Simulation of the Effects of Acoustic Noise on MEMS Gyroscopes Except where reference is made to the work of others, the work described in this thesis is my own or was done in collaboration with my advisory committee. This thesis does not include proprietary or classi ed information. Grant Roth Certi cate of Approval: Dan Marghitu Professor Mechanical Engineering George T. Flowers, Chair Professor Mechanical Engineering Robert Dean Assistant Professor Electrical and Computer Engineering George T. Flowers Dean Graduate School Simulation of the Effects of Acoustic Noise on MEMS Gyroscopes Grant Roth A Thesis Submitted to the Graduate Faculty of Auburn University in Partial Ful llment of the Requirements for the Degree of Master of Science Auburn, Alabama August 10, 2009 Simulation of the Effects of Acoustic Noise on MEMS Gyroscopes Grant Roth Permission is granted to Auburn University to make copies of this thesis at its discretion, upon the request of individuals or institutions and at their expense. The author reserves all publication rights. Signature of Author Date of Graduation iii Vita Grant Roth, son of James and Rebecca Roth, was born on August 17, 1983, in Huntsville, AL. In 2001 he began attending Auburn University where he received his Bachelors of Software Engineering in 2005. He then enrolled in the Mechanical Engineering Masters program at Auburn University in 2006. iv Thesis Abstract Simulation of the Effects of Acoustic Noise on MEMS Gyroscopes Grant Roth Master of Science, August 10, 2009 (B.S., Auburn University, 2005) 120 Typed Pages Directed by George T. Flowers Recent advances in MEMS technology have resulted in relatively low cost gy- roscopes and accelerometers. The low cost of these devices has led to inexpensive inertial measurement systems, opening up a wide variety of possible applications for inertial measurement units (IMUs) with environmental conditions ranging from mild to harsh. This study focuses on MEMS gyroscopes, which are based upon vibratory, rather than rotational designs, that have been proven susceptible to the e ects of acoustic noise. In some aerospace environments this is particularly true. In these environments the noise levels can reach higher than 120 dB with frequencies reaching in excess of 20kHz. These e ects can overwhelm the output signals causing them to become extremely contaminated and even completely saturated. A model is developed to simulate testing the gyroscope exposed to high frequency noise. The model also simulates the use of several types of acoustic foams to mitigate the ef- fect of high frequency noise. The model simulates high frequency noise as high frequency vibration. Samples of the foams will be tested to determine their ability to mitigate simulated high frequency noise. The information gathered from these tests will be used in the model to mitigate the e ects of the high frequency acoustic v noise simulated by use of high frequency vibration applied to the gyroscope in the model. Following testing with the model, several ADXRS300 gyroscopes were tested in an acoustically harsh environment to determine the e ects of the high frequency acoustic noise. The foams were also tested to provide data to validate the model. vi Acknowledgments The author would like to thank Dr. George T. Flowers for his encouragement, guidance, and invaluable help at every stage of this work. The author also wishes to acknowledge Dr. Robert Dean and Dr. Dan Marghitu for their evaluation and advice with this work. The author would also like to express his gratitude to Simon Castro for help during the course of testing. The author would also like to thank his friends and family for their continued support during this e ort. vii Style manual or journal used Journal of Approximation Theory (together with the style known as \aums"). Bibliography follows van Leunen?s A Handbook for Scholars. Computer software used The document preparation package TEX (speci cally LATEX) together with the departmental style- le aums.sty. viii Table of Contents List of Tables xi List of Figures xii 1 Introduction 1 1.1 Motivations & Objectives . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Review of Previous Works . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Organization of this Thesis . . . . . . . . . . . . . . . . . . . . . . . . 4 2 Background 5 2.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.4 Acoustic Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3 Simulation 10 3.1 Modeling of Gyroscope Dynamics . . . . . . . . . . . . . . . . . . . . 10 3.2 Model Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 4 Experimental Setup 18 4.1 Experimental Test Setup . . . . . . . . . . . . . . . . . . . . . . . . . 18 4.2 Test Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 4.2.1 Baseline Testing . . . . . . . . . . . . . . . . . . . . . . . . . . 