, the 
expression is  
 
()
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
+
+
++=
2121
2
21
118
ln1)(
DDPVV
a
VVP?  (4.27) 
where 2a is the length of contact between the cylinders, P is the compressive force 
per unit length applied to the cylinders, D
i
 is the diameter of each cylinder, and V is a 
 
 101 
 
material property defined as 
E?
?
2
1?
, where ? is Poisson?s ratio and E is Young?s 
modulus. For the special case of two cylinders of the same material and diameter, the 
relation simplifies to 
 
?
?
?
?
?
?
?
?
?
?
?
?
+=
VPD
a
PV
2
8
ln12?  (4.28) 
This force deflection relationship was verified for the shock sensor 
dimensions by using a three-dimensional finite element model composed in ANSYS 
[56] (Figure 4.4). Solid95 elements (a structural element with mid-side nodes) were 
used to conduct a static (steady-state) analysis of the contact with a constant applied 
force. Half-cylinders were used to allow a uniform pressure to be applied 
perpendicular to a flat edge. The flat edge of one half-cylinder was fully constrained 
and the load applied to the flat edge of the other half-cylinder. The loads chosen are 
 
Figure 4.4. Finite element model of latch contact. 
 
 102 
 
in the range of the expected contact forces predicted by the dynamic shock sensor 
model presented above. The deflection of the edge where the load was applied is 
equal to the apparent interpenetration used in the model. The finite element results are 
compared with Eq. (4.28) in Figure 4.5. 
The solutions of Eq. (4.26) are numerically determined for the chosen initial 
conditions and the given external acceleration profile to which the chip is subjected.  
The chip acceleration serves as excitation to the system. Although the model can 
accommodate any arbitrary function for the chip displacement y
c
, in this dissertation 
this forcing profile is considered to be a half-sine pulse for all simulations with the 
magnitude and duration of the half-sine pulse as noted in each case. 
The relevant parameters for the two sensor designs considered here are shown 
in Table 4.1. The results obtained from a representative simulation conducted for one 
sensor design and acceleration profile are shown in Figure 4.6. The latch initially 
does not undergo any motion because the mass has not contacted it. Contact is made 
 
 
Figure 4.5. Comparison of contact forces predicted by Eq. (4.28) and finite 
element model over the relevant range of applied force. 
 
 103 
 
at 1.1 ms in this case. At 1.4 ms, the mass has moved completely past the latch. After 
this event, in the simulations, the motions of the shock sensor mass and latch are 
treated as uncoupled. The latch undergoes free vibrations, while the mass bounces 
back and forth as it alternately contacts the extension frame and flat edge of the latch 
(see Figure 4.7). The motion of the latch in the post-contact region is treated as a free 
harmonic oscillation with an initial displacement equal to the position of the latch 
when the mass has just moved past it. As shown in Figure 4.6, the frequency of 
vibration of the latch (44.7 kHz) is much higher in this post-contact region than that 
of the mass bounce frequency (4.0 kHz). This is due to the much lower mass and 
higher spring constant of the latch. The frequency of the mass bouncing back and 
forth after latching is primarily dependent on the speed of the mass after latching. 
 
Table 4.1. Design parameters for shock sensor relevant to the lumped parameter 
model (corresponding physical dimensions are given in Table 3.1). 
Parameter Description Design 1 Design 2 
M (kg) sensor mass 3.42e-7 2.65e-7 
k (N/m) sensor stiffness 0.601 1.324 
f (Hz) natural frequency of sensor (uncoupled) 211 356 
m (kg) latch effective mass 5.64e-11 5.64e-11 
k
L
 (N/m) latch stiffness 4.44 4.44 
f
L
 (Hz) natural frequency of latch (uncoupled) 44,700 44,700 
k
c
 (N/m) contact stiffness 5.57e5 5.57e5 
y
0
 (?m) initial sensor travel to latch 150 150 
r (?m) latch radius 40 40 
d
i
 (?m) horizontal offset 15 15 
crit
y&&  (g) nominal acceleration for latching ? 
threshold levels 
50 100 
 
 104 
 
 
Figure 4.6. Design 2 sensor results obtained  from two DOF model for 
displacement of mass (y
m/c
) and displacement of latch (x) when ?=0.1, and 
acceleration pulse is 120 g for 5.9 ms. 
 
 
Figure 4.7. Alternating contact points which cause bouncing of mass after 
latching. 
 
 105 
 
The contact force can be extracted from the 2 degree-of-freedom model at 
each time step to help analyze the behavior. The results for a Design 1 device are 
shown in Figure 4.8. Since  the contact and the latch itself both have an inherent 
stiffness, the contact force is observed as a decaying oscillatory function. The decay 
is due to the changing direction of the contact normal, from just over 45 degrees from 
the direction of travel at the moment of first contact to 90 degrees to the direction of 
travel just before latching. Early on in the contact, the model predicts intermittent 
contact due to the latch bouncing away from the mass. This is more evident in Figure 
4.9, which shows the position of the latch during the initial phase of contact. There is 
a clear bounce evident generated by the contact force pushing the latch away from the 
mass. This effect is dependent on the time steps chosen in the model solution, with 
Force
 (
m
N)
0
0.5
1
1.5
2
2.5
Time (ms)
1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4
 
Figure 4.8. Time history of contact force for a Design 2 sensor as predicted by 
2 DOF model when ?=0.1, and acceleration pulse is 120 g for 5.9 ms. 
 
 106 
 
larger bounces predicted at larger time steps. This makes sense because with a larger 
time step the apparent penetration when contact is first detected is larger, and 
therefore the contact force will be larger and the acceleration of the latch away from 
the mass will also be larger. However, as the time step is decreased towards zero, the 
magnitude of the bounce approaches a limit of about 0.5 ?m ? smaller time steps no 
longer decrease the amplitude of the bounce. For the Design 1 sensor, the initial 
bounce causes a loss of contact for 0.7 ?s. Each successive bounce gets smaller and 
the loss of contact shorter until after about 40 ?s there is no longer any loss of 
contact. 
 
L
a
t
ch
 D
i
sp
l
a
ce
me
n
t
 (
?
m)
0
1
2
3
4
5
Time (ms)
1.06 1.07 1.08 1.09 1.1
6
7
8
A
B
C
 
Figure 4.9. Position of latch during initial phase of contact, displaying bounce of 
up to 0.4 ?m (blowup of the contact section in the latch position graph from 
Figure 4.6). Point A is where initial contact is made, points B are where contact 
is lost during a bounce, and points C are where contact is reestablished after a 
bounce. 
 
 107 
 
One interesting effect evident from the force time-history shown in Figure 4.8 
is that the maximum contact force does not occur during the initial contact, but rather 
after several bounces (on the fourth bounce for the case shown in Figure 4.8). This is 
because the acceleration pulse applied was a half-sine pulse with a pulsewidth of 
5.9 ms, and the sensor makes initial contact at 1.059ms, while the acceleration is still 
increasing. Figure 4.10 shows the kinetic energy of the mass and the latch during the 
latching event for the same case plotted in Figure 4.6 and Figure 4.8. The maximum 
kinetic energy for the latch is achieved at 1.070 ms, just after the third bounce. The 
Initial 
contact
1.05 1.1 1.15 1.251.2 1.3 1.35 1.4
Ki
n
e
tic E
n
er
g
y
 of
 
Latc
h
 
(
n
J)
3.5
3
4.5
4
5.5
5
6
30
20
40
10
0
Ki
netic E
n
erg
y
 of
 
Mass
 
(nJ)
Time (ms)
 
Figure 4.10.  Kinetic energy of mass and latch during latching event for Design 2 
sensor, when ?=0.1 and acceleration pulse is 120 g for 5.9ms. 
 
 108 
 
kinetic energy for the mass reaches its maximum value at 1.079ms, just after the sixth 
bounce. In other words, the relative approach velocity of the mass and latch is largest 
for the fourth and fifth bounce. After this the relative approach velocities of each start 
to decrease. This explains why the contact force is highest for the fourth bounce. 
 
4.3  Equivalent Stiffness for Suspension Springs and Latch 
The reduced order model approximates the array of suspension springs as a 
single spring with a single stiffness value k. The equivalent lumped stiffness of the 
suspension spring is calculated by adding the four identical folded-flexure springs in 
parallel. The stiffness of each folded-flexure spring is in turn calculated by adding the 
stiffness of the individual meanders in series. The individual members are each 
equivalent to a clamped-guided beam, with a stiffness of 12EI/L
3
 [57], where E is the 
Young?s modulus for the beam material, L is the length of the beam, and I is the area 
moment of inertia of the cross-section of the beam. The total equivalent spring 
constant for the sensor suspension springs can be found as 
 ,
12
3
33
tw
L
E
n
p
L
EI
n
p
k ==  (4.29) 
where n is the number of beams in series in each folded flexure, p is the number of 
folded flexures in parallel, t is the device thickness, and w is the width of each folded 
flexure beam. The latch is approximated as a cantilever, with an equivalent spring 
constant of  
 .
4
3
3
3
3
L
L
L
L
Etw
L
EI
k ==  (4.30) 
 
 109 
 
4.4 Comparison of 1DOF and 2DOF Model Results 
By way of comparison between the one degree-of-freedom model and the two 
degree-of-freedom model, the displacement time-history is plotted for one case in 
Figure 4.11. The difference between the two models is in the treatment of the contact 
force between the latch and the mass. The one degree-of-freedom model used the 
displacement of the mass to calculate the position of the latch (assuming contact is 
maintained). The position of the latch and the resultant spring force were then used to 
calculate the normal force on the mass. The two degree-of-freedom model does not 
assume a position for the latch, and the contact forces are calculated based on the 
relative positions of the mass and latch and the contact stiffness derived from 
Hertzian contact theory. The results match exactly before contact is made because the 
two models are functionally identical in this region.  
1DOF Model
2DOF Model
Ma
ss P
o
si
ti
o
n
 (
?
m)
0
25
50
75
100
125
Time (ms)
0 0.5 1 1.5 2 2.5 3
150
175
200
Latched 
position
 
Figure 4.11. Comparison of single degree-of-freedom results from [9] to two 
degree-of-freedom model results for Design 2 sensor, ? = 0.1, 
acceleration = 120 g for 5.9 ms.  
 
 110 
 
During contact the results diverge. The coefficient of friction affects the mass 
motion more strongly in the one degree-of-freedom model because the latch and mass 
are in constant contact, whereas in the two degree-of-freedom model the latch 
bounces away from the mass many times during the nominal contact period. This is 
even more interesting when the contact forces are compared (Figure 4.12). Although 
the magnitude of the contact force in the one degree-of-freedom model is only about 
10% of the maximum magnitude of the contact force in the two degree-of-freedom 
model, the work done by the friction is higher in the one degree-of-freedom model 
because the distance travelled during contact is much larger. 
The increased dissipation in the one degree-of-freedom model due to the 
larger work done by friction results in a longer time to latch and lower velocity of the 
mass after latching in Figure 4.11. The predicted frequency of the oscillations after 
2DOF Force
1DOF Force
Fo
rce
 (
m
N
)
0
0.5
1
1.5
2
2.5
Time (ms)
1 1.2 1.4 1.6 1.8 2
 
Figure 4.12. Comparison of normal forces associated with 1DOF and 2DOF 
models. 
 
 111 
 
latching are therefore lower with the one degree-of-freedom model than with the two 
degree-of-freedom model. Perhaps more importantly, it also results in dramatically 
different predictions of the threshold acceleration to latch for a given design. This is 
illustrated in Figure 4.13, in which the predictions from both models are plotted for  
Design 1 and Design 2 devices with various assumed friction coefficients. In every 
case, the one degree-of-freedom model predicts significantly higher acceleration 
thresholds before a device latches. The difference is about 30% for Design 1 and 
ranges from 10% to 20% for Design 2. 
The results presented in this section illustrate the difference in treatment of the 
contact interaction between the one degree-of-freedom and the two degree-of-
freedom models. The implication is that while a one degree-of-freedom model may 
be used for rough order of magnitude calculations, at least two degrees of freedom 
0
20
40
60
80
100
120
140
160
180
200
? = 0.1 ? = 0.3 ? = 0.6
Friction Coefficent 
P
r
edic
t
e
d T
h
r
e
s
hold
A
cce
l
e
r
a
t
i
o
n
 (
g
)
1DOF Design 1
2DOF Design 1
1DOF Design 2
2DOF Design 2
 
Figure 4.13. Comparison of predicted threshold acceleration to latch using one 
degree-of-freedom and two degree-of-freedom models for Design 1 and Design 2 
devices with various friction coefficients. Applied acceleration is a half-sine pulse 
with duration of 8 ms for Design 1 and 5.9 ms for Design 2. 
 
 112 
 
must be used for detailed predictions of device performance. Predictions from the two 
degree-of-freedom model are expected to be more reliable, given the more realistic 
treatment of the interaction forces. The two degree-of-freedom model is validated in 
Section 5.2 by using the results from the high speed video measurements of the 
sensor position time history presented in Section 3.1.2. 
4.5 Sensor Parametric Studies 
To provide insights into the design of latching threshold acceleration sensors, 
the author undertook several parametric studies. These demonstrate the effect of one 
parameter on the device performance by holding all other parameters constant while 
varying only the parameter of interest. There are a large number of independent 
design parameters that can be manipulated to create designs with different latching 
threshold acceleration levels. For instance, the sensor mass M can be changed by 
modifying the length or width of the mass, or changing the size or spacing of the etch 
holes. The stiffness of the sensor k can be changed by modifying the length or width 
of the springs, the number of meanders, or the number of suspension springs used. 
The following studies are not intended to be exhaustive, but rather to provide a 
framework for understanding how the design can be manipulated to achieve the 
desired latching threshold. The baseline case used was the Design 2 sensor, which 
nominally latches around 100 g. The baseline values for each design parameter along 
with the mass and stiffness values calculated from these parameters are shown in 
Table 4.2. 
 
 113 
 
 
The first parameters examined are the height of the mass and the length of a 
flexure. The mass of the sensor varies linearly with the height of the mass, which 
causes the acceleration to latch to decrease with increasing H
m
 as shown in Figure 
4.14. The length of the flexure is a little more complicated, because the stiffness k is 
proportional to the inverse cubed length, and the width of the mass is also linked to 
the length of the flexure. In the design and in the model, the width of the mass is 
equal to  
 ?m 1102 +=
fm
LW  (4.31) 
The result is that as the length of the flexure increases, the stiffness k decreases and 
the effective mass of the sensor M also increases. Since both of these effects 
contribute to decrease the acceleration to latch as the length of the flexure increases, 
Table 4.2. Baseline parameters used for parametric studies (taken from Design 2 
sensor). 
Parameter Units Description Baseline Value for 
Parametric Studies 
H
m
  mm height of mass 2.025 
W
m
  mm width of mass 3.1 
t ?m thickness of device 20 
M  kg equivalent mass of sensor (including 
flexures) 
2.65e-7 
L
f 
 mm length of suspension flexure 1.495 
W
f
 ?m width of suspension flexure 12 
k  N/m total suspension stiffness 1.324 
w
L
 ?m width of latch cantilever 8 
l
L
 ?m length of latch cantilever 460 
r ?m radius of latch and mass contact 
surfaces 
40 
d
i
 ?m horizontal offset between latch and 
mass contact surfaces 
15 
m kg equivalent mass of latch 5.64e-11 
k
L
 N/m bending stiffness of latch cantilever 4.44 
k
c
 N/m contact stiffness 5.57e5 
y
0
 ?m initial sensor travel required to latch 150 
 
 114 
 
the dependence shown in Figure 4.15 is far more pronounced than when the mass 
alone was varied.  
0
50
100
150
200
250
300
0 0.5 1 1.5 2 2.5 3 3.5
Height of Mass H
m
 (mm)
Lat
c
hing T
hr
es
hold 
(
g)
0
0.07
0.14
0.21
0.28
0.35
0.42
Ma
ss (
m
g
)
Threshold
Mas s
 
Figure 4.14. Effect of vertical mass dimension on acceleration to latch. All other 
parameters are the same as those for the Design 2 sensor. 
 
0
100
200
300
400
500
600
700
800
00.511.522.533.5
Length of Flexure L
f
 (mm)
La
t
c
hing T
h
r
e
s
hold (
g
)
0
0.6
1.2
1.8
2.4
3
3.6
4.2
4.8
S
ti
ffn
e
s
s
 (
N
/m
)
Threshold
Flexure Stiffness
 
Figure 4.15. Effect of suspension spring length on acceleration to latch. All other 
parameters are the same as those for the Design 2 sensor. 
 
 
 115 
 
The stiffness of the flexures can also be changed by modifying the flexure 
width W
f
. The stiffness changes with the cube of the flexure width, so the acceleration 
to latch increases as the flexure width increases, as shown in Figure 4.16.  
The acceleration to latch is not only affected by the mass and stiffness of the 
sensor; the characteristics of the latch must also be considered since the interaction 
forces are non-trivial in magnitude. The stiffness of the latch is affected primarily by 
the length and width of the latch cantilever. As the stiffness of the latch decreases, the 
interaction forces decrease and the threshold acceleration consequently also 
decreases. Therefore, as the latch cantilever gets longer the threshold acceleration 
decreases (Figure 4.17). Similarly, as the latch cantilever width decreases the 
threshold acceleration decreases (Figure 4.18).  
0
50
100
150
200
250
0 3 6 9 121518
Width of Flexure W
f
 (um)
Lat
c
h
ing T
h
r
e
s
hold (
g
)
0
0.6
1.2
1.8
2.4
3
S
t
i
f
fne
s
s
 (
N
/m
)
Threshold
Stiffness
 
Figure 4.16. Effect of flexure width on acceleration to latch. All other 
parameters are the same as those for the Design 2 sensor. 
 
 116 
 
0
100
200
300
400
500
600
700
800
900
1000
0 200 400 600 800
Length of Latch l
L
 (um)
La
t
c
h
i
n
g
 T
h
r
e
s
h
old (
g
)
0
6
12
18
24
30
36
42
48
54
60
S
ti
ffn
e
s
s
 (
N
/
m
)
Threshold
Stiffness
 
Figure 4.17. Effect of latch length on acceleration to latch. All other parameters 
are the same as those for the Design 2 sensor. 
 
0
50
100
150
200
250
300
350
400
450
500
0 3 6 9 12 15 18
Width of Latch w
L
 (um)
La
t
c
hing T
hr
e
s
hold (
g
)
0
3
6
9
12
15
18
21
24
27
30
S
ti
ffn
e
s
s
 (
N
/m
)
Threshold
Stiffness
 
Figure 4.18. Effect of latch width on acceleration to latch. All other parameters 
are the same as those for the Design 2 sensor. 
 
