WWeesstteerrnn UUnniivveerrssiittyy SScchhoollaarrsshhiipp@@WWeesstteerrnn Electronic Thesis and Dissertation Repository 8-14-2014 12:00 AM UUnncceerrttaaiinnttyy QQuuaannttiifificcaattiioonn ffoorr aa CCllaassss ooff MMEEMMSS--bbaasseedd VViibbrraattoorryy AAnngguullaarr RRaattee SSeennssoorrss Nujhat Abedin, The University of Western Ontario Supervisor: Dr. S. F. Asokanthan, The University of Western Ontario A thesis submitted in partial fulfillment of the requirements for the Master of Engineering Science degree in Mechanical and Materials Engineering © Nujhat Abedin 2014 Follow this and additional works at: https://ir.lib.uwo.ca/etd Part of the Acoustics, Dynamics, and Controls Commons, and the Electro-Mechanical Systems Commons RReeccoommmmeennddeedd CCiittaattiioonn Abedin, Nujhat, "Uncertainty Quantification for a Class of MEMS-based Vibratory Angular Rate Sensors" (2014). 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Uncertainty Quantification for a Class of MEMS-based Vibratory Angular Rate Sensors By Nujhat Abedin Graduate Program in Engineering Science Department of Mechanical and Material Engineering Submitted in partial fulfilment of the requirements for the degree of Master of Engineering Science School of Graduate and Postdoctoral Studies The University of Western of Ontario London, Ontario, Canada August 2014 © Nujhat Abedin 2014 i ABSTRACT Numerical schemes that are suitable for predicting response statistics of mass-spring and ring gyroscopes are developed when this class of vibratory gyroscopes are subjected to certain system parameters as well as environment uncertainties. The emphasis is placed on the steady-state part of the response since it is more critical to the operation of a gyroscope. A peak-picking approach which simulates the demodulation process which is used in practice is employed first before applying the Monte Carlo simulation method to predict the response statistics. A number of simulation trials to predict response statistics have been performed for mass-spring and ring-type gyroscopes in an effort to ascertain the optimal temporal points as well as sample paths for the impending uncertainty quantification study. Based on the optimal temporal and sample paths, uncertainties in input angular rate, mass/frequency mismatch and damping have been quantified. Keywords: MEMS based gyroscope, General coordinate, Uncertainty quantification, Monte Carlo method, Numerical prediction, Ensemble mean, Mass mismatch, Frequency mismatch, Quality factor, Dynamic response. ii ACKNOWLEDGEMENTS This thesis would not have been come to light without the cooperation of the my thesis supervisor Dr. S. F. Asokanthan. I am deeply thankful to him for introducing me to this research area and for his continuous guidance, encouragement and expertise and valuable contribution to this thesis. I would like to express my gratitude to those who provided me with guidance and support during the course of this thesis. My thanks also go to my colleagues and friends for their helpful and friendly behavior. I would like to express special thanks to Dr. Quazi Mehbubar Rahman and Muhammad Bashar for their guidance and inspiration. Finally, I would like to thank organizations such as the National Sciences and Engineering Research Council (NSERC) of Canada discovery grant, The University of Western Ontario's Academic Development fund/Small Grants and Western Graduate Research Scholarship (WGRS) from the University of Western Ontario, as this research work was partly funded by them. iii The work is dedicated to my beloved mother Zinnat Zahanara I am who I am because of my mom Whatever I have achieved, I owe to my mom iv Table of Contents ABSTRACT .................................................................................................................................... ii ACKNOWLEDGEMENTS ........................................................................................................... iii Table of Contents ............................................................................................................................ v List of Tables ............................................................................................................................... viii List of Figures ................................................................................................................................ ix Nomenclature ............................................................................................................................... xiv Chapter 1 ......................................................................................................................................... 1 1. Introduction and literature review ........................................................................................ 1 1.1. Introduction .................................................................................................................. 1 1.2. Literature review ........................................................................................................... 2 1.3. Motivation .................................................................................................................... 7 1.4. Aims of the thesis ......................................................................................................... 9 1.5. Thesis Outline ............................................................................................................. 10 Chapter 2 ....................................................................................................................................... 12 2. Dynamic Response Analysis for Mass-Spring Gyroscopes.............................................. 12 2.1. Introduction ................................................................................................................ 12 2.2. Model description ....................................................................................................... 13 2.3. Equations of Motion ................................................................................................... 14 2.4. Simulation of Deterministic Time Response .............................................................. 17 2.4.1. Introduction ......................................................................................................... 17 2.4.2. Numerical Simulations ........................................................................................ 17 2.4.2.1. Time response without input angular motion ................................................. 19 2.4.2.2. Time response with input angular motion ...................................................... 21 2.4.2.3. Frequency mismatch....................................................................................... 24 2.5. Simulation of Random Time Response ...................................................................... 26 2.5.1. Introduction ......................................................................................................... 26 2.5.2. Monte Carlo Simulation ...................................................................................... 27 2.5.3. Robustness of simulation .................................................................................... 29 2.5.3.1. Stochastic response simulation after peak-picking ........................................ 29 v 2.5.3.2. Optimal number of points along time response.............................................. 30 2.5.3.3. Discrete time steps.......................................................................................... 37 2.6. Closure ........................................................................................................................ 37 Chapter 3 ....................................................................................................................................... 38 3. Uncertainty Quantification for Mass-spring Gyroscope .................................................... 38 3.1. Introduction ................................................................................................................ 38 3.2. Optimal number of Samples ....................................................................................... 38 3.3. Uncertainty quantification .......................................................................................... 