20 4.2.2 Foam Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 5 Conclusion 73 Bibliography 75 Appendices 77 A Derivation of Equations of Motion 78 ix B Simulation Code 80 B.1 Plotting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 B.2 Initialization Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 B.2.1 init Data.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 B.2.2 init Data2.m . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 B.2.3 init Data3.m . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 B.3 Files That Run ode45 . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 B.3.1 sim run.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 B.3.2 sim run2.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 B.3.3 sim run3.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 B.4 Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 B.4.1 sim try.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 B.4.2 sim try2.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 B.4.3 sim try3.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 x List of Tables 3.1 Initial data for simulations . . . . . . . . . . . . . . . . . . . . . . . . 11 3.2 Simulations performed . . . . . . . . . . . . . . . . . . . . . . . . . . 12 4.1 List of resonant frequencies of gyroscopes . . . . . . . . . . . . . . . . 21 4.2 Sound levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 4.3 The absolute di erence in output voltage between baseline of no noise to that of noise during tests with foams for the reference gyro . . . . 24 xi List of Figures 2.1 Simple vibratory gyroscope . . . . . . . . . . . . . . . . . . . . . . . . 6 3.1 Simulation results without noise and for 130dB with 0dB, 8dB, 10dB, and 15dB of isolation with noise . . . . . . . . . . . . . . . . . . . . . 13 3.2 Comparison of simulation output without noise and with 130dB of simulated noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.3 Comparison of simulation output for tests at 130dB without isolation and with 8dB of isolation . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.4 Comparison of simulation output for tests at 130dB without isolation and with 10dB of isolation . . . . . . . . . . . . . . . . . . . . . . . . 16 3.5 Comparison of simulation output for tests at 130dB without isolation and with 15dB of isolation . . . . . . . . . . . . . . . . . . . . . . . . 17 4.1 Gyroscopes mounted to the rate table . . . . . . . . . . . . . . . . . . 25 4.2 Gyroscope mounting plate . . . . . . . . . . . . . . . . . . . . . . . . 25 4.3 Basic con guration of the test setup . . . . . . . . . . . . . . . . . . . 26 4.4 Control area directly external to the acoustic chamber . . . . . . . . . 27 4.5 Block diagram of con guration for monitoring sound levels . . . . . . 28 4.6 Reference gyro frequency sweep from 6kHz to 18kHz . . . . . . . . . 28 4.7 Output response from the reference gyroscope at resonant frequency of 13907.77Hz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 4.8 Output response from gyro 2 at resonant frequency . . . . . . . . . . 30 4.9 Output response from gyro 3 at resonant frequency . . . . . . . . . . 31 xii 4.10 Output response from gyro 4 at resonant frequency . . . . . . . . . . 32 4.11 Output response from gyro 5 at resonant frequency . . . . . . . . . . 33 4.12 Output response from gyro 7 at resonant frequency . . . . . . . . . . 34 4.13 Output response from gyro 8 at resonant frequency . . . . . . . . . . 35 4.14 Output response from gyro 9 at resonant frequency . . . . . . . . . . 36 4.15 Output response from gyro 12 at resonant frequency . . . . . . . . . . 37 4.16 Output response from gyro 12 at resonant frequency with y-scale enlarged . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 4.17 Output response from gyro 13 at resonant frequency . . . . . . . . . . 39 4.18 Output response from gyro 13 at resonant frequency with y-scale enlarged . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 4.19 Output response from gyro 14 at resonant frequency . . . . . . . . . . 41 4.20 Output response from gyro 15 at resonant frequency . . . . . . . . . . 42 4.21 Output response from gyro 15 at resonant frequency with y-scale enlarged . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 4.22 Output response from gyro 16 at resonant frequency . . . . . . . . . . 44 4.23 Output response from gyro 18 at resonant frequency . . . . . . . . . . 45 4.24 Output response from gyro 19 at resonant frequency . . . . . . . . . . 46 4.25 Output response from gyro 19 at resonant frequency with y-scale enlarged . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 4.26 Output response from gyro 20 at resonant frequency . . . . . . . . . . 48 4.27 Output response from gyro 20 at resonant frequency with y-scale enlarged . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 4.28 Output response from gyro 21 at resonant frequency . . . . . . . . . . 50 4.29 Output response from gyro 21 at resonant frequency with y-scale enlarged . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 xiii 4.30 Output response from gyro 22 at resonant frequency . . . . . . . . . . 52 4.31 Output response from gyro 23 at resonant frequency . . . . . . . . . . 53 4.32 Output response from gyro 23 at resonant frequency with y-scale enlarged . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 4.33 Output response from gyro 24 at resonant frequency . . . . . . . . . . 55 4.34 Photograph of foam samples used during the experimental testing . . 56 4.35 Output responses from the reference gyro without foam and with gray foam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 4.36 Output responses from the reference gyro without foam and with black foam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 4.37 Output responses from the reference gyro without foam and with pink foam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 4.38 Output responses from the reference gyro without foam and with blue foam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 4.39 Output responses from the reference gyro without foam and with green foam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 4.40 A comparison of the reference gyro?s output signal for all foams . . . 62 4.41 Output responses from gyro 14 without foam and with gray foam . . 63 4.42 Output responses from gyro 14 without foam and with black foam . . 64 4.43 Output responses from gyro 14 without foam and with pink foam . . 65 4.44 Output responses from gyro 14 without foam and with blue foam . . 66 4.45 Output responses from gyro 14 without foam and with green foam . . 67 4.46 Output responses from gyro 16 without foam and with gray foam . . 68 4.47 Output responses from gyro 16 without foam and with black foam . . 69 4.48 Output responses from gyro 16 without foam and with pink foam . . 70 4.49 Output responses from gyro 16 without foam and with blue foam . . 71 4.50 Output responses from gyro 16 without foam and with green foam . . 72 xiv Chapter 1 Introduction Microelectromechanical systems (MEMS) are devices within the scale of 1mm to 1 m that combine electrical components with mechanical systems [1]. The small size of MEMS devices has opened up a wide variety of applications to sensing technology that had been previously limited due to size constraints. The MEMS gyroscopes in this study have recently become more widely used due to this reduction in size and the decrease in manufacturing costs. Applications for MEMS transducers in general have environments that range from completely benign to extremely harsh. Examples of harsh environments include high frequency vibration, extreme temper- atures, mechanical shock, and high frequency, high power acoustic noise. Recent studies have focused on the e ects of high frequency vibration, extreme tempera- tures and mechanical shock on such devices. Little research has been performed in the area of high frequency acoustic noise. This work is an investigation into mitiga- tion techniques, with the objective of providing the ground work for further research in the eld and further development of noise mitigation techniques for the MEMS gyroscope. 1.1 Motivations & Objectives This thesis is the result of a desire to investigate what e ects high frequency, high power acoustic noise can have on MEMS gyroscopes, and the attempt to miti- gate the adverse e ects caused by such noise. This is not meant to be an exhaustive 1 search of mitigation techniques, but rather a focused study of one particular device using speci c mitigation strategies. The major objectives are: ? To demonstrate that high frequency, high power acoustic noise has an e ect on a MEMS gyroscope ? To develop a model of a MEMS gyroscope exposed to high frequency, high power acoustic noise ? To examine strategies for isolating this model from simulated high frequency, high power acoustic noise ? To experimentally validate mitigation techniques of the e ects of the acoustic noise on the MEMS gyroscope. 