Each of the trends presented here are obvious to the casual observer (e.g., as 
stiffness increases, the acceleration required to latch should also increase), but the 
exact relationships are not easily found by inspection or simple calculations. The 
 
 117 
 
parametric studies presented so far shed some light on the exact relationships of the 
design parameters, as well as the relative importance of one parameter as compared to 
the others. This is intended to aid in the design process ? in attempting to design a 
device that meets a certain threshold, one can refer to these relationships and 
determine which parameters to change. Some modeling will clearly still have to be 
undertaken for the best results, but the parametric studies should help decide where to 
start. 
Because the coefficient of friction can vary somewhat due to the fabrication 
process and over a number of cycles (see Section 3.3), the acceleration required to 
latch the device is plotted over a range of friction coefficients in Figure 4.19. The 
acceleration required to latch shows a strong dependence on friction coefficient, more 
than doubling over the range of 0.1 < ? < 0.8 (from 89 to 183 g). Because the friction 
coefficient can depend on the sidewall roughness (see Section 3.3 or [51]), good 
control over the fabrication process is critical with this particular shock sensor design 
0
50
100
150
200
0 0.2 0.4 0.6 0.8 1
Friction Coefficient 
?
P
r
edi
ct
ed T
h
r
e
shol
d
A
c
c
e
l
er
a
t
i
on (
g)
Design 2
 
Figure 4.19. Effect of friction coefficient on latching level for Design 2 sensor. 
 
 118 
 
to ensure that every device latches at the same threshold level. 
The bounce of the latch is an important aspect never reported before for 
latching shock sensors. The two degree-of-freedom model allows further study of the 
latch bounce phenomenon. The bounce dynamics are determined by the stiffness of 
the latch, the effective mass of the latch and the velocity of the mass when it hits the 
latch. Figure 4.20 shows how the magnitude of the first bounce (which is also the 
larrgest bounce) changes with changing stiffness and mass of the latch. The stiffness 
of the latch was changed by varying the width of the latch. The effective mass was 
calculated from the latch dimensions and the density of silicon, and assuming 
vibration in the first natural modeshape of a cantilever. The mass of the latch was 
changed directly to match each of the total effective masses calculated by changing 
the latch width (equivalent to adding a point mass at the end of the latch). When the 
Max
i
m
u
m
 Bounc
e
 (
?
m)
0.1
0.2
0.3
0.5
Latch Width (?m)
6789
0.6
10
0
0.4
42.3 49.3
5
56.4 63.4
5
70.5
Latch Effective Mass (pg)
Varying wl
Varying m, wl=8um
Varying w
L
, m=56.4pg
Varying m, w
L
=8?m
 
Figure 4.20. Effect of varying latch width and effective mass of latch on 
magnitude of bounce for Design 2 sensor. Applied acceleration pulse is 140 g for 
5.9 ms. 
 
 119 
 
latch width increases, the stiffness increases with the cube of the width and the mass 
increases linearly with the width. The result is that the bounces get smaller as the 
width of the latch decreases because the inertia decreases more slowly than the force 
keeping the latch in contact with the mass. When the mass of the latch increases while 
keeping the latch stiffness constant, the inertia increases and the force keeping the 
latch in contact with the mass remains the same, so the magnitude of the bounce 
increases. 
The model results outlined so far have assumed the only dissipative force is 
the friction in the contact between the mass and latch. There may in fact be some 
viscous damping due to air resistance or material damping due to stretching of the 
springs present throughout the entire sensor operation. This can be considered in the 
model by adding a constant damping term into the equation of motion for the mass 
(Eq. (4.26)): 
 
()
???
?
?
cos2)sgn(                                           
2
/
///
c
cm
ccmcmcmc
ky
y
kkyycyyM
&
&&&&&
?=
?
?
++++
 (4.32) 
 The effect of constant damping on the response of the mass is shown in 
Figure 4.21. As expected, the time to latch increases as the damping factor increases, 
and the acceleration required to latch also increases as the damping factor increases. 
The bouncing of the mass after latching is also dissipated when damping is added. 
The oscillations after latching decrease in frequency with increased damping because 
the velocity of the mass after latching decreases. 
 
 120 
 
zeta=0
zeta=0.1
zeta=0.2
zeta=0.3
M
a
ss P
o
si
t
i
o
n
 (
?
m)
0
25
50
75
100
125
Time (ms)
0 0.5 1 1.5 2 2.5 3
150
175
200
Latched 
position
zeta=0
zeta=0.1
zeta=0.2
zeta=0.3
 
Figure 4.21. Modeled response of shock sensor with various damping factors. 
 
4.6 Sensor Optimization Studies and Robust Design 
Various uncertainties in the properties of the fabricated devices can have a 
significant effect on the device performance. These uncertain properties can take the 
form of small deviations from the designed dimensions, uncertainty in the precise 
material properties, or in the case of the friction coefficient, variations in the surface 
roughness from device to device, wafer to wafer, or even changes over time due to 
wear, as illustrated by the measurements made with the friction test structure in 
Section 3.3 and the repetitive shock cycling tests of the sensor itself discussed in 
Section 3.1.1. The ideal way to account for variations and uncertainty is to create a 
robust design that exhibits very little variations in the performance over an expected 
range of a given parameter. This can be done through optimization of the design, with 
sensitivity to a particular parameter or set of parameters defined as the objective 
function. 
 
 121 
 
Given the observed dependence of latching threshold accelerometers on 
friction, and the variability in roughness (and therefore friction coefficients) of the 
sidewalls in the shock sensor design under consideration, optimization of  the design 
so that it is relatively insensitive to variations in the friction coefficient is examined in 
a case study. Similar optimizations can be performed with respect to other 
parameters, such as the width of the springs or latches. 
The optimization problem for friction sensitivity can be stated as a 
minimization of the slope of the latching threshold as a function of the friction 
coefficient, subject to the constraint that the latching threshold for a particular friction 
coefficient should not vary from the target threshold by more than an acceptable 
error. In mathematical terms, this can be written as: 
 
()?? ??
?
Target)Thresh(abs       s.t.
)][Thresh(     min
0
?
?
 (4.33) 
where the friction sensitivity is defined as: 
 
()
1
0.3Thresh
)Thresh(0.6
)][Thresh( ?=? ?  (4.34) 
To simplify the problem somewhat, the slope was evaluated by using only two 
different friction coefficients, 0.3 and 0.6. The constraint was evaluated using 
?
0
 = 0.3. The acceptable error ? between the target threshold and the actual threshold 
in an individual design was defined as 3% of the target threshold, or 3g for a target 
threshold of 100 g. For designs that differ by more than 3%, the constraint is not 
satisfied and the design is not considered feasible. 
 
 122 
 
Finding the threshold for an individual design is in itself an optimization 
problem. It is not tractable to come up with a closed form expression, and when the 
sensor either latches or does not latch, the gradient is undefined. Therefore a simple 
guess-and-test search algorithm was written to find the latching threshold. The 
algorithm is shown as a block diagram in Figure 4.22. Bounds on the search space are 
established, and the position/time response of the device is calculated by using the 
two degree-of-freedom model presented in Section 4.2.2. If the model predicts that 
the device will latch at the guessed acceleration value, the guess value is decreased 
and the simulation is run again. If the model predicts that the device will not latch, the 
guess value is increased and the simulation is run again. This process is repeated 
iteratively, updating the minimum and maximum bounds on the search at each step 
Latched? yn
Update max 
thresh
Update min 
thresh
Increase 
guess
Run 
simulation
Establish Min/Max 
Threshold Bound 
and initial guess
Decrease 
guess
 
Figure 4.22. Algorithm to find the acceleration threshold required to latch any 
individual design. 
 
 123 
 
until the difference between the minimum bound and maximum bound reaches an 
acceptable value (0.1g for the studies presented here). The final value of the 
maximum bound is used as the threshold acceleration required to latch, since the 
device is predicted to latch at this value and predicted not to latch just below this 
value. 
The independent design parameters used are as shown in Table 4.3. There are 
6 independent design variables here. Some other variables could be chosen, including 
the radius of the latch r, the initial horizontal offset d
i
, and the total distance to latch 
y
0
. Only six parameters were chosen to keep the problem somewhat tractable without 
limiting the results any more than necessary. Even so, this represents a tremendous 
number of potential designs, far more than could ever be experimentally investigated. 
This of course is the point of numerical optimization studies ? to identify potentially 
promising designs in a large design space without having to make and experimentally 
study each design. 
The difficulty of the latching shock sensor optimization problem is that 
finding the latching threshold is an iterative process that requires guessing an 
acceleration and solving the model to determine whether the sensor will latch. The 
Table 4.3. Design parameters used for optimization studies. All other parameters 
are the same as that for Design 2. 
Optimization 
Parameter 
Description Units Lower Bound Upper Bound 
t device thickness ?m 1 50 
L
f
 length of 
flexure 
mm 1 2 
W
f
 width of flexure ?m 3 20 
H
m
 height of mass mm 1 3 
w
L
 width of latch ?m 3 18 
l
L
 length of latch ?m 200 600 
 
 124 
 
result is binary: either the sensor latches or it does not. If it latches, the guessed 
acceleration is lowered and the process is repeated. If it does not latch, the guess is 
raised and the process is repeated, until the latching threshold is found within an 
acceptable error range. This is essentially a line-search technique where the gradient 
is zero everywhere except right at the latch threshold, where it is infinite. On a typical 
desktop computer, each  guess/test iteration of the model takes on the order of 40 
seconds, so the process of finding the latching threshold for any one design can take 
several minutes. Additionally, this optimization problem requires finding the latching 
threshold for two different friction coefficients. The computational time can therefore 
quickly become prohibitive on a desktop computer. Parallel processing, wherein 
multiple designs are evaluated simultaneously on separate processors, relieves this 
problem somewhat. 
Two MATLAB [58] optimization tools are used to solve the optimization 
problem, fmincon (function call: ?fmincon?) and the genetic algorithm tool (function 
call ?gatool?). Fmincon attempts to find the minimum of a nonlinear multivariable 
problem subject to linear and/or nonlinear constraints. Fmincon is a gradient-based 
technique, but the gradient of the objective function does not need to be known before 
beginning the optimization. The MATLAB function begins by changing each of the 
variables slightly to calculate the gradient, then moves in the most promising 
direction. With each step, the gradient is updated. Unfortunately, if the search reaches 
the boundary of the parameter space, the minimization is terminated rather than 
searching in another direction. This makes the success of the Fmincon optimization 
for this problem highly dependent on the user-specified initial point. This led to 
 
 125 
 
several unsuccessful optimization runs which took several hours to complete but did 
not significantly improve on the initial design. This is illustrated by the optimization 
run in Figure 4.23, solved by fmincon by using the Design 2 parameters as the initial 
point. Each successive point attempted by the fmincon solver is plotted in sequence. 
It is clear that the friction sensitivity does not improve, and the optimization 
terminated after trying 19 different points and reaching the edge of the parameter 
space. Most of the points attempted are not feasible (that is, the acceleration to latch 
with a friction coefficient of ? = 0.3 is not within 3g of the target acceleration to latch, 
100 g) and not a single point has measurably lower friction sensitivity than the initial 
design. 
Genetic algorithms are pseudo-evolutionary models used for optimization in 
cases when the gradient cannot easily be determined [59]. They are more of a brute-
force technique, and very computationally intensive compared to gradient-based 
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0 5 10 15 20
Optimization Step
F
r
ic
t
i
on S
ens
i
t
iv
i
t
y
Feasible Points
Infeasible Points
 
Figure 4.23. Fmincon optimization run using Design 2 as the initial point. 
 
 126 
 
techniques. Bounds are applied on the design variables and a population of multiple 
points is chosen inside the bounds. The objective function (and any constraints) must 
be calculated for every member of the population. The results are used to create a new 
population through mutation and cross-breeding. Each successive population is called 
a generation. The mutation and cross-breeding favor the fittest members of the 
population, theoretically improving the population with each successive generation. 
The quality of the results depends strongly on the population size and the number of 
generations used. The randomness of the cross-breeding and mutation help ensure 
that the optimization does not get stuck at local minima. 
Given a sufficiently high number of points in the population and enough 
generations, a genetic algorithm stands a very good chance of success even for 
problems (like this one) that are challenging for gradient-based techniques. The 
genetic algorithm was initially more successful than fmincon, although each 
generation of the genetic algorithm can take several hours to evaluate the constraints 
and objective function. The genetic algorithms were run using the vectorized 
population option, in which the genetic algorithm function passes the entire 
population to the objective function at once rather than in sequence. This allows the 
use of the MATLAB parallel for loop construct, in which multiple processing cores 
can be used to independently and simultaneously evaluate different members of the 
population. These simulations were run on the Army Research Laboratory?s (ARL) 
high performance computing center?s supercomputer. ARL?s MATLAB license has a 
limit of 8 processors, so this effectively speeds up the computations by a factor of 8. 
Even so, with a population of 100 (which may even be low given the 6 different 
 
 127 
 
independent design parameters used) each generation took over 12 hours to evaluate 
on the supercomputer. 
The results obtained for the first seven generations, with the initial population 
generated by the random uniform distribution function provided by MATLAB 
(genetic algorithm option CreationFcn set to @gacreationuniform), are shown in 
Figure 4.24. There was essentially one step improvement in friction sensitivity, from 
about 0.56 in generation 3 to about 0.19 in generation 4. This is somewhat of an 
improvement on the 0.3 friction sensitivity of the Design 2 sensor and that found by 
the first fmincon optimization run. 
Given the time required for the genetic algorithm runs, the author decided to 
try using the best point from the genetic algorithm as a starting point for another 
fmincon optimization rather than run the genetic algorithm for more iterations. (While 
it was unknown to the author at the time of running the simulations, the use of  a 
0
0.1
0.2
0.3
0.4
0.5
0.6
02468
Generation
F
r
ic
t
i
on S
ens
it
iv
it
y
 
Figure 4.24. Best value of objective function (friction sensitivity) from first 7 
generations of genetic algorithm by using 100 points in population. The first 
generation had no feasible points.  
 
 128 
 
genetic algorithm to find promising directions and then following it with the gradient-
based fmincon tool is now an automated option (beginning with MATLAB R14 
service pack 3). This is available by setting the Hybrid Function option of the genetic 
algorithm solver to fmincon). It had also become clear from studying the genetic 
algorithm results that the thickness of the sensor had no effect on the latch threshold 
or the friction sensitivity (this is because both the mass and stiffness scale linearly 
with thickness and effectively cancel each other out), so that parameter was removed 
from consideration. The results are shown in Figure 4.25. This time, the fmincon 
function succeeded at improving the friction sensitivity dramatically, from 0.19 to 
0.006. The final design is feasible and has almost negligible change in the latching 
threshold when the friction coefficient changes from 0.3 to 0.6 (98g to 98.6g). 
The parameter values for this optimized design are given in Table 4.4. The 
obvious change from Design 2 is the width and length of the latch cantilever ? both 
are taken to the edge of the defined parameter space to minimize the stiffness of the 
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0 1020304050
Optimization Step
F
r
ic
t
i
on S
ens
it
iv
it
y
Feasible Points
Infeasible Points
 
Figure 4.25. Fmincon optimization run using best point found from genetic 
algorithm as starting point. 
 
 129 
 
latch. This minimizes the interaction forces, including the frictional force, as 
compared to the restoring force of the spring. The stiffness of the suspension springs 
also increases even though the flexures get longer, because they also get wider. This 
increases the restoring force at a given deflection, further diluting the effect of the 
frictional forces on the latching threshold. The size of the mass is increased, which 
compensates for the increase in stiffness of the flexures and increases the inertia of 
the sensor before it hits the latches, making it more difficult for the latches to stop the 
sensor during the contact phase. Each of these changes to the design makes sense, but 
all of them combine to produce a design that has virtually no dependence on friction. 
The tradeoff is that the optimized design is about 30% wider and 10% taller than the 
Design 2 device. Also, the changes in the latch dimensions will make the latch more 
fragile and more likely to fracture during operation. Notably, neither the genetic 
algorithm nor the fmincon gradient-based optimization approach arrived at this design 
on their own ? it was only when the two approaches were combined that the 
optimization study led to a successful completion. 
Table 4.4. Optimized Design for low sensitivity to friction-coefficient changes. 
Optimization 
Parameter 
Description Units Design 2 
value 
Optimum 
Value 
t device thickness ?m 20 20 
L
f
 length of flexure mm 1.495 2 
W
f
 width of flexure ?m 12 20.5 
H
m
 height of mass mm 2.025 2.25 
w
L
 width of latch ?m 8 3 
l
L
 length of latch ?m 500 600 
M equivalent mass of sensor ?g 265 396 
k equivalent stiffness of 
flexures 
N/m 1.324 2.76 
m equivalent mass of latch ng 56.4 80 
k
L
 equivalent stiffness of 
latch 
N/m 4.44 0.1 
 
 130 
 
The changes in the threshold acceleration to latch were calculated for the 
optimized design over a wide range of friction values and compared to those for 
Design 2 as a baseline. The results are shown in Figure 4.26. It is clear from the graph 
that Design 2 is highly sensitive to changes in the friction coefficient, with the 
acceleration threshold changing by more than a factor of two over the range of 
0.1 < ?  < 0.8. The threshold for the optimized design changes by less than 2% 
(increasing from 98.2 to 99.8 g) over the same range. This optimized design will be 
fabricated and evaluated experimentally in future work. 
4.7 V-Beam Actuator Model 
The V-beam structure can be modeled by using the same basic procedure as 
that used for the offset beam actuator, by using the transverse component of the 
developed force and the actuator stiffness to find the actuator displacement. To 
develop an analytical expression for the actuator transverse stiffness, Castigliano?s 
0
50
100
150
200
0 0.2 0.4 0.6 0.8 1
Friction Coefficient 
?
P
r
edi
ct
ed T
h
r
e
shol
d
A
ccel
e
r
a
t
i
on (
g
)
Design 2
Optimized Design
 
Figure 4.26. Comparison between the sensitivity of Design 2 and optimized 
design to changes in the friction coefficient.  Note that optimized design has 
almost no change in threshold acceleration over a wide range of friction 
coefficients. 
 
 131 
 
theorem [60] is employed because the structure is statically indeterminate. 
Castigliano?s theorem can be summarized as the displacement at the point of 
application of a force in the direction of that force is equal to the partial derivative of 
the structure?s complementary energy with respect to the force. For Hookean 
materials, the strain energy is equal to the complementary energy, so the relationship 
becomes  
 
F
U
u
F
?
?
=  (4.35) 
where F is the applied force and u
F
 is the displacement at the point of application of F 
in the direction of F. Referring to Figure 4.27, the reactions can be solved by 
recognizing that the displacement and rotation at the clamped ends of the structure are 
equal to zero: 
 0)0( =
?
?
=
AX
F
U
u  (4.36) 
P
F
AY
F
BY
F
AX
F
BX
M
A
M
B
x
?
L
F
AY
F
AX
M
A
x
?
M(x)
V
N
 
 
Figure 4.27. Notation and geometry used for V-beam actuator stiffness model. 
 