43 3.4. Uncertainty Quantification Results and Discussion ................................................... 45 3.4.1. Uncertainty in Input Angular Rate ...................................................................... 46 3.4.2. Uncertainty in Frequency Mismatch ................................................................... 47 3.4.3. Uncertainty in Quality Factor .............................................................................. 49 3.5. Frequency response .................................................................................................... 51 3.6. Closure ........................................................................................................................ 62 Chapter 4 ....................................................................................................................................... 64 4. Dynamic Response Analysis for Ring-based Gyroscopes ................................................. 64 4.1. Introduction ................................................................................................................ 64 4.2. Model description ....................................................................................................... 64 4.3. Equation of motion .................................................................................................... 65 4.4. Simulation of Deterministic Time Response .............................................................. 70 4.4.1. Introduction ......................................................................................................... 70 4.4.2. Natural frequency variation ................................................................................. 70 4.4.3. Numerical simulation .......................................................................................... 72 4.4.3.1. Time response without input angular motion ................................................. 73 4.4.3.2. Time response with input angular motion ...................................................... 75 4.4.3.3. Mass mismatch ............................................................................................... 78 4.5. Simulation of Random Time Response ...................................................................... 80 4.5.1. Introduction ......................................................................................................... 80 4.5.2. Robustness of simulation .................................................................................... 81 4.5.2.1. Stochastic response simulation after peak-picking ........................................ 81 4.5.2.2. Optimal number of points along time response.............................................. 82 vi 4.5.2.3. Discrete time steps.......................................................................................... 88 4.6. Closure ........................................................................................................................ 88 Chapter 5 ....................................................................................................................................... 90 5. Uncertainty Quantification for Ring-based Gyroscopes .................................................... 90 5.1. Introduction ................................................................................................................ 90 5.2. Optimal number of Samples ....................................................................................... 90 5.3. Uncertainty Quantification Results and Discussion ................................................... 95 5.3.1. Uncertainty in Input Angular Rate ...................................................................... 96 5.3.2. Uncertainty in Mass Mismatch ........................................................................... 97 5.3.3. Uncertainty in Damping Ratio .......................................................................... 100 5.4. Frequency response .................................................................................................. 102 5.5. Closure ...................................................................................................................... 113 Chapter 6 ..................................................................................................................................... 115 6. Conclusions ...................................................................................................................... 115 6.1. Summary of the thesis .............................................................................................. 115 6.2. Thesis contributions .................................................................................................. 117 6.3. Recommendations for future research ...................................................................... 117 References ................................................................................................................................... 119 Appendices .................................................................................................................................. 122 Curriculum Vitae ........................................................................................................................ 152 vii List of Tables Table 2- 1. Parameters of Mass-spring Gyroscope for the Numerical Simulations ..................... 18 Table 4- 1. Ring Parameters for the Numerical Calculations ....................................................... 71 viii List of Figures Figure 1-1. Analog MEMS Vibratory Gyroscope (reproduced from Giunta at el., 2006) ............. 3 Figure 1-2. Delphi‘s metal ring gyroscope (reproduced from the website of Silicon Sensing Systems Japan Ltd.) ........................................................................................................................ 4 Figure 2-1. Translation-based single-axis vibratory gyroscope .................................................... 13 Figure 2-2. Motion of a particle in body-fixed frame that rotates relative to an inertial frame .... 14 Figure 2-3. Radial displacement in the (a) driving direction and (b) sensing direction without input angular rate .......................................................................................................................... 20 Figure 2-4. Input angular rate time-profile ................................................................................... 22 Figure 2-5. Radial displacement in the (a) driving direction and (b) sensing direction with 𝛺 = 2π rad/sec input angular rate ................................................................................................ 23 Figure 2-6. Variation of radial displacement in the (a) driving direction and (b) sensing direction when frequency mismatch values change from 0 to 0.03% while one frequency is fixed another is changing for 𝛺 = 2π rad/sec input angular rate ....................................................................... 25 Figure 2-7. Time response after peak-picking for mass-spring gyroscope (𝛺=2𝜋 rad/sec) ......... 30 Figure 2-8. Number of points (time) Vs Mean along the time response for mass-spring gyroscope (a) without drift (b) with drift (𝛺=2𝜋 rad/sec) ............................................................................. 32 Figure 2-9. Number of points Vs standard deviation along the time response for mass-spring gyroscope (a) without drift (b) with drift (𝛺=2𝜋 rad/sec) ............................................................ 33 Figure 2-10. Radial displacement in the sensing direction with input angular rate (100 samples) ....................................................................................................................................................... 34 Figure 2-11. Number of samples Vs Ensemble Mean (a) without drift and (b) with drift (100 samples along path axis and 𝛺=2𝜋 rad/sec) ................................................................................. 35 Figure 2-12. Number of points Vs Standard deviation (a) without drift and (b) with drift (100 samples along path axis and 𝛺=2𝜋 rad/sec) ................................................................................. 36 Figure 3-1. Time response after peak-picking for mass-spring gyroscope (𝛺=2𝜋 rad/sec) ......... 39 Figure 3-2. Radial displacement in the sensing direction after peak-picking (50 samples) ......... 39 ix