1.2 Review of Previous Works A review of previous research in the eld of high frequency, high power acoustic noise on gyroscopes yielded few results. This section will provide an overview and summary of what was found and other similar studies in MEMS applications. Weinberg [2] discussed potential error sources in vibratory tuning fork gyro- scopes. The error sources discussed in this paper focus mainly on design aspects. Weinberg presented the following reasons for vacuum packaging: it reduces the re- quired driving force of the device; it increases bandwidth of the device; the resolution is enhanced due to lower damping; and it reduces the hydrodynamic lift. As an error source, Weinberg also cited mechanical quadrature caused by imperfections in the manufacturing process and electrical coupling of capacitance which causes false out- put signals. Saukoski discussed sources of zero-rate output errors identifying these 2 sources as manufacturing defects, interface electronics, environmental vibration and temperature. The article includes several methods of compensating for quadrature error, as well as methods for distinguishing the sources[13]. Pryputniewicz used an optoelectronic laser interferometric microscope to com- pare thermal expansion of attachment methods such as gold bump, gold/tin braze, and mechanical interposer. The deformation of each method was measured at di er- ent temperatures. This research indicated that the mechanical interposer produced the least deformation at higher temperatures of all attachment methods examined in the study. During the course of this research, it was also discovered that the anchors connecting the proof mass to the frame could have deformations of up to 40nm dependent on the fabrication process[3]. Dean, et al. proved that vibration at or near the resonant frequency of a MEMS gyroscope has an adverse e ect on the output signal. It was also proven that these e ects can be mitigated by integrating a low-pass lter into the die level packaging, by suspending the article between two membranes of rubbery or soft material, and by creating a low-pass lter in printed circuit board laminate[6]. This research was extended in another study using electrostatic actuators to create an active lter, allowing the lters to have a variable damping coe cient and spring constant. These active lters provided increased performance over the passive lters in the mitigation of vibration[7]. Another study by Dean proved that wide-band high power acoustic noise can have an e ect on MEMS gyroscopes. This study used four commercially available gyroscopes, the ADXRS300, the SiRRS01, the QDARS and the Tokin gyroscope. At noise levels approaching 100dB, it was seen that the gyroscope?s noise oors were increasing, and at levels near 130dB, the gyroscopes yielded no reliably usable 3 data[4]. In a continuation of the study by Dean, Castro subjected an inertial mea- surement unit containing multiple MEMS gyroscopes to the same wind tunnel noise used in the previous study[5]. The inertial measurement unit was isolated from the noise by surrounding it with di erent types and thicknesses of foams. The foams provided su cient isolation to prevent complete corruption of the output data. 1.3 Organization of this Thesis This thesis is organized as follows: Chapter 2 contains background information of gyroscopes including MEMS gy- roscopes. Chapter 3 describes the development of a simulation of MEMS gyroscopes and the simulation results. Chapter 4 discusses the experimental test procedures and results. Chapter 5 summarizes the results and discusses the observations and conclusions of this research. 4 Chapter 2 Background This chapter provides a brief description of the function of traditional gyro- scopes. It also provides a description of MEMS gyroscopes and applications for which these sensors are best suited. 2.1 History The inertial gyroscope dates back to the early 1800?s. Over the years, there have been a number of major advances in gyroscope technology. Until the development of MEMS devices, there were two major types of conventional gyroscopes- mechanical and optical. Traditional mechanical gyroscopes incorporate a spinning rotor free to rotate about a spin axis which is contained within a frame called a gimbal that is allowed to freely move in one or two axes[8]. Traditional optical gyroscopes utilize the Sagnac e ect using two beams of laser generated light traveling in two di erent directions through a ring of optical ber; the rate of rotation is calculated by the determining the distance the light has traveled[8]. The optical types of gyroscopes are not as a ected by harsh environments as are mechanical gyroscopes on the other hand; optical gyroscopes are not as compact as MEMS gyroscopes. 2.2 Background MEMS gyroscopes are fabricated using micromachining techniques. For those interested in speci c details, an excellent discussion and overview is provided in 5 Chapters 15-23 of The MEMS Handbook[1]. MEMS gyroscopes do not rely on a spinning rotor as used in conventional mechanical gyroscopes because fabricating rotating parts with signi cant useful mass is di cult at the micro level[10]. Instead, a mechanical member is driven to resonance in one axis which excites a secondary vibration in the same structure or a vibration in secondary structure, due to the Coriolis e ect[9]. Figure 2.1 shows a basic diagram of a vibratory gyroscope. Figure 2.1: Simple vibratory gyroscope The Coriolis acceleration and Coriolis force are