 
 132 
 
The general form for the strain energy in a beam is: 
 dx
GA
V
AE
N
EI
M
U
L
? ?
?
?
?
?
?
?
?
++=
2
0
2
5
6
22
2
)(
22
2  (4.37) 
where M, N, and V are the internal moment, normal force, and shear in the beam, 
respectively, E is the Young?s modulus of the beam material, A is the cross-sectional 
area, I is the moment of inertia of the cross-section about the bending axis centroid, 
and G is the shear modulus of the material. Differentiating Eq. (4.37) with respect to 
the reaction force F
AX
 leads to 
 dx
F
V
GA
V
F
N
AE
N
F
M
EI
M
F
U
L
AXAXAXAX
? ?
?
?
?
?
?
?
?
?
?
+
?
?
+
?
?
=
?
?
2
0
5
6
)(
2  (4.38) 
The internal forces and moments can be determined as 
 
?
?
?
?
?
?
?
??+
??+
=
=
?
?
?
?
?
?
?
??
???
=
L/2x0              tan
2
L/2x0                        tan
2
)(
)(
LxL/2            
2
L/2x0         
2
)(
?(L-x)F(L-x)
P
-M
?xFx
P
-M
xM
FxN
P
P
xV
AXA
AXA
AX
 (4.39) 
The derivatives of the internal forces and moments can then be written as: 
   .0   ,tan   ,1 =
?
?
=
?
?
=
?
?
AXAXAX
F
V
x
F
M
F
N
?  (4.40) 
After substituting these relations into Eq. (4.38), and taking advantage of symmetry to 
avoid the discontinuity at x = L/2, it is found that  
 0tantan
2
1
2
2
0
=
?
?
?
?
?
?
?
?
+
?
?
?
?
?
?
+?=
?
?
?
dx
AE
F
xxFx
P
M
EIF
U
L
AX
AXA
AX
??  (4.41) 
 
 133 
 
Integrating and setting (4.41) equal to zero, a relation between the reaction moment 
and x-direction component of the reaction force can be obtained in terms of the 
applied axial force P.  
 
AE
LFLFPLLM
EI
AXAXA
2
tan
24
tan
48
tan
8
1
0
2
332
+
?
?
?
?
?
?
?
?
+?= ???  (4.42) 
The second equilibrium equation is obtained using the fact that the slope of the 
deflection profile must be zero at the clamped end. Applying Castigliano?s theorem 
again, leads to 
 0)0( =
?
?
=
A
M
U
?  (4.43) 
The derivative of the strain energy with respect to the reaction moment can be written 
as: 
 dx
M
V
GA
V
M
N
AE
N
M
M
EI
M
M
U
L
AAAA
? ?
?
?
?
?
?
?
?
?
?
+
?
?
+
?
?
=
?
?
2
0
5
6
)(
2  (4.44) 
From Eq. (4.38), the partial derivatives on the right-hand side of Eq. (4.44) are 
determined as  
 0   ,1   ,0 =
?
?
=
?
?
=
?
?
AAA
M
V
M
M
M
N
 (4.45) 
On substituting  Eq. (4.45) into Eq. (4.44), and solving Eq. (4.44) results in 
 
8
tan
162
0
22
?LFPLLM
AXA
+?=  (4.46) 
Combining the two equilibrium equations (4.46) and (4.42), the reactions can be 
obtained as 
 
 134 
 
 
A
I
L
PL
F
AX
96
tan2
tan
22
2
+
=
?
?
 (4.47) 
 
?
?
?
?
?
?
?
?
?
?
?
?
+
?=
A
I
L
PLLPL
M
A
96
tan2
tan
4
tan
8 22
2
?
??
 (4.48) 
Next, the transverse stiffness of the actuator is defined as 
 
()
2
L
w
P
k =  (4.49) 
Now, Castigliano?s theorem can be used one more time to obtain 
 () dx
P
V
GA
V
P
N
AE
N
P
M
EI
M
P
U
w
L
L
?
?
?
?
?
?
?
?
?
+
?
?
+
?
?
=
?
?
=
2
0
5
6
2
)(
2  (4.50) 
After substituting the reaction force and moment into the relations for the internal 
forces and moments, the partial derivatives on the right hand side can be written as: 
 
2
1
 
 ,
96
tan2
tan
 ,
96
tan2
tan
2
1
384
tan8
tan
8
22
2
22
22
22
23
?=
?
?
+
=
?
?
+
+?
+
?=
?
?
P
V
A
I
L
L
P
N
x
A
I
L
L
x
A
I
L
LL
P
M
?
?
?
?
?
?
 (4.51) 
After substituting, integrating, and solving, Eq. (4.50) becomes: 
 
()
GA
PL
A
I
L
L
AE
PL
A
I
L
L
EI
PL
P
U
L
w
20
6
96
tan2
tan
                                            
96
tan2
tan
2
1
48
2
2
22
2
2
22
223
+
?
?
?
?
?
?
?
?
?
?
?
?
+
+
?
?
?
?
?
?
?
?
?
?
?
?
+
?=
?
?
=
?
?
?
?
 (4.52) 
 
 135 
 
Solving for the stiffness from Eq. (4.49) leads to 
 
GA
L
A
I
L
L
AE
L
EI
L
A
I
L
L
k
20
6
96
tan2
tan
48
96
tan2
tan
2
1
1
2
22
23
2
22
22
+
?
?
?
?
?
?
?
?
?
?
?
?
+
+
?
?
?
?
?
?
?
?
?
?
?
?
+
?
=
?
?
?
?
 (4.53) 
For slender beams, the shear term can be neglected without substantially changing the 
results. For a straight beam (? = 0), if the shear term is neglected, this formula 
converges to the stiffness formula for a traditional clamped-clamped beam [57]: 
 
3
192
L
EI
k =  (4.54) 
 
By using typical parameters for the V-beam actuators used in this research 
(Table 4.5), the analytical stiffness values obtained from Eq. (4.53) closely match the 
finite-element predictions of the stiffness along the entire range of theta from 0 to 4 
degrees, and both converge to the clamped-clamped value as the angle is decreased to 
zero, as shown in Figure 4.28. The match is very good, with a maximum difference 
between the analytical predictions and finite-element stiffness calculations of 1.2%. 
Therefore Eq. (4.53) is validated over this range. This is the first reported analytical 
expression for the stiffness of these common actuators, and may help maximize 
efficiency.  
Table 4.5. Parameters used in FEA and numerical calculations 
Parameter Value 
P 
100?N 
L 
1200?m 
E 169e9 
b 
20?m 
t 
5?m 
 
 136 
 
0
50
100
150
200
250
300
01234
Bend Angle (degrees)
k (
N
/
m
)
Analytical Spring
Constant
FEA Spring Constant
 
Figure 4.28. Comparison of analytical and finite element spring constants for V-
beam actuator. 
 
4.8 Offset-beam Rotational Thermal Actuator Model 
The actuators under consideration for this dissertation work are electro-
thermal actuators. Current driven through the conductive actuator beams causes Joule 
heating, which induces thermal expansion. Through geometric constraints, the 
thermal expansion is amplified and converted into usable motion. The thermal 
actuators are to be modeled by using an analytical electro-thermal model to calculate 
the expected free thermal expansion followed by the use of a mechanical strain-
displacement model to convert this thermal expansion into actuator displacement.  
The electro-thermal model for the offset-beam rotational actuator is developed 
in a similar manner as in reference [42]. The geometry is shown in Figure 4.29. For 
the electro-thermal model, it is assumed that there is no temperature variation through 
the beam cross-section and that convection and radiation effects are negligible effects 
compared to conduction as has been demonstrated for other microscale thermal 
 
 137 
 
actuators [45]. It is further assumed that the anchors remain at the substrate 
temperature. The rotational actuator is modeled as a beam suspended over a substrate, 
for which the one-dimensional steady-state heat equation is given by 
 0
)()(
2
2
2
=
?
?+
?
gh
TxT
SkJ
dx
xTd
k
as
?  (4.55) 
 
where x is the variable along the direction of heat flow. The first term in Eq. (4.55) 
corresponds to heat flow into the element from the adjacent elements, the second 
corresponds to heat generation in the element via joule heating, and the final term 
corresponds to conductive heat flow from the element through the surrounding air 
into the substrate. The parameters in this equation are as defined in Table 4.6.  
The shape factor S in equation (4.55) describes the ratio of heat loss from the 
bottom of the beam to the sides of the beam. Examining a cross-section of a thermal 
actuator, the shape factor can be expressed in terms of the geometric parameters, the 
applied current and resistivity, and the temperature difference between the substrate 
and the beam cross-section as 
P
r
2
r
1
X
Y
L
 
Figure 4.29. Offset beam rotational thermal actuator parameter definition. 
 
 138 
 
 
()
?
?
=
TTg
hJ
S
FEA
?
2
 (4.56) 
An empirical expression for shape factor was extracted from finite element 
simulations by Lin and Chiao for a polysilicon beam suspended over a substrate in 
[61], but this does not scale up to beam heights of 10 ?m or more. Maloney used a 
similar technique to derive the following expression for shape factor for 50 ?m thick 
SOI beams [43]:  
 ( ) 1101
4
6
+?+=
?
g
w
S  (4.57) 
This expression does not include the beam height, and is therefore only useful 
for beams exactly 50 ?m height. The same technique has been applied in this 
dissertation work to develop a more general relationship that can be used for many 
different beam heights.  Beam heights of 10-50 ?m were used, along with gaps from 
0.25-3 ?m and beam widths of 2-20 ?m. The new shape factor approximation derived 
from this parametric study includes the beam height, and reduces to Maloney?s shape 
factor when a beam height of 50 ?m is used: 
Table 4.6. Electro-thermo-mechanical model parameter definition 
Parameter Definition 
? 
thermal expansion coefficient of silicon 
? 
electrical resistivity of silicon 
g Air gap between beam and substrate 
h height of beam 
J current density in beam 
k
a 
thermal conductivity of air 
k
s 
thermal conductivity of silicon 
L length of actuator 
S thermal shape factor 
T
?
 ambient temperature 
w width of beam 
 
 
 139 
 
 
 1
50
4
+
?
?
?
?
?
?
+=
h
g
w
S  (4.58) 
This relationship approximates the shape factor calculated from the finite-element 
analysis to within 5% for beam height to gap ratios in the range of 7 to 40 with less 
than 5% error (see Figure 4.30).  
Equation (4.55) must be solved iteratively because the thermal conductivity of 
silicon, the thermal conductivity of air, and the electrical resistivity of silicon are all 
temperature dependent. The thermal conductivity of silicon has previously been as a 
function of temperature is approximated in reference [62] as  
 
 [W/mK]   )(
28.12ln28.1 +?
=
T
s
eTk  
(4.59) 
 
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
- 20.00 40.00 60.00 80.00 100.00 120.00
h/g
Erro
r
h=20
h=10
h=30
h=50
 
 
Figure 4.30. Error in shape factor approximation with respect to finite-element 
calculations. 
 
 
 140 
 
A table with thermal conductivity of air at various temperatures is given in reference 
[63]. Applying a second-order polynomial fit to these values, the thermal conductivity 
of air can be approximated within 1% over the range of 100K to 950K as 
 
[W/mK]  101.05                                     
T10 9.62T103.06- )(
4-
-52-8
?+
?+?=Tk
a
 (4.60) 
The temperature dependence of the electrical resistivity of the low resistivity 
silicon used to make the actuators was measured using an offset-beam actuator. The 
device was wirebonded to an electronic package and placed on a hotplate. The 
temperature of the hotplate was increased gradually and the resistance of the device 
was measured at various temperatures. Because the temperature of the hotplate and 
the silicon are not the same, a thermocouple with digital readout was placed in 
contact with the top of the silicon chip. The temperature/resistance profile is shown in 
Figure 4.31. As can be seen from the chart, the first cycle displays a resistance drop 
0
50
100
150
200
250
300
0 100 200 300 400 500
Temperature (?C)
R
e
si
st
an
ce (
O
h
m
s)
Cycle 1
Cycle 2
 
Figure 4.31. Temperature dependence of resistance for an offset beam actuator. 
 
 141 
 
as the temperature increases up to 150?C, then the resistance begins to increase as 
expected for silicon. This is due to annealing of the contacts, decreasing the contact 
resistance. The resistance was measured during cooling of the hotplate as well, and 
the cooling trend follows a linear slope. A second cycle of heating and cooling 
follows the cooling trend of the first cycle very closely. The temperature coefficient 
of resistivity is found by using the device dimensions to convert the data to resistivity, 
then finding the slope of the temperature-resistivity plot. While the resistivity may 
vary somewhat from wafer to wafer, the temperature coefficient of resistivity is 
expected to be relatively stable. The temperature coefficient of resistivity found from 
this data was 4.7x10
-8
 ?-m/K. 
Without knowledge of the final temperature distribution, the precise values of 
the thermal conductivity of air and silicon as well as the electrical resistivity of silicon 
are unknowns; hence, equation (4.55) cannot be solved directly. An iterative solution 
procedure must be used by assuming an initial temperature distribution and 
corresponding values for the thermal conductivities, then calculating the temperature 
distribution by using those thermal conductivity values, updating the thermal 
conductivity values, and repeating this process until the temperature change from step 
to steps is negligible. A finite difference approximation to equation (4.55) which 
allows such an iterative solution procedure can be written as 
 
 
()
?
?+
??=?
?
?+
T
ghk
Sk
k
J
T
ghk
Sk
x
TTT
s
a
s
i
s
aiii
?
2
2
11
2
 (4.61) 
where the subscript i, i-1, and i+1 correspond to the current element, the previous 
element, and the adjacent elements. Equation (4.61) can be solved using the matrix 
 
 142 
 
inversion technique presented in reference [63] with the material properties updated 
at each element based on the temperature calculated for that element in the previous 
iteration.  A uniform temperature of 298K can be assumed for the first iteration. The 
convergence criteria is subjective, but a maximum temperature change of any element 
of less than 1x10
-3 
K from one step to the next is a reasonable criterion, and this is 
typically achieved in 6 to 8 iterations. 
The thermal expansion over an actuator beam is then calculated as 
 [],)()(
0
dxTxT
L
?
??=??  (4.62) 
where the thermal expansion coefficient of silicon, ?, is also a temperature-dependent 
material property  that can be approximated as [64] 
  
()
( ) []
-1101241088.56
K   10548.5110  725.3)(
3
TeT
T ?????
?+??=
?
?  (4.63) 
Because the temperature distribution is discrete due to the approximation of 
the beam as a set of elements, a trapezoidal rule approximation is used to evaluate the 
integral (4.62). The temperature of each element is used to assign an element-specific 
thermal expansion coefficient. Once the thermal expansion is known, an equivalent 
force which would produce the same increase in length can be determined as 
 
L
EA
F
?
=  (4.64)  
The actuator stiffness can be derived, and this force can be applied to 
determine the actuator displacement. For the rotational actuator, the stiffness is 
obtained by approximating the structure as two clamped-pinned beams connected at 
the pinned end. An additional stiffness term is used to account for the extension of 
each beam due to the vertical offset r
1
 (see Figure 4.32). The actuation of this device 
 
 143 
 
is assumed to be due to a pure moment due to symmetry; so the torsion stiffness is 
used. The torsion stiffness of a clamped-pinned beam is given by  
  
L
EI
k
cp
4
=  (4.65) 
The torsion stiffness term due to the extension of each beam segment is 
 
2
1
r
L
EA
k
ext
=  (4.66) 
The full actuator spring constant is obtained by taking the two spring 
constants in parallel for each segment, and realizing that the segments are also in 
parallel with each other the total torsion spring constant is 
 ( )
2
1
4
2
22 ArI
L
E
kkk
extcp
+=+=
?
 (4.67) 
If used as a translational actuator, the equivalent spring constant is given by 
 
?
2
2
r
M
x
F
k ==  (4.68) 
 ( )
2
1
2
2
2
2
4
2
ArI
Lr
E
r
k
k +==
?
 (4.69) 
=
2 x
r
1
 
Figure 4.32. Spring constant model for rotational actuator. 
 
 144 
 
The electro-thermal model can be modified for the vacuum case by 
eliminating the term representing the heat loss through the air into the substrate. 
Equation (4.55) then becomes 
 
.0
)(
2
2
2
=+ ?J
ds
sTd
k
s
 (4.70) 
The solution to (4.70) can be found by separating variables and applying the 
boundary conditions T = T
?
 at s = 0, L. 
 
?
++?= Ts
k
LJ
s
k
J
sT
ss
??
2
2
2
2
)(
 (4.71) 
The resulting thermal expansion and free angular deflection in each actuator 
beam under vacuum conditions become 
 
?
?
?
?
?
?
?
?
=
6
32
L
k
J
s
??
?  (4.72) 
 ( )
2
1
22
1
4
3
2
ArI
k
LJAE
r
s
+=
??
?  (4.73) 
For comparison, the temperature profiles for an actuator with L = 400 ?m, 
w = 5 ?m are shown in Figure 4.33 under vacuum and atmospheric conditions. The 
predicted maximum temperature is nearly 90 ?C higher in vacuum than at 
atmospheric pressure with the same applied current. This clearly will result in much 
larger actuator displacement if the actuator is used at low pressure (for instance, in a 
vacuum-sealed electronic package). 
 
 145 
 
4.8.1 Parametric Studies of Rotational Thermal Actuator 
To aid in device design, parametric studies were undertaken using the 
electrothermo-mechanical model developed above, including the temperature 
dependent material properties. The geometric parameters h, w, l, and r
1
 were each 
varied, and the results are presented here in terms of free rotation and blocked 
moment for a constant power consumption of 50mW. Figure 4.34 shows the effect of 
actuator width w and neutral axis offset r
1
 on the free rotation of the actuator. It can 
be seen that the actuator width w should be minimized within the constraints of the 
fabrication process in order to maximize the free deflection. This is expected because 
smaller widths correspond to lower actuator stiffness. There is also clearly an 
optimum value of the offset r
1
 once the actuator width w has been determined. This 
optimum value of r
1
 does not depend on the height or length of the actuator beams, 
and the optimum can be represented by a linear relationship between r
1
 and w (see 
Figure 4.35): 
T
e
m
p
er
at
ur
e (?C
)
Position (?m)
 
Figure 4.33. Vacuum and atmosphere temperature profiles for an actuator with 
L = 400 ?m, w = 5 ?m, and an applied current of 5 mA. 
 
 146 
 
 00105.0577.0
1
+= wr  
(4.74) 
There is no optimum length or height which maximizes free deflection for a 
given input power found within the parameter range used (Figure 4.36). L was varied 
from 100 ?m to 800 ?m and h was varied from 10 ?m to 50 ?m (corresponding to the 
 
Figure 4.34. Optimization of r
1
 for maximum free rotation, with h = 20 ?m and 
L = 400 ?m. 
 
 
Figure 4.35. Optimum value of  netrual axis offset r
1
 vs beam width w. 
 
 147 
 
height range over which the shape factor approximation was found to be valid). From 
anywhere in the parameter space, increasing the actuator length or decreasing the 
actuator height gives a corresponding increase in free deflection. These trends 
correspond primarily to decreasing the actuator stiffness, which is expected to result 
in larger free deflections. 
When optimizing for maximum blocked moment at constant driving power, 
the results are somewhat more complicated. An optimum length to height relationship 
is found which can be fit with a quadratic curve, but the optimum length/height 
relationship changes when the width is varied. The relationship is shown for one 
particular width value in Figure 4.37 and Figure 4.38. The blocked moment is also 
found to increase linearly with r
1
, which directly results from the increase in the 
moment arm of the actuator. 
 
Figure 4.36. Effect of L and h on free deflection, at constant 50 mW power. 
 
 148 
 
The possible failure modes of the actuator when overdriven are buckling (as if 
both beams are in line), fracture of the actuator beam due to exceeding the fracture 
stress, or plastic deformation at temperatures above about 550?C [65, 66]. In testing, 
free deflection was generally limited by buckling, as shown in Figure 4.39. Some 
 
Figure 4.37. Optimization of L and h for maximum blocked moment, with 
r
1
 = 5 ?m and w = 5 ?m. 
 
Figure 4.38. Optimum relationship between L and h for w = 5 ?m, r
1
 = 5 ?m. 
 
 149 
 
devices did fail with plastic deformation, as shown in Figure 4.40, but this typically 
occurred when the actuator was driven past the current required for maximum 
deflection. 
The maximum deflection that can be achieved as a function of beam width 
and length is shown in Figure 4.41. The fracture strength of silicon is taken to be 7 
GPa [67]. Plastic deformation is not considered because there is no reliable 
relationship between yield stress and temperature available. Fracture is expected to 
 
Figure 4.39. Rotational actuator limited by buckling. 
 
 
Figure 4.40. Plastically deformed rotational actuator. 
 
 150 
 
dominate over buckling at shorter actuator lengths and larger actuator widths, which 
are less susceptible to buckling. There is an optimum line on the plot that allows 
maximum deflection before failure if power is not a consideration. 
The maximum deflection that can be achieved is also plotted as a function of 
actuator width and beam offset r
1
 in Figure 4.42. L and h are held constant in this 
plot, and the predicted failure mode for the entire range of w and r
1
 is buckling. The 
optimum ratio of r
1
 to w is found to be 0.578. 
 
Figure 4.41. Maximum deflection achievable from a rotational actuator as a 
function of L and w, with r
1
 = 5 ?m and h = 20 ?m. 
 
 151 
 
 
Figure 4.42. Maximum deflection achievable from a rotational actuator as a 
function of r
1
 and w, with L = 500 ?m and h = 20 ?m. 
 
4.9 Summary and Contributions 
In this chapter, reduced-order models have been developed for the shock 
sensor, the reset actuators, and the friction measurement device. Two models have 
been presented for the shock sensor, a one degree-of-freedom model that is very 
efficient to solve, but one which is based on the assumption that the sensor moves 
smoothly past the latch without losing contact, and a more computationally intensive 
two degree-of-freedom model that allows for inertia of the latch and loss of contact. 
Both models attempt to account for interaction forces between the mass and latch, 
which have been neglected in the previous studies of this class of device [18-24]. 
Comparisons between the models illustrate that the single degree-of-freedom model, 
 
 152 
 
while more realistic than models used in previous work, does not allow for a full 
representation of  the dynamics of latching and the interaction forces. The prediction 
of loss of contact due to the inertia of the latch by the two degree-of-freedom model is 
an illustration of an aspect that cannot be captured with the one degree-of-freedom 
model.. This phenomenon has not been predicted before in latching acceleration 
switches because of the limitations of the single degree-of-freedom approach. 
Parametric studies of the effect of changing various design parameters on the 
two degree-of-freedom model predictions were also presented as a quick-reference 
tool for future designs of latching acceleration switches. An optimization approach 
was presented and used to optimize the design to reduce the sensitivity to changes in 
friction coefficient that can result from variations in the sidewall roughness of 
fabricated devices. The optimized design is predicted by the model to result in 
negligible changes in the acceleration threshold over a wide range of friction 
coefficients. This is expected to result in significantly more repeatable experimental 
results, which has been one of the barriers to carry out a wide study of latching 
MEMS devices. 
An electro-thermomechanical model was also presented for the rotational 
thermal actuator proposed in Section 2.2. This builds on previous models of thermal 
actuators by accounting for the temperature dependence of thermal expansion 
coefficient of silicon and by presenting a more general estimate of the shape factor 
that describes the loss of heat through the sides and bottom of the thermal actuators 
than presented in previous work [43, 61]. Parametric studies of the actuator design 
were also presented as a design tool for future versions of this type of actuator. 
 
 153 
 
Finally, a closed-form expression for the stiffness of a bent-beam actuator has 
been presented.  Since optimal actuator performance occurs when the stiffness of the 
actuator is matched to the stiffness of the resisting load, this is a valuable design tool. 
While these actuators are quite common in MEMS devices, a simple closed-form 
expression for the stiffness has never been reported before. 
Taken together, the models, parametric design and optimization studies 
presented in this chapter provide a complete framework for the design of MEMS 
latching shock sensor systems. 
 
 
 
 154 
 
5. Results and Discussion 
In this chapter, comparisons between the experimental measurements and 
model predictions are made for the shock sensor and the thermal actuator for the 
purpose of verifying the proposed models. Portions of this section have been adapted 
from the author?s journal articles on the latching MEMS shock sensor [38, 48] and 
the rotational actuator [39]. 
5.1 Shock Testing  Model Fit 
To explore the changes in the sensor response observed over multiple cycles, 
the average acceleration to latch measured over the first 50 cycles was compared to 
model results. There are no position time-histories for these measurements, and only 
the minimum acceleration required to latch the sensor was measured. The effective 
friction coefficient (?) in the model was changed to obtain the same threshold 
acceleration values observed in the experiments. All other parameters were kept fixed 
from cycle to cycle in the model, and the parameters used in the model are shown in 
Table 5.1. The two degree-of-freedom reduced-order model presented in Section 
Table 5.1. Parameter values used in simulations of  reduced-order model. 
Geometric Parameters Calculated Parameters 
t 20 ?m k (Design 1) 0.529 N/m 
w 11 ?m k (Design 2)  
w
L
 7 ?m k
L
 2.98 N/m 
L
f
 (Design 1) 1945 ?m M (Design 1) 342 ?g 
L
f
 (Design 2) 1495 ?m M (Design 2) 265 ?g 
L
L
 460 ?m m 97 ng 
r 40 ?m Material Properties 
d
i
 15 ?m E 169 GPa 
y
o
 150 ?m density 2.33 g/cm
3
 
n 5   
p 4  
 
 155 
 
4.2.2 was used to obtain the results presented in this section. 
The widths of the springs and latches is an important parameter because 
deviations have a significant effect on the results. The average width was found by 
matching the first cycle results for Design 1 and Design 2 devices from the same 
wafer, and was about 1?m narrower than the designed values. All of the other 
parameters values correspond to the design values. The resulting friction coefficient 
fits using these parameters are shown in Figure 5.1. 
For wafer 1, the effective friction coefficient fit using the Design 1 devices 
(average threshold data from Figure 3.7) decreased from 0.77 to 0.58 over the first ten 
cycles (a 25% decrease), and then increased to 0.65. The fit obtained by using the 
Design 2 devices from the same wafer (average threshold data from Figure 3.8) 
started at 0.76 and decreased to 0.72 after seven cycles (Design 2 devices were not 
tested beyond seven cycles because no change was observed in the average threshold 
acceleration). For Design 1 devices from wafer 2 (acceleration data from Figure 3.6), 
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 102030405060
Number of Cycles
F
r
i
c
ti
o
n
 C
o
e
ffi
c
i
e
n
t
Design 1 - Wafer 1
Design 1 - Wafer 2
Design 2 - Wafer 1
 
Figure 5.1. Friction coefficients found from model fit. 
 
 156 
 
the effective friction coefficient fit decreased from 0.76 to 0.55 over the first ten 
cycles.  
It is notable that the friction coefficients obtained in this way are nearly 
identical for Design 1 devices on wafers 1 and 2, indicating good repeatability from 
wafer to wafer. The values obtained for all devices also match reasonably well with 
the friction coefficient of 0.7 measured by using the friction test devices discussed in 
Section 3.3. The friction test devices were produced by using the same fabrication 
process, and hence,  it is expected that the friction coefficients would match. The fact 
that they actually do match is encouraging for verification of the model. 
The 25% decrease in the effective friction coefficient over the first ten cycles 
for Design 1 devices suggests that contact conditioning of about ten cycles might be 
required for consistent results from cycle to cycle with this design. This is not as 
important for Design 2 devices, which are less sensitive to changes in the friction 
coefficient. Furthermore, measures to eliminate the effect of stiction might be 
necessary for consistent results after 10 cycles. Since the stiction is most likely 
precipitated by electrostatic attraction, these could include increasing the gap between 
the device and the substrate, increasing the thickness of the device layer to increase 
the out-of-plane stiffness of the springs, or minimizing the effective substrate area 
under the mass by etching through holes in the substrate. 
5.2 Comparison of High-Speed Video Images with Model Predictions 
The primary validation of the proposed two degree-of-freedom shock sensor 
model involved matching the high speed video measurements of the position time-
history of the sensor presented in Section 3.1.2. Two different sensors were tested and 
 
 157 
 
filmed on high-speed camera, representing two different nominal threshold levels ? 
50 g (Design 1) and 100 g (Design 2). The only differences between the two designs 
are the length of the springs and the size of the mass. The width of the springs, the 
size and stiffness of the latches, and the travel required to latch the sensor are all 
identical. The video was captured at 4261 frames per second, which is the maximum 
framerate of our camera with a reasonable pixel resolution. 
The parameters used in the model predictions are the same as those used in 
Section 5.1 to extract the friction coefficients from the repetitive shock cycling 
experiments. The same 1?m spring narrowing was used for both the springs and the 
latch, with all other parameters kept as designed. The damping coefficients for the 
sensor and latch extracted from the harmonic measurements were also used in the 
model predictions. The friction coefficients extracted in Section 5.1 did not give a 
reasonable match to the experimental data, perhaps because the high speed video 
measurements were performed on devices from a different wafer. The model response 
was therefore obtained using different values of the friction coefficient for Design 1, 
and the closest match was obtained with a friction coefficient of ? = 0.1. This value 
was then also used for the Design 2 comparison of the model prediction to 
experimental results. 
The various frames from one high-speed video capture for the Design 1 sensor 
are shown in Figure 5.2 along with line drawings of the latch interaction generated 
using the two degree of freedom model results at the same timesteps. The timesteps 
are shown on the line drawings and correspond to the framerate of the high-speed 
video capture. An examination of the line drawings and the high-speed video frames 
 
 158 
 
shows a very good qualitative agreement between the two, including an identical 
number of frames with the mass contacting the latch (frames 9-11) before pushing 
past. 
 
2.347
2.112
1.878
1.643
1.408
1.173
0.939
0.704
0.469
0.235
0 ms
3.051
2.816
2.582
mass
latch
latch
mass
 
Figure 5.2. Time-stamped images generated from model results and from high-
speed video for Design 1 sensor. 
 
 159 
 
 
For a quantitative measure of the model performance, the model results were 
also compared with the relative motion of the mass extracted from the high-speed 
video. The results are shown in Figure 5.3 and Figure 5.4 for the Design 1 sensor and 
the Design 2 sensor, respectively. A friction coefficient of ? = 0.1 was used for the 
comparison. The model of the Design 1 sensor matches the high-speed video 
measurements very well. There are not as many data points from the high-speed video 
for the Design 2 sensor because the response is faster, and the match is not quite as 
good as the Design 1 sensor. However, the overall time to latch matches well and the 
predicted position before latching is fairly close to the high-speed video results, 
although outside the error bars for the third and fourth frames. Altogether this 
confirms that the model predicts the device performance very well. 
 
High Speed Video
Model Prediction
M
a
ss D
i
s
p
l
ac
e
m
ent (
?
m)
Time (ms)
01 23 43.52.51.50.5
0
25
175
75
100
125
150
50
200
-25
Latched 
Position
 
Figure 5.3. Comparison of model with high-speed video for Design 1 sensor. 
 
 
 160 
 
M
a
s
s
 D
i
s
p
l
ac
e
m
ent (
?
m)
Time (ms)
0122.51.50.5
0
25
175
75
100
125
150
50
200
-25
High Speed Video
Model Prediction
Latched 
Position
 
Figure 5.4. Comparison of model with high-speed video for Design 2 sensor. 
 
5.3 Experimental Confirmation of Latch Bounce Effect Observed in Model 
Regarding the loss of contact, or bounce of the latch predicted by the two 
degree of freedom model in Section 4.2.2, there is no confirmation of this in the high-
speed video measurements because the magnitude and duration of the bounces are too 
small to be captured (0.5 ?m and 7 ?s for the Design 1 sensor). The video resolution 
is about 2.5 ?m/pixel and the time between frames is 0.23 ms, both of which are 
much too large to enable visualization of any bouncing that might be occurring. 
It is possible to confirm bouncing electrically, however. Recall that in the 
standard design, the flat surface of the latch is coated with metal to give a low 
resistance contact after the device has latched (see Figure 3.17). The rounded surface 
is typically uncoated to ensure that the electrical circuit does not close until the device 
has fully latched. By switching this metallization scheme and coating the rounded 
latch surface with metal instead of coating the flat latch surface, physical contact can 
 
 161 
 
be electrically monitored during the phase when the mass is sliding past the latch. 
This was accomplished by constructing a voltage divider circuit similar to that used 
for the electrical characterization of the latch performance in Section 3.1.3. This time 
a 10k? resistor was used in place of the 1M? resistor. The resistor was connected to 
one latch and grounded at the other end. The other latch was connected to a 5V DC 
power supply and the voltage across the 10k? resistor was monitored on an 
oscilloscope at 10MHz sampling frequency. When the two curved surfaces of the 
latch and mass come into contact, the 5V source is connected to ground through the 
contact resistance of the latches and the 10k? resistor. The voltage drop across the  
10k? resistor depends on the latch resistance. The lower value resistor (10k? as 
opposed to 1M?) makes the output voltage more sensitive to contact resistance 
changes, to illustrate variations in the contact pressure. 
One representative result showing the measured acceleration pulse (low-pass 
filtered at 400Hz) and the output voltage from the contact monitoring signal is shown 
in Figure 5.5. In this plot, the acceleration profile is from the data acquisition system 
provided with the shock table because there is included signal conditioning for lower 
noise. The output voltage is taken from an oscilloscope because the sampling rate of 
the data acquisition system is only 40kHz. To reduce the amount of data from the 
contact monitoring circuit, the periods before first contact and after the last contact 
have been reduced to the zero in the plot. The arrows in Figure 5.5b indicate times 
where contact is established/re-established after a bounce. At least 6 events are 
detected, with two other possible events where the voltage is only slightly above the 
noise floor. In addition, two of the events (at 2.35 ms and 2.4 ms) contain multiple 
 
 162 
 
voltage oscillations even though the voltage never goes to zero in between. 
Qualitatively, these results confirm both the model predictions of latch bounce with 
loss of contact and contact force oscillations without loss of contact. 
5.4  Rotational Thermal Actuator Results 
The electro-thermal model predictions determined on the basis of Eqs. (4.62) 
and (4.68-4.69) are plotted along with the experimental data in Figure 5.6 and Figure 
5.7, and there is good agreement in all cases. The model predictions are compared 
-10
0
10
20
30
40
50
60
2.25 2.3 2.35 2.4 2.45 2.5 2.55
Time (ms)
Ac
c
e
l
e
ra
t
i
o
n
 (g
)
-1
0
1
2
3
4
5
6
O
u
t
put
 V
o
lt
age (
V
)
Accelerometer
Contact Monitor
-10
0
10
20
30
40
50
60
012345678
Time (ms)
A
c
c
e
l
e
ra
t
i
o
n
 (g
)
-1
0
1
2
3
4
5
6
O
ut
p
ut
 V
ol
t
ag
e (
V
)
Accelerometer
Contact Monitor
b)
a)
 
Figure 5.5. Electrical monitoring of contact of latch for a Design 1 sensor: a) 
full time history, and b) expanded view of contact period. Positive spikes in 
voltage indicated the latch is in contact with the mass, zero voltage indicates 
no contact. Arrows indicate contact is reestablished after a bounce. 
 
 163 
 
with experimental results for 400 ?m actuator beam lengths with the three different 
actuator beam widths in Figure 5.6. The data for different actuator beam lengths with 
a fixed beam width of 7 ?m are compared to the model predictions in Figure 5.7. The 
experimental error is assumed as one-half of a vernier gradation, or 0.075 degrees of 
rotation. 
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
0 3 6 9 12 15 18 21 24
Current (mA)
R
o
t
a
t
i
o
n
 A
n
gle (
d
e
g
)
5?m exp
5?m model
7?m exp
7?m model
10?m exp
10?m model
 
Figure 5.6. Comparison of model free deflection predictions with experimental 
data: Actuators with L = 400 ?m and  varying actuator beam widths. 
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
0 3 6 9 12 15 18
Current (mA)
R
ot
at
i
on A
ng
l
e 
(
deg)
400?m exp
400?m model
500?m exp
500?m model
600?m exp
600?m model
 
Figure 5.7. Comparison of model free deflection predictions with experimental 
data for actuators with w = 7 ?m and  varying actuator beam lengths. 
 
 164 
 
The measurements match the model predictions fairly well. The experimental 
results deviate most from the model for the case when the beam width w is 10 ?m. 
This is likely because the measured actuator stiffnesses for 10 ?m beams do not 
match very well with the analytically determined spring constants (see Figure 5.8). 
The trends in Figure 5.6 and Figure 5.7 illustrate that, for a given applied current, the 
actuator rotation angle increases with decreasing beam width and increasing beam 
length. Both of these trends are tied to the decreasing actuator stiffness; this means 
that to minimize power and maximize deflection performance for low resisting loads, 
the actuator stiffness should be low. For the tested devices, the free deflection benefit 
of a less wide actuator is much larger than the benefit of a longer actuator. 
 
 
0.0E+00
2.0E-06
4.0E-06
6.0E-06
8.0E-06
1.0E-05
1.2E-05
0.0E+00 2.0E-06 4.0E-06 6.0E-06 8.0E-06 1.0E-05 1.2E-05
Kq (calc)
Kq (m
eas)
L = 400?m
L =500?m 
L =600?m
5?m 
width
7?m 
width
10 ?m 
width
M
eas
ur
ed K
?
(N?m
/
d
e
g
)
Predicted K
?
(N?m/deg)
 
Figure 5.8. Predicted versus measured torsional stiffness for the rotational 
offset-beam thermal actuators. 
 
 
 165 
 
5.5 Summary and Contributions 
Comparisons between experiments and model predictions have been presented 
in this chapter. First, the two degree-of-freedom model was validated by using 
repetitive shock cycling data and effective friction coefficients extracted from the 
model. The results showed good agreement between the effective friction coefficients 
for Design 1 and Design 2 devices on the same wafer, and between Design 1 devices 
from two different wafers. 
Next the high-speed video experiments were compared to model predictions 
with a good match for Design 1, and a reasonable match for Design 2 devices. The 
caveat is that the friction coefficients extracted by fitting the repetitive cycling 
acceleration threshold data were significantly higher than those that give a good fit to 
the high speed video measurements. It is possible that the surface roughness on the 
wafer used for the high speed video measurements was lower, resulting in a smaller 
effective friction coefficient. More work is needed to fully understand this 
discrepancy and whether it indicates variations in the fabricated devices or a 
limitation of the model itself. 
A qualitative experimental verification of the latch bounce/chatter effect 
predicted by the two degree-of-freedom model was also presented. Electrical 
monitoring of contact resistance on devices with metallization on the round contact 
surfaces was performed, and showed several successive contact/loss of contact events 
during the latching progression. This phenomenon, previously unreported in this class 
of device, was first predicted by the two degree-of-freedom reduced order model. 
Experimental verification provides justification for using at least two degrees of 
 
 166 
 
freedom in modeling latching acceleration threshold switches, because a one-degree-
of-freedom model simply cannot account for this type of behavior, nor the effect it 
has on the interaction forces and latching threshold levels. 
Finally, the measured free deflection of the novel rotational actuator design 
was compared to the prediction of the electro-thermomechanical model presented in 
Section 4.8. The model predictions agree very well with the experimental results. 
 
 167 
 
6. Concluding Remarks 
In this chapter, a summary of the work is presented, along with the 
contributions made to the field and suggestions for future research directions. 
6.1 Summary 
A uniaxial, bidirectional, latching, resettable microelectromechanical 
acceleration threshold switch has been studied in depth in this work. Rather than a 
device, this actually comprises a small-scale system with a sensor, latching 
mechanism, and reset actuators. Each component of the system has been studied 
experimentally and theoretically. Models for the operation of the various parts of the 
system have been developed and used to gain insight into the design and operation of 
this class of device. The body of literature on MEMS latching acceleration threshold 
switches is fairly thin, with only a handful of device reported in the literature. The 
literature that does exist contains very limited experimental data and overly simplistic 
models that neglect the interaction of the sensor with the mass in every case. 
The primary goal of this work has been to advance the understanding of the 
design and operation of this type of device. This has included extensive 
characterization of fabricated latching acceleration threshold switches, including the 
acceleration required to latch, changes to this threshold due to wear, and high-speed 
images of the latching event. It has also included lumped parameter models that 
capture the dynamics of the latching event and allow for insight into the contact 
forces and possibility of loss of contact during the latching event. The lumped 
parameter models have also been used in various parametric and optimization studies 
to gain insight into the design of latching acceleration threshold switches. As a 
 
 168 
 
byproduct of this work, a new type of rotational thermal actuator and a new friction 
measurement device suitable for friction measurements on deep reactive ion etched 
sidewalls have also been developed. 
6.2 Contributions 
The primary scholarly contribution of this work is a more in-depth 
understanding of latching shock sensors, with more thorough experimental data than 
ever presented before and a reduced-order model which for the first time allows 
detailed study of the shock sensor dynamics, including interactions with the latch. 
Along the way, the author has made various contributions to fabrication, design, 
testing, metrology, and modeling.  The author?s detailed contributions in each of 
these areas are as follows: 
In the area of fabrication,   the author?s contributions include the following: 
? Selectively patterned metallization on sidewalls of MEMS structures, used 
here to ensure high contact resistance before latching and low contact 
resistance after latching. This is a challenging process but the use of a thick 
negative photoresist, slow spin speeds, and increasing the exposure time by 
about 15% enabled photo-defined openings on 20?m sidewalls for sputter-
coating with metal. 
? A wafer level packaging process suitable for high aspect ratio structures while 
allowing wirebonding to the device contacts. This is a new idea which 
eliminates the need for filling of through-wafer vias for electrical contact to a 
devices encapsulated by wafer-bonding. 
 
 
 169 
 
In the area of actuator design and modeling,  the author?s contributions include the 
following: 
? Derivation of a closed-form expression for the stiffness of bent-beam 
actuators 
? Developing a novel rotational thermal actuator design and publication of 
experimental performance data 
? Parametric studies of the rotational thermal actuator design to show design 
tradeoffs and opportunities for improved performance 
? Developing a model for the rotational thermal actuator which incorporates 
temperature dependent thermal expansion coefficients and a new shape factor 
expression that fits a wider set of beam heights than previously published 
approximations, both of which can be applied to other types of thermal 
actuators 
 
In the area of MEMS metrology,  the author?s contributions include the following: 
? Developing a novel thermally-actuated MEMS friction test structure for 
measurement of friction on sidewalls of MEMS devices at high normal forces 
and demonstration of friction coefficient change due to wear of sidewall 
asperities from repetitive cycling 
 
In the area of latching acceleration sensor testing,  the author?s contributions include 
the following: 
 
 170 
 
? Publication of shock testing data of latching accelerometers for the first time, 
including acceleration thresholds required to latch sensors and changes to the 
thresholds over many cycles. These data enhance the understanding of the 
limitations of the device that can be used to improve the robustness of 
performance. For instance, the contacts should be conditioned by latching and 
unlatching the sensor a minimum of 10 times to allow the effective friction 
coefficient to settle in before the sensor is used. 
? High-speed video derived images of the sensor progression from resting state 
to latched under a shock pulse. This enhances the understanding of the 
dynamics of the mass/latch interaction, as well as the timing of the 
progression from resting, to contact, to latched, to closed electrical circuit 
(phases i-iv in Figure 4.1). 
 
Finally, in the area of latching acceleration sensor modeling,  the author?s 
contributions include the following: 
? Development of a two degree-of-freedom reduced order model for a latching 
acceleration threshold sensor which includes a separate degree of freedom for 
the mass and the latch, allowing in-depth study of the interaction of the mass 
and latch and the contact forces. Previously published papers either had no 
model at all or ignored the interaction forces for a very rough estimate of 
latching levels. 
? Use of the two degree-of-freedom model to predict ?bouncing? effect causing 
intermittent loss of contact between the mass and latch during latching. This is 
 
 171 
 
a new observation made possible by the two degree-of-freedom model, and 
subsequently the author experimentally confirmed this phenomenon by 
electrically monitoring a device with contact metallization on the rounded 
surface of the latches. 
? Use of the two degree-of-freedom model to optimize the design for reduced 
sensitivity to the friction coefficient. 
 
This work has also led to a total of seven conference and journal publications. 
In particular, two journal articles have been published in a leading MEMS journal, 
Sensors and Actuators A. In the first article [38] (published in 2008), the authors 
described the sensor design and fabrication process, introduced the single degree of 
freedom lumped model, and compared the model results to latching threshold 
measurements taken using a shock table. This was the first publication in the open 
literature of a model for this type of sensor which took into account the interaction 
forces between the sensor and the latch, albeit with a limiting assumption prohibiting 
loss of contact between the sensor and the latch. In the second paper [48], (accepted 
for publication in March 2010), the authors introduced the two degree of freedom 
model, eliminating this assumption and allowing study of the dynamics of the contact 
between the latch and sensor. The second paper also presented for the first time 
pictures of the progression from resting to latched state for this type of sensor, and 
compared the progression to that predicted by the 2DOF model. The author also 
published an article in the journal Advances In Science and Technology [39] on the 
rotational thermal actuator, presenting the design, numerical model, and test results 
 
 172 
 
from various configurations of the design. The author has also presented this research 
at 5 conferences, including the ASME International Mechanical Engineering 
Conference and Exposition, the International Workshop on Structural Health 
Monitoring, the Shock and Vibration Symposium, and the 3
rd
 International 
Conference on Smart Materials, Structures and Systems (CIMTEC 2008). Each of 
these presentations was also accompanied by a manuscript published in the respective 
proceedings [39, 51, 49, 50, 68]. 
6.3 Possible Improvements and Future Directions 
The shock sensor itself may be improved in a few ways. Most notably, stiction 
failures are common if the sensor is exposed to moderate or high humidity levels for 
extended periods of time. This can be alleviated either by including a hermetic seal, 
as in the packaging process presented in Section 0, or by coating the sensor with a 
hydrophobic material. While the packaging process has been developed, the 
hermeticity has not been tested, nor has the influence of the package on the operation 
of the sensor (if there is any). For instance, the amount of gas in the package and the 
confinement might affect the damping factors for the sensor and/or latch. Coating 
with a hydrophobic material has been done for many other MEMS devices [69], and 
is fairly well understood, but this device is more susceptible to stiction given the large 
surface area of the mass, the small gap between the mass and the substrate, and the 
long, compliant suspension springs. 
The reset operation of the shock sensor makes use of thermal actuators. These 
could be replaced by in-plane piezoelectric actuators, which draw far less current (nA 
as compared to mA). The fabrication challenges with integrating a piezoelectric 
 
 173 
 
material are nontrivial in general, and compounded in this case by the hydrofluoric 
acid release process. Hydroflouric acid aggressively attacks the most attractive 
piezoelectric actuator material, PZT. ARL has demonstrated fabrication and release of 
piezoelectric actuators on SOI in a process compatible with the shock sensor, but the 
shock sensor release process is much longer and this will make protecting the PZT 
more challenging [70]. 
Shock sensor designs for other acceleration threshold levels should also be 
studied. The dynamics of the latching event are likely to be qualitatively different for 
much higher or much lower acceleration levels. 
The response of the shock sensor to different shape acceleration pulses is also 
an avenue of future research. Impulse, half-sine, step function, or triangular 
acceleration profiles could be studied experimentally and through the sensor model. 
Modifying the shape of the acceleration profile for the model is very simple; although 
possible, it is considerably more involved to obtain different acceleration profiles 
with a shock table  [71]. 
This work has treated the acceleration pulses as being applied precisely along 
the axis of the sensor. How the sensor responds to both cross-axis shocks and slight 
variations in the angle of the applied shock from the sensor axis is an important topic 
to investigate because the sensor is likely to encounter both of these in real-world use. 
The model could be extended to include two or even three degrees of freedom for the 
mass and for the latch to study these effects from a theoretical perspective. From an 
experimental perspective, variable angle jigs or mounting blocks could be constructed 
to change the angle of the sensor with respect to the shock table.  
 
 174 
 
Further experimental study of latch bouncing can be conducted by using the 
devices with the metallization on the rounded edge of the latch, monitoring the 
resistance between the mass and latch during an acceleration event. Variations in the 
contact resistance are expected to track the contact pressure, and when contact is 
actually lost, the contact resistance increases dramatically. Further study of the time 
in contact, the time between bounces, and the contact pressure as it relates to the 
contact resistance could yield further insight into the validity of the model 
predictions. The metal coating on the latch will change the contact stiffness and 
friction coefficient, however, so the results may not be conclusive. 
Finally, another useful optimization study would be a constrained multi-
objective optimization of the friction sensitivity and the area footprint of the sensor, 
since it was found in Section 4.6 that the changes that reduce the friction sensitivity 
tend to increase the physical size of the sensor. Understanding the tradeoff completely 
requires knowledge of the Pareto front for these two objectives, with the constraint of 
latching threshold level being within some acceptable range. 
 
 175 
 
A. Appendix 
The fabrication process flow and detailed recipes to make the device is 
presented as Appendix A.1. The various MATLAB scripts used to run models and 
simulations throughout this work are provided in Appendix A.2. The ANSYS scripts 
used for the shape factor calculation for the rotational thermal actuator model and the 
contact stiffness calculation for the two degree-of-freedom sensor model are 
presented as Appendix A.3. 
A.1 Fabrication Details 
A.1.1 Detailed Fabrication Process Flow 
 
 The following is the fabrication process flowsheet used to track the fabrication 
process. It includes every step used in the fabrication of the shock sensors, as 
discussed in Section 2.3. 
 
Device Wafer Process: 
? Bondpad Metal:              Completed by:        Date:  
1. Wafer lot number: _____________ 
Wafer number (scribe on back):_______ 
  
2. Measure:  Resistivity (thickness = 20um): 
______________?-cm 
  
3. Photo 1 (Bondpad metal) AZ 5214 resist Reverse 
Image - Basic Wafer Flat Alignment 
  
4. Descum in Metroline M4L oxygen plasma 
(5214.RGP.descum.5min) 
  
5. 1 minute 6:1 buffered oxide etch immediately 
before loading into evaporator 
  
6. E-beam evaporation for bondpads in CHA SEC-
600-RAP (400? Cr/ 1000? Pt/ 4000? Au) 
  
7. Liftoff in Baker PRS-3000 photoresist stripper at 
80?C 
  
 
 
 176 
 
? Waferbond Ring:   
8. Photo 2 (Bondring metal) Futurex NR5-8000 
negative resist (1000rpm spin, 6.8s exposure time, 
~40sec develop) ? alignment to layer 1 
  
9. Descum in Metroline M4L oxygen plasma 
(5214.RGP.descum.5min) 
  
10. E-beam evaporation for bondrings in CHA SEC-
600-RAP (200? Cr/ 1.5um Au) 
  
11. Liftoff in Futurex RR4 photoresist stripper at 
110?C 
  
 
? Device DRIE etch 
12. Photo 3 (Device DRIE etch) AZ 5214 ? 
alignment to layer 1 
  
13. Descum in Metroline M4L oxygen plasma 
(5214.RGP.descum.5min) 
  
14. DRIE in VLR-700 (ARL_Via for ~14+ min)   
15. Strip P/R with Acetone/IPA/DI water and finish 
with Metroline M4L 5214.RGP.5min ash 
  
 
? Cleave Assist Streets Etch 
16. Photo 5 (Cleave streets DRIE) Futurex NR5-8000 
resist on backside of wafer (front/back alignment) 
align to layer 1 
  
17. Etch in VLR for 110 minutes (ARL_Via_Hold)   
18. Strip photoresist in Acetone/IPA/DI rinse   
19. Metroline M4L oxygen plasma ash to remove 
residue if necessary (5214.RGP.10min) 
  
 
? Latch metallization 
20. Photo 4 (Latch metal) Futurex NR5-8000 resist ? 
align to layer 1 
  
21. Sputter Cr/Au in CVC (200A/4000A) ? 2.8min for 
Cr, 34 min for Au 
  
22. Liftoff in Acetone ? do not use ultrasonic agitation   
23. Oxygen plasma strip ? 5min to get rid of any 
photoresist residue 
  
24. Anneal on hotplate to improve contact resistance  ? 
ramp to 400C and hold for 1min on hotplate 
  
25. Keep in nitrogen drybox and in conductive wafer 
carrier from this point on 
  
 
? Vapor HF release 
26. Release in Primaxx MEMS-CET Vapor HF Etcher 
?start with ARL Clear4 & UC4 recipe, continue 
with more undercut steps if necessary 
  
 
 177 
 
27. Inspect under infrared microscope for full release   
 
Wafer Bonding Process: 
? Wafer Standoff:      Completed By: Date:  
1.  Begin with double-side polished silicon wafer   
2. Deposit isolation oxide in Plasmatherm 790 
PECVD reactor (12:30 ~ 5000A) 
  
3. Photo 6 (bond ring) with Futurex NR5-8000 
photoresist) 
  
4. Descum in Metroline M4L Oxygen plasma 
(5214.RGP.descum.5min) 
  
5. E-beam evaporation of bondring in CHA evaporator 
    (200? Cr/ 500? Au/1.5um AuSn, 500? Au) 
  
6.  Liftoff in Futurex RR4 photoresist stripper at 110?   
 
? Silicon Cap Wafer Via:  
7.  Photo 7 (Via step 1) Futurex NR5-8000 resist on 
backside of cap wafer (front/back alignment) align 
to layer 1 
  
8.  DRIE in Unaxis VLR-700 (ARL_Via) for 70min 
(timed etch ? approx 200um)  
  
9.  Strip Resist (Acetone/IPA/DI water, followed by 
Metroline M4L oxygen plasma 5214.RGP.5min if 
necessary) 
  
10  Spin AZ9245 photoresist on backside of capwafer 
(2500 rpm) and softbake, then spin AZ9245 on 
front of capwafer (2500rpm) and softbake 
supported by scrap pieces to keep backside resist 
from sticking to hotplate 
  
11  Photo 8 (Via step 2) AZ9245 resist, 2500rpm 
process, expose for 12sec (align to layer 1) 
  
13  DRIE (VLR ? ARL_Via) for 50min (3,000s) ? 
check for through etch completion under 
microscope 
  
14  Strip resist ? Acetone and IPA, followed by 
Metroline M4L oxygen plasma (5214.RGP.5min) 
if necessary 
  
 
? Wafer Bonding    
15  Align patterned silicon cap wafer to SOI device 
wafer in Karl Suss MA/BA6 ? silicon/silicon 
program  (device wafer loaded first, then cap 
wafer) 
  
16  Bond patterned silicon cap wafer to top of SOI 
device wafer in Karl Suss SB6 Wafer Bonder 
(AuSn_315C_10min_50vac.rcp) 
  
 
 178 
 
 
A.1.2. Process Recipe Details 
? Photolithography Recipes: 
? AZ 5214 (~2 ?m thick) 
? Positive Tone 
Spread: 500rpm for 5 seconds 
Spin: 2000 rpm for 40 seconds 
Softbake: 110 C for 60 seconds on hotplate 
Exposure: 2.3 seconds at 1000mJ/cm
2
 
Develop: 80 seconds in AZ300 developer 
? Negative Tone 
Spread: 500rpm for 5 seconds 
Spin: 2000 rpm for 40 seconds 
Softbake: 110 C for 60 seconds on hotplate 
Exposure: 2.3 seconds at 1000mJ/cm
2
 
Post-Exposure Bake: 30 seconds at 120C on hotplate 
Flood Exposure (no mask): 2.8 seconds at 1000mJ/cm
2
 
Develop: 80 seconds in AZ300 developer 
 
? Futurex NR5-8000 (~15 ?m thick ) 
? Standard Process 
Spread: 500rpm for 5 seconds 
Spin: 1000rpm for 40 seconds 
Softbake: 110C for 20 sec, followed by 150C for 60 sec 
Exposure: 6.8 seconds at 1000mJ/cm
2
 
Post-exposure bake: 110C for 120 seconds on hotplate 
Gradual cool: 120 seconds on pyrex dish (to keep resist from 
cracking if it cools too fast 
Develop: 40 seconds in Futurex RD6 developer 
 
? Latch Metallization (over 20 ?m topography) 
Same as above except use 7.8 second exposure 
  
? Etch Recipes: 
? Primaxx MEMS-CET Vapor HF etch recipe 
Clear Cycle 
(to remove exposed oxide 
slowly) 
Pressure: 125 T 
HF flowrate: 250 sccm 
Methanol flowrate: 
500 sccm 
N2 flowrate: 1050 sccm 
Undercut Cycle 
(to undercut quickly) 
Pressure: 125 T 
HF flowrate: 600 sccm 
Methanol flowrate: 
450 sccm 
N2 flowrate: 1000 sccm
 
 179 
 
 
? Unaxis VLR-700 ICP DRIE 
Etch Step 
Pressure: 25 mT 
SF6 flowrate: 100 sccm 
Ar flowrate: 40 sccm 
Plasma power: 850 W 
Platen Power: 16W 
 
Deposition Step 
Pressure: 20 mT 
C4F8 flowrate: 70 sccm 
Ar flowrate: 40 sccm 
Plasma Power: 850 W 
Platen Power: 1 W 
 
 
? Silicon Dioxide RIE 
Pressure: 5 mT 
CHF3 flowrate: 5 sccm 
CF4 flowrate: 15 sccm 
He flowrate: 28.3 sccm 
Plasma Power: 500 W 
Platen Power: 5 W 
 
? Deposition Recipes: 
? CHA SEC-600-RAP E-beam evaporator: 
o  Metal evaporation rates: 
Chromium: 1 ?/s 
Platinum: 2.5 ?/s 
Gold: 4 ?/s 
AuSn (80/20): 5 ?/s 
? CVC Sputtered gold (3? target) 
Power Ramp: 100W/minute 
Power: 100W 
Ar flow: 45sccm 
Rotation: 6 rpm 
 
? Wafer-bonding recipe: 
Pump/purge chamber 3x 
Backfill with H
2
N
2
 (forming gas) to 50 Torr 
Remove spacers to place wafers in contact 
Tool Pressure: 4000 N 
Heat to 315C for 10 minutes 
Cool to <200C before removing from chamber 
 
 
 180 
 
A.2 MATLAB Scripts 
A.2.1 Rotational Actuator MATLAB Model 
%therm_rot_iter2.m 
%12/29/09 
%calculates temperature profiles and free deflections for rotational  
%offset actuator, updating thermal conductivity and iterating 
  
clear all 
curr=linspace(0,12e-3,13);  %input current 
p=(5.00e-3)/100;            %room-temp resistivity 
  
%geometric properties 
L=600e-6;               %length of one beam - total length = 2L 
x=linspace(0,2*L);      %define grid of nodes 
g=2e-6;                 %gap between beam and substrate 
w=7e-6;                 %width of beam 
h=20e-6;                %height of beam (i.e., thickness of SOI) 
A=w*h;                  %cross-sectional area of beam 
I=1/12*h*w^3;           %bending moment of inertia 
offset=5e-6;            %offset between each beam and pivot point 
  
%material properties      
E=169e9;                %Young's modulus 
  
%S=4/w*(1e-6+g)+1;       %Maloney's shape factor 
S=4/w*(g+h/50)+1;        %Currano's shape factor 
T_inf=298;              %boundary temperature 
  
%calculate spring constant of actuator 
k=2*E/L*(4*I+A*offset^2); 
  
%convert current to current density in A/m^2 
J=curr/A; 
  
%Step through various current densities 
for j=1:length(J) 
    N=length(x)/2;      %N = number of elements in single beam 
    T=ones(N,1)*T_inf;  %initialize temperature of each element 
    Told=zeros(N,1); 
    %initialize matrix 
    B=2*eye(N); 
    for r=1:N-1 
        B(r,r+1)=-1; 
        B(r+1,r)=-1; 
    end 
    B(N,N-1)=-2; 
  
    dx=max(x)/(length(x)-1); 
    %loop through iterative solution procedure 
    iter=0; 
    while abs(max(T-Told))>1e-3 
        iter=iter+1 
 
 181 
 
        Told=T; 
  %element-wise temp-dependent coefficients of thermal       
  %conductivity: 
        ks=exp(-1.28*log(T)+12.28); 
        ka=-3.06e-8*T.^2+9.62e-5*T+1.05e-4; 
        %element-wise temp-dependent resistivity 
        rho=p+4.7e-8*(T-T_inf); 
        %finite difference soln to heat Eq 
        C=B.*repmat(ks,1,N)/dx^2+S*diag(ka)/g/h; 
        D=S*ka*T_inf/g/h+J(j)^2*rho; 
        D(1)=D(1)+T_inf*ks(1)/dx^2; 
        T=C\D;      %solve for temperature distribution 
    end 
    average_resistivity=mean(rho) 
    max_temp=max(T); 
  
    %calculate thermal expansion in beam, summing over all elements 
    delta=0; 
    total_resistance=0; 
    for s=2:N 
        %temperature in element assumed to be average of temperature  
        % at the nodes: 
        avg_T=(T(s)+T(s-1))/2; 
        %temperature dependent coefficient of thermal expansion 
        alpha=(3.725*(1-exp(-5.88e-3*(avg_T-124)))+? 
5.548e-4*avg_T)*1e-6; 
        delta=delta+alpha*(x(s)-x(s-1))/2*((T(s)-T_inf)+? 
(T(s-1)-T_inf)); 
        elem_resistance=(p+4.7e-8*(avg_T-T_inf))*dx/A; 
        total_resistance=total_resistance+elem_resistance; 
    end 
  
    %Calculate force/moment from thermal expansion 
    F=delta*E*w*h/L; 
    M=2*F*offset; 
  
    %calculate rotation angle in degrees 
    theta(j)=M/k*180/pi; 
end 
theta' 
A.2.2. 1DOF Shock Sensor MATLAB Model 
%predicts transient response of latching shock sensor to steady 
%acceleration beginning at time = 0 
%does not handle impulse acceleration (i.e., acceleration pulse 
%lasts for shorter than 1/2 of device period 
function shock_sensor 
clear all 
format short e; 
hold off; 
  
global chip_accel k m kl ybar r di mu yc; 
  
%initialize global variables to describe system 
%75g sensor 
 
 182 
 
%chip_accel=-103.7*9.81; 
%k=1.324; 
%m=2.65e-7; 
  
%37.5g sensor 
chip_accel=-63*9.81; 
k=0.601; 
m=3.42e-7; 
  
kl=4.44; 
ybar=150e-6; 
r=40e-6; 
di=15e-6; 
%no friction 
%mu=0; 
%best fit friction coefficient 
mu=0.29; 
  
yc=91.9e-6;         %distance to initial contact 
yc2=ybar;           %distance to latch/distance till contact is lost 
  
%define the frequency and period of the basic spring-mass system 
freq=sqrt(k/m)/(2*pi); 
period=1/freq; 
  
%Solve differential equation for period before contact 
%set options to find where contact occurs and stop integration 
refine=20; 
options=odeset('Events',@events,'Refine',refine); 
initial=[0;0] 
[T1,Y1,TE1,YE1,IE]=ode45(@shock_init,[0 period],initial,options) 
subplot(2,1,1),plot(T1,Y1(:,1)); 
%subplot(2,1,2),plot(T1,zeros(length(T1),1)*chip_accel/9.81); 
  
hold on; 
  
%Solve differential equation for period while mass is in contact  
%with latch 
%find where contact ends and stop integration 
yc=yc2; 
options=odeset('Events',@events,'Refine',refine); 
initial2=[YE1(1,1);YE1(1,2)]; 
[T2,Y2,TE2,YE2,IE2]=ode45(@latchfr,[TE1 
period+TE1],initial2,options) 
subplot(2,1,1),plot(T2,Y2(:,1),'g:'); 
%subplot(2,1,2),plot(T2,ones(length(T2),1)*chip_accel/9.81); 
  
%Solve differential equation for period after mass breaks contact  
%with latch 
initial3=[YE2(1,1);YE2(1,2)] 
[T3,Y3]=ode45(@shock_init,[TE2 period+TE2],initial3); 
subplot(2,1,1),plot(T3,Y3(:,1),'r'); 
%subplot(2,1,2),plot(T3,ones(length(T3),1)*chip_accel/9.81); 
  
%Labels and legends for graph of data 
 
 183 
 
title('Displacment vs. Time of Shock Sensor') 
xlabel('Time (s)') 
ylabel('Displacement (m)') 
legend('pre-contact','contact','post-contact') 
  
%------------------------------------------------------------------- 
function dy=shock_init(t,y) 
  
%this function defines the differential equation before the shock  
%sensor makes contact with the latch 
%y(1) is the displacement, y(2) is the velocity 
%ypp is the acceleration function, F is the mass-latch interaction  
%(friction & normal force) 
  
global chip_accel k m; 
  
F=0; 
ypp=-chip_accel-k*y(1)/m-2*F/m; 
  
dy=[y(2);ypp]; 
  
%-------------------------------------------------------------------  
function dy=latchfr(t,y) 
  
%this function defines the differential equation while the shock  
%sensor is in contact with the latch 
%y(1) is the displacement, y(2) is the velocity 
%ypp is the acceleration function, F is the mass-latch interaction  
%(friction & normal force) 
  
global chip_accel k m kl ybar r di mu yc; 
  
yl=ybar-y(1);                     %remaining travel 
theta=asin(yl/(2*r));            %contact angle 
F=2*kl*(2*r*cos(theta)-r-di)*(sin(theta)+? 
   mu*cos(theta))/(cos(theta)-mu*sin(theta)); 
  
ypp=-chip_accel-k*y(1)/m-2*F/m; 
  
dy=[y(2);ypp]; 
  
%------------------------------------------------------------------- 
  
function dy=shock_final(t,y) 
  
global chip_accel k m; 
  
%this function defines the differential equation after the shock  
%sensor leaves contact with the latch 
%y(1) is the displacement, y(2) is the velocity 
%ypp is the acceleration function, F is the mass-latch interaction  
%(friction & normal force) 
  
 
 184 
 
F=0; 
ypp=-chip_accel-k*y(1)/m-2*F/m; 
  
dy=[y(2);ypp]; 
  
%------------------------------------------------------------------- 
  
function [value,isterminal,direction]=events(t,y) 
global yc; 
    %this function compares the distance the device has travelled to  
    %a critical distance (either distance to contact or 
    %distance to loss of contact) to determine when to switch the  
    %differential equation used 
    value=abs(y(1))-abs(yc); 
    isterminal=1; 
    direction=1; 
 
A.2.3. 2DOF Shock Sensor MATLAB Model 
% 2DOF shock sensor model with deflection of latch included 
%predicts transient response of latching shock sensor to half-sine  
%acceleration function beginning at time = 0 
function [state]=shock_sensor_contact3() 
%clear all 
format short e; 
hold off; 
  
global chip_accel duration k kaxial M m kl ybar r di mu yc ca_fn?  
Fymon1 Fymon2 Fymon3 Fmon; 
%Fymon2=0; Fymon3=0; 
E=169e9;    %Young's Modulus 
wl=8e-6;    %width of latch 
ll=460e-6;  %length of latch 
tl=20e-6;   %thickness of latch 
%stiffness of a single latch - multiplied by 2 in force function ? 
%nominal 4.44 
kl=3*E*(1/12*tl*wl^3)/ll^3;        
kaxial=E*wl*tl/ll;          %axial stiffness of latch 
dens=2300;                  %density of silicon 
m=1/3*ll*wl*tl*dens;        %mass of latch 
freql=sqrt(kl/m)/(2*pi)     %natural frequency of latch 
  
%initialize global variables to describe system 100 g sensor 
chip_accel=-120*9.81;    %103.72 for hs video 
duration=5.9e-3; 
k=0.887; 
M=2.65e-7; 
  
%50 g sensor chip_accel=-62.56*9.81;      
%duration=8.24e-3;    
%k=0.601;     
  
ybar=150e-6;        %total travel to latched position 
r=40e-6;            %radius of latch 
di=15e-6;          %initial horizontal offset between latch and mass 
 
 185 
 
mu=0.1;             %friction coefficient 
yc=91.9e-6;         %distance to initial contact 
yc2=ybar;           %distance to latch/distance till contact is lost 
  
%define the frequency and period of the basic spring-mass system 
freq=sqrt(k/M)/(2*pi); 
period=1/freq; 
  
%Solve differential equation for period before contact set options  
%to find where contact occurs and stop integration 
refine=20; 
options=odeset('Events',@events,'Refine',refine); 
initial=[0;0;0;0]; 
[T1,Y1,TE1,YE1,IE]=ode45(@shock_init,[0 period],initial,options); 
subplot(3,1,1),plot(T1,Y1(:,1)); 
hold on 
subplot(3,1,2),plot(T1,Y1(:,3)); 
hold on; 
  
%Solve differential equation for period while mass is in contact  
%with latch find where contact ends and stop integration 
refine=1; 
yc=yc2; 
maxstep=1e-7; 
initstep=0.5e-9; 
%full version - includes storage of force information at each 
%timestep, but runs slowly 
%options=odeset('Events',@events,'Refine',refine,'MaxStep'? 
%,maxstep,'OutputFcn',@Fstore,'Stats','on','InitialStep',initstep);  
 
%simple version - no storage of force data - much faster execution 
options=odeset('Events',@events,'Refine',refine,'MaxStep',maxstep,'I
nitialStep',initstep); 
initial2=[YE1(1,1);YE1(1,2);YE1(1,3);YE1(1,4)]; 
[T2,Y2,TE2,YE2,IE2]=ode45(@latchfr,[TE1 duration],initial2,options); 
%if desired, code can be interrupted here to simply report whether  
%the device latched or not 
%if IE2==2  
    %this corresponds to the velocity going to zero (didn't latch) 
%    state=0; return; 
%else state=1; 
    %otherwise the device did latch 
%    return; 
%end 
subplot(3,1,1),plot(T2,Y2(:,1),'g:'); 
subplot(3,1,2),plot(T2,Y2(:,3),'g:'); 
%Solve differential equation for period after mass breaks contact  
%with latch 
initial3=[YE2(1,1);YE2(1,2);YE2(1,3);YE2(1,4)] 
[T3,Y3]=ode45(@shock_final,[TE2 duration],initial3); 
subplot(3,1,1),plot(T3,Y3(:,1),'r'); 
%Labels and legends for graph of data 
title('Displacment vs. Time of Shock Sensor') 
xlabel('Time (s)') 
ylabel('Displacement (m)') 
legend('pre-contact','contact','post-contact') 
 
 186 
 
grid on 
axis([0 .006 0 200e-6]) 
subplot(3,1,2),plot(T3,Y3(:,3),'r'); 
axis([0 .006 -50e-6 50e-6]) 
%Labels and legends for graph of data 
title('Latch Displacement') 
xlabel('Time (s)') 
ylabel('Displacement (m)') 
grid on 
%plot x displacement 
figure 
plot(T2,Y2(:,3))      %plot x-displacement of latch 
title('Latch Displacement') 
xlabel('Time (s)') 
ylabel('Displacement (m)') 
grid on 
figure 
plot(Fmon(:,1),Fmon(:,5))  %plot y force on latch 
title('Normal Force on Latch') 
xlabel('Time (s)') 
ylabel('Force (N)') 
grid on 
%write out position/time data for animation 
%fid=fopen('T_data.txt','wt'); 
%T=[T1;T2;T3]; fprintf(fid,'%15.6e\n',T); fclose(fid); 
%fid=fopen('Y_data.txt','wt'); Y=[Y1(:,1);Y2(:,1);Y3(:,1)]; 
%fprintf(fid,'%15.6e\n',Y); fclose(fid); 
%fid=fopen('X_data.txt','wt'); 
%X=[Y1(:,3);Y2(:,3);Y3(:,3)]; fprintf(fid,'%15.6e\n',X); 
%fclose(fid); 
  
%------------------------------------------------------------------- 
function dy=shock_init(t,y) 
  
%this function defines the differential equation before the shock  
%sensor 
%makes contact with the latch y(1) is the displacement, y(2) is the 
%velocity ydd is the Eq of motion for the mass, F is the mass-latch 
%interaction (friction & normal force) 
  
global chip_accel duration k m M Fymon1; 
  
F=0; 
Fy=F; 
Fx=F; 
if t<=duration 
    ydd=-chip_accel*sin(t/duration*pi)-k*y(1)/M-2*F/M; 
else 
    ydd=-k*y(1)/M-2*F/M; 
end 
xdd=0; 
dy=[y(2);ydd;y(4);xdd]; 
  
%------------------------------------------------------------------- 
function dy=latchfr(t,y) 
  
 
 187 
 
%this function defines the differential equation while the shock  
%sensor is 
%in contact with the latch y(1) is the displacement, y(2) is the  
%velocity 
%ypp is the acceleration function, F is the mass-latch interaction 
%(friction & normal force) 
  
global chip_accel duration k m kl kaxial ybar r di mu yc M Fymon2 Fx 
Fy F_normal; 
  
yl=ybar-y(1);                     %remaining travel 
  
%calculations of surface deflection using various equations for  
%cylinders 
%reasonable range for normal force 0-5mN 
L=20e-6; 
P=linspace(1e-6,5e-3); 
r=40e-6; 
v=0.28; 
E=169e9; 
%Eq. 4 - for two cylinders in contact 
V4=(1-v^2)/(pi*E); 
P4=P/L; 
a=L/2; 
d=2*P4.*V4.*(1+log((4*a^2)./(V4*P4*2*r))); 
  
int2=2*r-sqrt((ybar-y(1))^2+(r+di+y(3))^2); 
theta=asin(yl/(2*r-int2));            %contact angle 
if int2<0 
    int2=0; 
end 
  
%approx. stiffness of contact 
kc=P(100)/d(100); 
%effective latch stiffness in direction of contact 
kl_eff=1/(cos(theta)^2/kl+sin(theta)^2/kaxial); 
keff=kl_eff+kc; 
  
F_normal=keff*int2; 
  
Fx=F_normal*cos(theta); 
Fy=F_normal*sin(theta)*2+2*mu*F_normal*cos(theta); 
  
%equation of motion for mass, ydd means y double dot 
if t<=duration 
    ydd2=-chip_accel*sin(t/duration*pi)-k*y(1)/M-2*F_normal*(ybar-
y(1))/sqrt((ybar-y(1))^2+(r+di+y(3))^2)/M-
sign(y(2))*2*mu*F_normal*(r+di+y(3))/(2*r-int2)/M; 
else 
    ydd2=-k*y(1)/M-2*F_normal*(ybar-y(1))/sqrt((ybar-
y(1))^2+(r+di+y(3))^2)/M-sign(y(2))*2*mu*F_normal*(r+di+y(3))/(2*r-
int2)/M; 
  
end 
  
 
 188 
 
%equation of motion for latch, xdd means x double dot 
  
xdd2=(-kl*y(3)+kc*int2*(r+di+y(3))/sqrt((ybar-
y(1))^2+(r+di+y(3))^2)-sign(y(4))*mu*F_normal*sin(theta))/m; 
%return state for each time step 
dy=[y(2);ydd2;y(4);xdd2]; 
  
%------------------------------------------------------------------- 
function dy=shock_final(t,y) 
  
global chip_accel ca_fn duration k kaxial kl m M ybar Fymon3 mu; 
  
%this function defines the differential equation after the shock  
%sensor 
%leaves contact with the latch y(1) is the displacement, y(2) is the 
%velocity ypp is the acceleration function, F is the mass-latch  
%interaction 
%(friction & normal force) 
zm=0; 
cm=2*zm*sqrt(k/M); 
if y(1)>ybar&&y(1)<170e-6 
    F=0; 
elseif y(1)>170e-6 
    F=kaxial*(y(1)-170e-6); 
else 
    F=kaxial*(y(1)-ybar); 
end 
if t<=duration 
    ydd=-chip_accel*sin(t/duration*pi)-k*y(1)/M-cm*y(2)/M-2*F/M; 
else 
    ydd=-k*y(1)/M-cm*y(2)/M-2*F/M; 
end 
zeta=0; 
cl=2*zeta*sqrt(kl*m); 
%equation of motion for latch, xdd means x double dot 
%xdd=-kl*y(3)/m-sign(y(4))*mu*F/m; 
xdd=-kl*y(3)/m-cl*y(4)/m; 
dy=[y(2);ydd;y(4);xdd]; 
  
%------------------------------------------------------------------- 
  
function [value,isterminal,direction]=events(t,y) 
global yc; 
    %this function compares the distance the device has travelled to 
a 
    %critical distance (either distance to contact or distance to 
loss of 
    %contact) to determine when to switch the differential equation 
used 
    value=[abs(y(1))-abs(yc);y(2)]; 
    isterminal=[1;1]; 
    direction=[1;-1]; 
     
%-------------------------------------------------------------------
function status=Fstore(t,y,flag) 
 
 189 
 
  global Fx Fy Fmon F_normal; 
if strcmp(flag,'init') 
  Fmon=[Fmon;t(1) y(1) Fx Fy F_normal]; 
elseif strcmp(flag,'') Fmon=[Fmon;t(1) y(1) Fx Fy F_normal]; 
end 
  status=0; 
  
A.2.4. Shock Sensor Parametric Study Code 
%self-contained code to vary design paprameters and solve for  
%corresponding latching thresholds 
function [thresh_array]=find_thresh_param() 
    %define baseline parameter values 
    t=20e-6; 
    Lf_baseline=1495e-6; 
    Wf_baseline=12e-6; 
    Hm_baseline=2.025e-3; 
    wl_baseline=8e-6; 
    ll_baseline=460e-6; 
     
    %define range to vary each parameter and array to store results 
    Hm=linspace(1e-3,3e-3,21); 
    Hm_thresh=zeros(length(Hm),1); 
    Lf=linspace(1e-3,3e-3,21); 
    Lf_thresh=zeros(length(Lf),1); 
    Wf=linspace(5e-6,15e-6,21); 
    Wf_thresh=zeros(length(Wf),1); 
    wl=linspace(3e-6,15e-6,13); 
    wl_thresh=zeros(length(wl),1); 
    ll=linspace(200e-6,600e-6,9); 
    ll_thresh=zeros(length(ll),1); 
  
    %define initial guess for threshold acceleration for each array 
    Hm_guess=500;        
    Lf_guess=500;       
    Wf_guess=50;         
    wl_guess=80;         
    ll_guess=350; 
    mu=0.3; 
     
    %step through array of parameter values and solve for threshold 
    %comment out all but parameter being currently investigated 
    for i=1:length(ll) 
        %assemble full parameter to pass 
        %Hm_pass=[Hm_guess mu t Lf_baseline Wf_baseline Hm(i) wl ll] 
        %Hm_thresh(i)=find_thresh(Hm_pass) 
        %Hm_guess=Hm_thresh(i); 
         
        %Lf_pass=[Lf_guess mu t Lf(i) Wf_baseline Hm_baseline wl ll] 
        %Lf_thresh(i)=find_thresh(Lf_pass) 
        %Lf_guess=Lf_thresh(i); 
         
        %Wf_pass=[Wf_guess mu t Lf_baseline Wf(i) Hm_baseline wl ll] 
        %Wf_thresh(i)=find_thresh(Wf_pass) 
        %Wf_guess=Wf_thresh(i); 
 
 190 
 
        %wl_pass=[wl_guess mu t Lf_baseline Wf_baseline Hm_baseline 
  %wl(i) ll_baseline]; 
        %wl_thresh(i)=find_thresh(wl_pass) 
        %wl_guess=wl_thresh(i); 
        ll_pass=[wl_guess mu t Lf_baseline Wf_baseline Hm_baseline 
wl_baseline ll(i)]; 
        ll_thresh(i)=find_thresh(ll_pass) 
        ll_guess=ll_thresh(i); 
    end 
    %print out results 
    ll' 
    ll_thresh 
  
%------------------------------------------------------------------- 
%Uses internal version of ssc3_optim to find threshold acceleration 
%for a given design 
function [thresh]=find_thresh(ft) 
   %set parameters of particular design 
   guess=ft(1); 
   mu=ft(2); 
   t=ft(3); 
   Lf=ft(4); 
   Wf=ft(5); 
   Hm=ft(6); 
   wl=ft(7); 
   ll=ft(8); 
    global chip_accel 
    incr=20;    %initial adjustment increment 
    chip_accel=-guess     %apply initial guess 
    ca_low=0;              %apply initial bounds 
    ca_hi=5000; 
%loop through until threshold is found within 1g 
    while (ca_hi-ca_low)>=1                
 latched=ssc3_optim(t,Lf,Wf,Hm,wl,ll,mu) %test if latches 
        if latched==0           %if doesn't latch, increase accel 
            ca_low=-chip_accel;    %update lower bound on search 
            if abs(chip_accel-incr)>=ca_hi       %don?t overshoot 
                incr=incr/2;     %decrease increment 
            end 
            chip_accel=chip_accel-incr;         %adjust guess 
            continue            %and retry 
  
        else                        %if it did latch, decrease  
            ca_hi=-chip_accel;            %modify upper bound 
            if abs(chip_accel+incr)<=ca_low       %don?t overshoot 
                incr=incr/2;     %decrease increment if you would  
            end 
            chip_accel=chip_accel+incr         %adjust guess 
            continue            %and retry 
        end 
    end     %end while loop 
    mu 
    thresh=ca_hi      %set threshold to lowest accel resulting in 
%latched event 
     
     
 
 191 
 
%------------------------------------------------------------------- 
% 2DOF shock sensor model with deflection of latch included 
%predicts transient response of latching shock sensor to steady 
%acceleration beginning at time = 0 
%also predicts steady state maximum deflection 
%does not handle impulse acceleration (i.e., acceleration pulse 
%lasts for shorter than 1/2 of device period 
function [state]=ssc3_optim(t,Lf,Wf,Hm,wl,ll,mu_pass) 
%clear all 
format short e; 
hold off; 
  
global chip_accel duration k kaxial M m kl ybar r di mu yc ca_fn 
Fymon1 Fymon2 Fymon3 Fmon max_bounce; 
E=160e9; 
radius=40e-6; 
%define wl,ll,t outside this m-file 
%calculate mass and stiffness of latch 
kl=3*E*(1/12*t*wl^3)/ll^3;       %stiffness of a single latch - 
%multiplied by 2 in force function -nominal 4.44, 3.66 for 7.5um 
%wide latch 
kaxial=E*wl*t/ll; 
dens=2330; 
m=33/140*(ll+40e-6)*wl*t*dens+pi*radius^2/4*t*dens; 
freql=sqrt(kl/m)/(2*pi); 
  
%initialize global variables to describe system 
%chip_accel=-120*9.81;       %chip_accel comes from input at runtime 
duration=5.9e-3;                 % 
Np=4; 
Ns=5; 
  
%define t,Lf,Wf,Hm outside of this m-file 
%calculate mass and stiffness of springs and mass of Mass 
k=E*t*Wf^3/(Lf^3*Ns)*Np; 
Wm=Lf*2+110e-6; 
num_holes=ceil((Hm-40e-6)/(60e-6))*ceil((Wm-40e-6)/(60e-6)); 
Mm=dens*t*(Hm*Wm-120e-6*120e-6-num_holes*20e-6*20e-6);  %mass of 
%sensor 
Mf=Lf*Wf*t*dens*Ns*Np/3;                                 %equivalent 
%mass of springs 
M=Mm+Mf; 
  
ybar=150e-6; 
r=40e-6; 
di=15e-6; 
  
mu=mu_pass;            %mu defined by input at runtime 
yc=91.9e-6;         %distance to initial contact 
yc2=ybar;           %distance to latch/distance till contact is lost 
k2=kl;                %axial stiffness of latch 
  
%define the frequency and period of the basic spring-mass system 
freq=sqrt(k/M)/(2*pi); 
period=1/freq; 
 
 192 
 
  
%Solve differential equation for period before contact 
%set options to find where contact occurs and stop integration 
options=odeset('Events',@events); 
initial=[0;0;0;0]; 
[T1,Y1,TE1,YE1,IE]=ode45(@shock_init,[0 period],initial,options); 
if IE==2 
    state=0;    %unlatched 
    return; 
end 
  
%Solve differential equation for period while mass is in contact 
%with latch 
%find where contact ends and stop integration 
yc=yc2; 
maxstep=10e-7;      %1e-8 used for model match 
initstep=5e-8;     %0.5ns for model match 
refine=1; 
  
options=odeset('Events',@events,'Refine',refine,'MaxStep',maxstep,'S
tats','off','InitialStep',initstep); 
initial2=[YE1(1,1);YE1(1,2);YE1(1,3);YE1(1,4)]; 
[T2,Y2,TE2,YE2,IE2]=ode45(@latchfr,[TE1 
duration+period],initial2,options); 
%max_bounce 
if IE2==2 
    state=0;    %unlatched 
    return; 
else state=1;   %latched 
    return; 
end 
  
%-------------------------------------------------------------------
function dy=shock_init(t,y) 
  
%this function defines the differential equation before the shock 
%sensor makes contact with the latch 
%y(1) is the displacement, y(2) is the velocity 
%ydd is the Eq of motion for the mass, F is the mass-latch 
%interaction (friction & normal force) 
  
global chip_accel duration k m M Fymon1; 
  
F=0; 
Fy=F; 
Fx=F; 
if t<=duration 
    ydd=-chip_accel*9.81*sin(t/duration*pi)-k*y(1)/M-2*F/M; 
else 
    ydd=-k*y(1)/M-2*F/M; 
end 
xdd=0; 
dy=[y(2);ydd;y(4);xdd]; 
  
%------------------------------------------------------------------- 
 
 193 
 
function dy=latchfr(t,y) 
  
%this function defines the differential equation while the shock 
%sensor is in contact with the latch 
%y(1) is the displacement, y(2) is the velocity 
%ypp is the acceleration function, F is the mass-latch interaction 
%(friction & normal force) 
  
global chip_accel duration k m kl kaxial ybar r di mu yc M Fymon2 Fx 
Fy F_normal max_bounce; 
  
yl=ybar-y(1);                     %remaining travel 
  
%calculations of surface deflection using various equations for 
%cylinders 
%reasonable range for normal force 0-5mN 
L=20e-6; 
P=linspace(1e-6,5e-3); 
r=40e-6; 
v=0.28; 
E=169e9; 
%Eq. 4 - for two cylinders in contact 
V4=(1-v^2)/(pi*E); 
P4=P/L; 
a=L/2; 
d=2*P4.*V4.*(1+log((4*a^2)./(V4*P4*2*r))); 
  
int2=2*r-sqrt((ybar-y(1))^2+(r+di+y(3))^2); 
theta=asin(yl/(2*r-int2));            %contact angle 
  
if int2<0 
    if abs(int2)>max_bounce 
        max_bounce=abs(int2); 
    end 
    int2=0; 
end 
  
%approx. stiffness of contact 
kc=P(100)/d(100); 
%effective latch stiffness in direction of contact 
kl_eff=1/(cos(theta)^2/kl+sin(theta)^2/kaxial); 
keff=kl_eff+kc; 
  
F_normal=keff*int2; 
  
Fx=F_normal*cos(theta); 
Fy=F_normal*sin(theta)*2+2*mu*F_normal*cos(theta); 
  
if t<=duration 
    ydd2=-chip_accel*9.81*sin(t/duration*pi)-k*y(1)/M-
2*F_normal*(ybar-y(1))/sqrt((ybar-y(1))^2+(r+di+y(3))^2)/M-
sign(y(2))*2*mu*F_normal*(r+di+y(3))/(2*r-int2)/M; 
else 
 
 194 
 
    ydd2=-k*y(1)/M-2*F_normal*(ybar-y(1))/sqrt((ybar-
y(1))^2+(r+di+y(3))^2)/M-sign(y(2))*2*mu*F_normal*(r+di+y(3))/(2*r-
int2)/M; 
end 
zeta=.1; 
cl=2*zeta*sqrt(kl*m); 
%equation of motion for latch, xdd means x double dot 
%xdd=(-kl*y(3)-cl*y(4)+Fx)/m 
xdd2=(-kl*y(3)+Fx)/m; 
xdd2=(-kl*y(3)+kc*int2*(r+di+y(3))/sqrt((ybar-
y(1))^2+(r+di+y(3))^2)-sign(y(4))*mu*F_normal*sin(theta))/m; 
%return state for each time step 
dy=[y(2);ydd2;y(4);xdd2]; 
  
%------------------------------------------------------------------- 
  
function dy=shock_final(t,y) 
  
global chip_accel ca_fn duration k kaxial kl m M ybar Fymon3 mu; 
  
%this function defines the differential equation after the shock 
sensor leaves contact with the latch 
%y(1) is the displacement, y(2) is the velocity 
%ypp is the acceleration function, F is the mass-latch interaction 
(friction & normal force) 
zm=3e-7; 
cm=2*zm*sqrt(k/M); 
if y(1)>ybar&&y(1)<170e-6 
    F=0; 
elseif y(1)>170e-6 
    F=kaxial*(y(1)-170e-6); 
else 
    F=kaxial*(y(1)-ybar); 
end 
if t<=duration 
    ydd=-chip_accel*9.81*sin(t/duration*pi)-k*y(1)/M-cm*y(2)/M-
2*F/M; 
else 
    ydd=-k*y(1)/M-cm*y(2)/M-2*F/M; 
end 
zeta=0.1; 
cl=2*zeta*sqrt(kl*m); 
%equation of motion for latch, xdd means x double dot 
%xdd=-kl*y(3)/m-sign(y(4))*mu*F/m; 
xdd=-kl*y(3)/m-cl*y(4)/m; 
dy=[y(2);ydd;y(4);xdd]; 
  
%------------------------------------------------------------------- 
  
function [value,isterminal,direction]=events(t,y) 
global yc; 
    %this function compares the distance the device has travelled to 
%a critical distance (either distance to contact or 
    %distance to loss of contact) to determine when to switch the 
%differential equation used 
 
 195 
 
    value=[abs(y(1))-abs(yc);y(2)];    %returns 2 if velocity is 
%zero, returns 1 if moves past latch dist 
    isterminal=[1;1]; 
    direction=[1;-1]; 
     
%------------------------------------------------------------------- 
 
A.2.5. Genetic Algorithm Codes 
%---------------------run_ga.m---------------------- 
%requires the following m-files: ssc3_optim.m GA_frict_thresh_obj.m, 
thresh_err_calc.m 
  
datestr(now) 
tic 
  
global THRESH_P3 TARGET_THRESH 
TARGET_THRESH=100; 
  
matlabpool local 8 
format('shorte'); 
% Start with the default options 
options = gaoptimset; 
% Modify options setting 
options = gaoptimset(options,'PopulationSize', 100); 
options = gaoptimset(options,'EliteCount', 2); 
options = gaoptimset(options,'CreationFcn', @gacreationuniform); 
options = gaoptimset(options,'Display', 'iter'); 
options = gaoptimset(options,'PlotFcns', { @gaplotscores }); 
options = gaoptimset(options,'Vectorized', 'on'); 
options = gaoptimset(options,'UseParallel', 'never'); 
options = gaoptimset(options,'SelectionFcn',@selectionstochunif); 
lb=[20e-6 1000e-6 3e-6 1e-3 8e-6 460e-6] 
ub=[20e-6 2000e-6 20e-6 3e-3 8e-6 460e-6] 
[x,fval,exitflag,output,population,score] = ... 
 ga(@GA_frict_thresh_obj,6,[],[],[],[],lb,ub,? 
 @thresh_err_calc,options); 
matlabpool close 
toc 
datestr(now) 
 
%-----------GA_frict_thresh_obj.m---------------------- 
%vectorized single-objective optimization 
%uses find_thresh, ssc3_optim to perform constrained single-
objective optimization 
%objective is: Low friction sensitivity 
% uses ndsort to assign fitness values at the end of each generation 
%intended as objective function in an optimization routine 
  
function fitness=GA_frict_thresh_obj(x) 
    %x is the vector of independent variables for the optimization 
    %definition of x=[t Lf Wf Hm wl ll] 
  
    global TARGET_THRESH 
    TARGET_THRESH=100; 
 
 196 
 
    persistent GEN pop_results 
    if isempty(GEN) 
        GEN=1;       
        PARETO=[]; 
    else GEN=GEN+1; 
    end 
    if isempty(pop_results) 
        pop_results=zeros(1,10) 
    end 
    pop_size=size(x,1) 
    %initialize results array 
    %array elements will be: 
    %[generation# x-values thresh_p3 thresh_p6] 
    current_results_p3=zeros(pop_size,1); 
    current_results_p6=zeros(pop_size,1); 
     
%step through population: 
parfor i=1:pop_size     
    %find low friction threshold for current design 
    mu_pass=0.3; 
    quick_test=1000; 
    guess=TARGET_THRESH; 
    par=[guess mu_pass x(i,1) x(i,2) x(i,3) x(i,4) x(i,5) x(i,6)]; 
    
latched=ssc3_optim(x(i,1),x(i,2),x(i,3),x(i,4),x(i,5),x(i,6),mu_pass
,quick_test);     
    if latched==1 
        thresh_p3=quick_test; 
    else 
        thresh_p3=find_thresh(par) 
   end 
    %compare to target 
    thresh_err=abs(thresh_p3-TARGET_THRESH); 
  
     %find high friction threshold for current design 
    if (thresh_p3<1000) 
        mu_pass=0.6; 
        guess=thresh_p3; 
        par=[guess mu_pass x(i,1) x(i,2) x(i,3) x(i,4)? 
   x(i,5) x(i,6)]; 
        thresh_p6=find_thresh(par) 
    else thresh_p6=10000 
    end 
    %compare to thresh_p3 
    frict_sens=abs(thresh_p6/thresh_p3-1); 
     
    %update current_results array 
    current_results_p3(i)=thresh_p3; 
    current_results_p6(i)=thresh_p6; 
end 
thresh_err_array=abs(current_results_p3-? 
 TARGET_THRESH*ones(pop_size,1)); 
frict_sens_array=abs(current_results_p6./current_results_p3-1); 
  
%assign final fitness values to current results 
%just using friction sensitivity 
 
 197 
 
fitness=frict_sens_array; 
     
%assemble current results array 
current_results=cat(2,GEN*ones(pop_size,1),x,current_results_p3,curr
ent_results_p6,fitness) 
  
%update the persistent results array 
    pop_results=cat(1,pop_results,current_results) 
    save('frict_thresh_results.mat','pop_results') 
  
%---------------find_thresh.m---------------------- 
%Uses ssc3_optim to find threshold acceleration for a given design 
function [thresh]=find_thresh(ft) 
   %set parameters of particular design 
   guess=ft(1);     %guess should be positive 
   mu=ft(2); 
   t=ft(3); 
   Lf=ft(4); 
   Wf=ft(5); 
   Hm=ft(6); 
   wl=ft(7); 
   ll=ft(8); 
    global chip_accel 
    %t=20e-6; 
    %Lf=1495e-6; 
    %Wf=12e-6; 
    %Hm=2.025*1e-3; 
    %wl=8e-6; 
    %ll=460e-6; 
    incr=2;    %initial adjustment increment 
quick_test=1000; 
   latched=ssc3_optim(t,Lf,Wf,Hm,wl,ll,quick_test,mu); 
if latched==0     
   thresh=1000 
   return; 
else 
    ca_low=0;              %apply initial bounds 
    ca_hi=1000; 
     
%loop through until threshold is found within 1g 
    while (ca_hi-ca_low)>=0.1   
 %test to see if sensor latches at this level 
       latched=ssc3_optim(t,Lf,Wf,Hm,wl,ll,guess,mu);      
  %if doesn't latch, increase chip accel 
if latched==0                       
  ca_low=guess;    %update lower bound on search 
        %make sure not to overshoot past bounds... 
            if abs(guess+incr)>=ca_hi        
                incr=incr/2;     %decrease increment if you would  
            end 
            guess=guess+incr;         %adjust guess 
            continue            %and retry 
  
 %if it did latch, decrease chip_accel   
 else                                     
       ca_hi=guess;            %modify upper bound on search 
 
 198 
 
            if abs(guess-incr)<=ca_low       %don?t overshoot 
                incr=incr/2;     %decrease increment if you would  
            end 
            guess=guess-incr;         %adjust guess 
            continue            %and retry 
        end 
    end     %end while loop 
end 
    mu 
    %set threshold to lowest accel resulting in latched event 
    thresh=ca_hi;       
end 
     
%-------------------ssc3_optim.m---------------------- 
% 2DOF shock sensor model with deflection of latch included 
%predicts transient response of latching shock sensor to steady 
%acceleration beginning at time = 0 
function [state]=ssc3_optim(t,Lf,Wf,Hm,wl,ll,guess,mu_pass) 
%clear all 
format short e; 
hold off; 
  
global chip_accel duration k kaxial M m kl ybar r di mu yc ca_fn 
Fymon1 Fymon2 Fymon3 Fmon max_bounce; 
E=160e9; 
%wl=8e-6;        %design width is 8um 
%ll=460e-6; 
%t=20e-6; 
radius=40e-6; 
%define wl,ll,t outside this m-file 
kl=3*E*(1/12*t*wl^3)/ll^3;       %stiffness of a single latch - 
%multiplied by 2 in force function -nominal 4.44, 3.66 for 7.5um 
%wide latch 
kaxial=E*wl*t/ll; 
dens=2330; 
m=33/140*(ll+40e-6)*wl*t*dens+pi*radius^2/4*t*dens; 
freql=sqrt(kl/m)/(2*pi); 
  
%initialize global variables to describe system 
%75g sensor 
chip_accel=-guess;         %chip_accel comes from input at runtime 
duration=5.9e-3;                 % 
%k=1.324;                          %design value 1.324, value with 
10.5um wide spring is 0.887 
%define t,Lf,Wf Ns outside of this m-file 
Np=4; 
Ns=5; 
k=E*t*Wf^3/(Lf^3*Ns)*Np; 
%M=2.65e-7;                          %design value 2.65e-7 
%define Hm outside this m-file 
Wm=Lf*2+110e-6; 
num_holes=ceil((Hm-40e-6)/(60e-6))*ceil((Wm-40e-6)/(60e-6)); 
%mass of sensor 
Mm=dens*t*(Hm*Wm-120e-6*120e-6-num_holes*20e-6*20e-6);   
%equivalent mass of springs 
Mf=Lf*Wf*t*dens*Ns*Np/3;                                  
 
 199 
 
M=Mm+Mf; 
  
%50 g sensor 
%chip_accel=-60*9.81;     %27.5g for hs video 
%duration=6e-3;   %7.52ms for hs video, 8.24ms for standard testing 
%k=0.601;    %actual value=0.601 - 0.529 for 11.5um wide spring, 
%0.403 for 10.5um wide spring 
%M=3.42e-7; 
  
ybar=150e-6; 
r=40e-6; 
di=15e-6; 
  
mu=mu_pass;            %mu defined by input at runtime 
yc=91.9e-6;         %distance to initial contact 
yc2=ybar;           %distance to latch/distance till contact is lost 
k2=kl;                %axial stiffness of latch 
  
%define the frequency and period of the basic spring-mass system 
freq=sqrt(k/M)/(2*pi); 
period=1/freq; 
%Solve differential equation for period before contact 
%set options to find where contact occurs and stop integration 
options=odeset('Events',@events); 
initial=[0;0;0;0]; 
[T1,Y1,TE1,YE1,IE]=ode45(@shock_init,[0 
duration+period],initial,options); 
if isempty(IE) 
    state=0; 
    return; 
elseif IE==2 
    state=0;    %unlatched 
    return; 
end 
  
max_bounce=0; 
%Solve differential equation for period while mass is in contact 
%with latch 
%find where contact ends and stop integration 
yc=yc2; 
maxstep=10e-7;      %1e-8 used for model match 
initstep=5e-8;     %0.5ns for model match 
refine=1; 
%options=odeset('Events',@events,'Refine',refine,'MaxStep',maxstep,'
%OutputFcn',@Fstore,'Stats','on','InitialStep',initstep); 
options=odeset('Events',@events,'Refine',refine,'MaxStep',maxstep,'S
tats','off','InitialStep',initstep); 
initial2=[YE1(1,1);YE1(1,2);YE1(1,3);YE1(1,4)]; 
if length(TE1)>1 
    TE1 
        YE1 
    TE1=max(TE1) 
end 
[T2,Y2,TE2,YE2,IE2]=ode45(@latchfr,[TE1 
duration+period],initial2,options); 
%max_bounce 
 
 200 
 
if isempty(IE2) 
    state=0; 
    return; 
elseif IE2==2 
    state=0;    %unlatched 
    return; 
else state=1;   %latched 
    return; 
end 
  
clear global F* 
  
%------------------------------------------------------------------ 
function dy=shock_init(t,y) 
  
%this function defines the differential equation before the shock 
%sensor makes contact with the latch 
%y(1) is the displacement, y(2) is the velocity 
%ydd is the Eq of motion for the mass, F is the mass-latch 
%interaction (friction & normal force) 
  
global chip_accel duration k m M Fymon1; 
  
F=0; 
Fy=F; 
Fx=F; 
if t<=duration 
    ydd=-chip_accel*9.81*sin(t/duration*pi)-k*y(1)/M-2*F/M; 
else 
    ydd=-k*y(1)/M-2*F/M; 
end 
xdd=0; 
dy=[y(2);ydd;y(4);xdd]; 
  
%------------------------------------------------------------------ 
function dy=latchfr(t,y) 
  
%this function defines the differential equation while the shock 
%sensor is in contact with the latch 
%y(1) is the displacement, y(2) is the velocity 
%ypp is the acceleration function, F is the mass-latch interaction 
%(friction & normal force) 
  
global chip_accel duration k m kl kaxial ybar r di mu yc M Fymon2 Fx 
Fy F_normal max_bounce; 
  
yl=ybar-y(1);                     %remaining travel 
  
%calculations of surface deflection using various equations for 
%cylinders 
%reasonable range for normal force 0-5mN 
L=20e-6; 
P=linspace(1e-6,5e-3); 
r=40e-6; 
v=0.28; 
 
 201 
 
E=169e9; 
%Eq. 4 - for two cylinders in contact 
V4=(1-v^2)/(pi*E); 
P4=P/L; 
a=L/2; 
d=2*P4.*V4.*(1+log((4*a^2)./(V4*P4*2*r))); 
  
int2=2*r-sqrt((ybar-y(1))^2+(r+di+y(3))^2); 
theta=asin(yl/(2*r-int2));            %contact angle 
  
if int2<0 
    if abs(int2)>max_bounce 
        max_bounce=abs(int2); 
    end 
    int2=0; 
end 
  
%approx. stiffness of contact 
kc=P(100)/d(100); 
%effective latch stiffness in direction of contact 
kl_eff=1/(cos(theta)^2/kl+sin(theta)^2/kaxial); 
keff=kl_eff+kc; 
  
F_normal=keff*int2; 
  
Fx=F_normal*cos(theta); 
Fy=F_normal*sin(theta)*2+2*mu*F_normal*cos(theta); 
  
if t<=duration 
    ydd2=-chip_accel*9.81*sin(t/duration*pi)-k*y(1)/M-
2*F_normal*(ybar-y(1))/sqrt((ybar-y(1))^2+(r+di+y(3))^2)/M-
sign(y(2))*2*mu*F_normal*(r+di+y(3))/(2*r-int2)/M; 
else 
    ydd2=-k*y(1)/M-2*F_normal*(ybar-y(1))/sqrt((ybar-
y(1))^2+(r+di+y(3))^2)/M-sign(y(2))*2*mu*F_normal*(r+di+y(3))/(2*r-
int2)/M; 
end 
zeta=.1; 
cl=2*zeta*sqrt(kl*m); 
%equation of motion for latch, xdd means x double dot 
%xdd=(-kl*y(3)-cl*y(4)+Fx)/m 
xdd2=(-kl*y(3)+Fx)/m; 
xdd2=(-kl*y(3)+kc*int2*(r+di+y(3))/sqrt((ybar-
y(1))^2+(r+di+y(3))^2)-sign(y(4))*mu*F_normal*sin(theta))/m; 
%return state for each time step 
dy=[y(2);ydd2;y(4);xdd2]; 
  
%------------------------------------------------------------------ 
function dy=shock_final(t,y) 
  
global chip_accel ca_fn duration k kaxial kl m M ybar Fymon3 mu; 
  
%this function defines the differential equation after the shock 
%sensor leaves contact with the latch 
%y(1) is the displacement, y(2) is the velocity 
 
 202 
 
%ypp is the acceleration function, F is the mass-latch interaction 
%(friction & normal force) 
zm=3e-7; 
cm=2*zm*sqrt(k/M); 
if y(1)>ybar&&y(1)<170e-6 
    F=0; 
elseif y(1)>170e-6 
    F=kaxial*(y(1)-170e-6); 
else 
    F=kaxial*(y(1)-ybar); 
end 
if t<=duration 
    ydd=-chip_accel*9.81*sin(t/duration*pi)-k*y(1)/M-cm*y(2)/M-
2*F/M; 
else 
    ydd=-k*y(1)/M-cm*y(2)/M-2*F/M; 
end 
zeta=0.1; 
cl=2*zeta*sqrt(kl*m); 
%equation of motion for latch, xdd means x double dot 
%xdd=-kl*y(3)/m-sign(y(4))*mu*F/m; 
xdd=-kl*y(3)/m-cl*y(4)/m; 
dy=[y(2);ydd;y(4);xdd]; 
  
%------------------------------------------------------------------- 
  
function [value,isterminal,direction]=events(t,y) 
global yc; 
    %this function compares the distance the device has travelled to 
%a critical distance (either distance to contact or 
    %distance to loss of contact) to determine when to switch the 
%differential equation used 
    value=[y(1)-yc;y(2)];    %returns 2 if velocity is zero, returns 
%1 if moves past latch dist 
    isterminal=[1;1]; 
    direction=[1;-1]; 
     
%------------------------------------------------------------------- 
function status=Fstore(t,y,flag) 
  global Fx Fy Fmon F_normal; 
if strcmp(flag,'init') 
  Fmon=[Fmon;t(1) y(1) Fx Fy F_normal]; 
elseif strcmp(flag,'') Fmon=[Fmon;t(1) y(1) Fx Fy F_normal]; 
end 
  status=0; 
 
A.2.6. Fmincon Optimization Routine 
Note: this optimization routine also requires the ssc3_optim.m and 
find_thresh.m files from section A.2.5 
   
%----------fmin_frict_thresh_opt.m----------- 
%define overall program flow here 
%requires the follwoing m-files: 
%ssc3_optim.m, fmin_frict_thresh_obj.m, thresh_err_constraint.m 
 
 203 
 
  
    global THRESH_P3 TARGET_THRESH 
    TARGET_THRESH=100; 
     
    %definition of x=[t Lf Wf Hm wl ll] 
    x0=[20e-6 1869e-6 18.4e-6 2.607e-3 6.36e-6 379e-6]; 
    lb = [20e-6 1000e-6 3e-6 1e-3 3e-6 200e-6];   %lower bounds on x 
    ub = [20e-6 2000e-6 20e-6 3e-3 18e-6 600e-6]; %upper bounds on x 
    options = optimset('DiffMinChange',.1); 
    options = optimset('TypicalX',[20e-6 1500e-6 12e-6 2.6e-3 8e-6 
400e-6]) 
    options = optimset('Algorithm','internal-point') 
    
[x,fval,exitflag]=fmincon(@fmin_frict_thresh_obj,x0,[],[],[],[],lb,u
b,@thresh_err_constraint) 
 
%-----------fmin_frict_thresh_obj.m------------- 
%FMINCON optimization 
%uses files: ssc3_optim to perform constrained optimization 
%objective is: Low friction sensitivity 
%constraint is: less than 3g variation from target acceleration 
%intended as objective function in an optimization routine 
  
%---------------OBJECTIVE FUNCTION-------------------- 
function frict_sens=fmin_frict_thresh_obj(x) 
    %x is the vector of independent variables for the optimization 
    %definition of x=[t Lf Wf Hm wl ll] 
    global THRESH_P3 TARGET_THRESH 
    x 
    persistent gen compiled_results 
    if isempty(gen) 
        gen=1;       
    else gen=gen+1; 
    end 
    if isempty(compiled_results) 
        compiled_results=zeros(1,11) 
    end 
  
    %initialize results array 
    %array elements will be: 
    %[generation# x-values THRESH_P3 thresh_p6] 
       
    %find low friction threshold for current design 
    mu_pass=0.3; 
    guess=TARGET_THRESH; 
    par=[guess mu_pass x(1) x(2) x(3) x(4) x(5) x(6)]; 
    THRESH_P3=find_thresh(par) 
    disp(['x-values are: ',num2str(x)]); 
    disp(['THRESH_P3 found from objective function is 
',num2str(THRESH_P3)]); 
    %compare to target 
    thresh_err=abs(THRESH_P3-TARGET_THRESH); 
  
     %find high friction threshold for current design 
    if (THRESH_P3<1000) 
 
 204 
 
        mu_pass=0.6; 
        guess=THRESH_P3; 
        par=[guess mu_pass x(1) x(2) x(3) x(4) x(5) x(6)]; 
        thresh_p6=find_thresh(par) 
    else thresh_p6=10000 
    end 
    %compare to THRESH_P3 
    frict_sens=abs(thresh_p6/THRESH_P3-1); 
     
%assemble current results array 
current_results=[gen x THRESH_P3 thresh_p6 thresh_err frict_sens] 
compiled_results=[compiled_results;current_results] 
end 
 
 
%---------thresh_err_constraint.m------------------ 
function [c,ceq]=thresh_err_constraint(x) 
    global THRESH_P3 TARGET_THRESH 
    ceq=[]; 
    disp(['x-values are: ',num2str(x)]); 
    disp(['THRESH_P3 used in constraint function is 
',num2str(THRESH_P3)]); 
    c=abs(THRESH_P3-TARGET_THRESH)-3 
end 
     
A.3. ANSYS Scripts 
A.3.1. Shape Factor Code 
!Ansys model for 
!Thermal actuator 
!shape factor calculation 
! 
/show,win32c 
/contour,all,128 
 
/prep7 
et,1,plane77 
et,2,infin110 
keyopt,2,1,2 !set infin element dof to temp 
 
rho=5.868e-3/100 !resistivity of beam 
 
gap=2e-6  !gap between beam and substrate 
h=20e-6   !height of beam 
w=10e-6   !width of beam 
diam=2*h  !location of infinite boundary elements 
 
J=1e8   !current density 
 
 205 
 
 
!silicon material properties 
mp,kxx,1,148  !thermal conductivity of silicon 
 
!air material properties 
mp,kxx,2,0.026  !thermal conductivity of air 
 
!define geometry 
cyl4,0,0,diam,0,,90 !air circle 
blc4,0,gap,0.5*w,h !silicon rectangle 
blc4,0,0,diam,-gap !substrate 
cyl4,0,0,diam,0,2*diam,90  !infinite element domain 
blc4,diam,0,diam,-gap !infinite element domain 
 
!glue areas together 
asba,1,2,,delete,keep 
asel,all 
aglue,all 
 
!assign material properties and element types to air volume 
asel,s,,,8 
aatt,2,0,1 
!assign material properties and element types to silicon volumes 
asel,s,,,2,3,1 
aatt,1,0,1 
!assign material properties and element types to infinite domain 
asel,s,,,1 
aatt,1,0,2 
asel,s,,,7 
aatt,2,0,2 
 
!size elements and mesh 
asel,all 
lsel,s,loc,x,0 
lsel,s,loc,y,gap/2 
lesize,all,,,3 
 
!infinite domain is one element thick 
lsel,s,length,,diam 
lsel,u,loc,x,diam/2 
lesize,all,,,1 
 
!number of infinite elements around the boundary 
lsel,s,radius,,diam 
lsel,a,radius,,2*diam 
lesize,all,,,60 
 
 206 
 
 
lsel,s,loc,x,diam,2*diam 
lsel,r,loc,y,-gap/2 
lesize,all,,,2 
 
asel,s,type,,2 
mshkey,1 
amesh,all 
 
asel,all 
mshkey,0 
esize,1e-6 
amesh,all 
 
!apply boundary conditions 
asel,s,loc,y,-gap,0 
asel,r,loc,x,0,diam 
da,all,temp,0 
lsel,s,loc,x,0 
dl,all,,symm 
 
asel,s,,,2 
bfa,all,hgen,J*J*rho 
 
lsel,s,radius,,2*diam 
lsel,a,loc,x,2*diam 
nsll,s,1 
sf,all,inf 
 
allsel 
/solution 
solve 
 
/expand,2,rect,half 
 
/post1 
plnsol,temp 
A.3.2. Latch Contact Force/Deflection Code 
!ansys contact force/displacement model 
 
/prep7 
et,1,solid95 
 
mp,ex,1,169e3 
mp,prxy,1,0.28 
 
 207 
 
 
radius=40 
depth=20 
cyl4,0,0,radius,0,,180,depth 
cyl4,0,2*radius,radius,0,,-180,depth 
 
vmesh,all 
 
asel,s,loc,y,0 
da,all,all,0 
 
P=5000 
pressure=P/(2*radius*depth) 
asel,s,loc,y,80 
sfa,all,,pres,pressure 
allsel 
 
!define contact pair and solve through GUI 
 
 
 
 
 
 
 208 
 
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