de ned as: ~ac = 2~ ~vr (2.1) ~Fc = 2m~ ~vr (2.2) The particular gyroscope design that is the focus of this research is a tuning fork vibratory gyroscope and uses a combdrive to drive the proof mass to resonance, resulting in a Coriolis coupling between the two degrees of freedom. The basic 6 equations of motion for a vibratory gyroscope are similar to that of a spring mass damper system. The derivation of the following equations of motion is shown in Appendix A. m x+cx _x+ (kx m _ 2)x 2m _ _y m _y = Fd (2.3) m y +cy _y + (ky m _ 2)y + 2m _ _x+m _x = 0 (2.4) m (x2 +y2 + 112(l2x +l2y)) + 2m _ (x_x+y_y) my x+mx y = 0 (2.5) where m is the mass, kx;ky;cx; and cy are de ned as follows, Q is the quality factor, !d is the drive axis resonant frequency, and !s is the resonant frequency of the sense axis: kx = !2dm (2.6) ky = !2sm (2.7) cx = m!dQ (2.8) cy = m!sQ (2.9) while Fd is de ned as: Fd = 2:28N TVdcVac sin!drivetg c (2.10) Here N is the number of capacitors created by ngers in the combdrive, is the dielectric constant, T is the thickness, Vdc and Vac are the driving voltages, and gc is the gap between the ngers in the combdrive[11]. 7 2.3 Applications Traditional gyroscopes have limitations in some of the applications where the fu- ture use of IMUs is concerned. These applications are better suited for IMUs based on MEMS technology. There have been considerable advancements in microma- chined mechanical transducers in recent years, which have allowed the size, power requirements, and production cost of these devices to decrease. Low production costs, power requirements, and small size have allowed the introduction of MEMS transducers into a variety of new applications. Examples of these applications are consumer products such as videogames and cell phones, automotive stability control and rollover sensors, and small aerospace vehicles. These transducers are exposed to a variety of environments depending on their application. These environments range from benign to severe, where severe applications have harsh characteristics that can have an adverse e ect on the ability of these devices to perform prop- erly. Recently, research has focused on MEMS gyroscopes and the e ects caused by mechanical shock, high frequency vibrations, and extreme temperatures. With the use of MEMS gyroscopes in both aerospace and commercial applications, it is not di cult to identify environments that contain high power, high frequency acoustic noise. This high power, high frequency acoustic noise can have detrimental e ects on the output of these devices. 2.4 Acoustic Noise The average person?s hearing range is approximately 20Hz - 20kHz. The level of noise present in certain environments can be as high as or higher than 150dB. Examples of the dB scale of acoustic noise are as follows: 30dB would be the average noise level in a quiet place such as a library; 60dB is the average conversation; 90dB 8 would be tra c near an interstate highway; 110dB is the output of pneumatic tools; 120dB is considered to be painful without the use of hearing protection; 150dB and higher cause immediate hearing loss.[12] The gyroscopes used in this study were found to be e ected at levels around 100dB [5]. The e ects of high frequency acoustic noise can be transmitted to the device through contact between the package of the device and through the uid surrounding the device similar to the manner in which the e ects of high frequency vibration are transmitted to the device through the chassis. If the acoustic energy frequency components are close to the natural frequency of the mechanical structure in the MEMS gyroscope, it can produce undesirable motion of the sensor proof mass resulting in a corruption of the output signal. 9 Chapter 3 Simulation This chapter describes the simulation model development and the simulation results. 3.1 Modeling of Gyroscope Dynamics As described in Chapter 2, recall the equations 2.3, 2.4, 2.5, and 2.10. These are the equations of motion for a tuning fork gyroscope. The simulation consists of several MATLAB les, with one containing the equations of motion for the gyroscope with an additional noise term. The equation used to simulate the acoustic noise as a vibration is shown in 3.1. Fnoise = nsin!senset (3.1) This simulated noise term is introduced into the x and y equations of motion. The level of noise, n, is calculated before it is introduced to the equations of motion, using a method to acquire the correct values to provide similar results to that of the physical gyroscope. The equation follows where sound level is equivalent to the sound level in the chamber reduced by 100 and noise isolation is equivalent to insertion loss of the simulated acoustic foams. n = :75(sound level noise isolation) (3.2) 10 The frequency of the vibration is equivalent to that of the sense direction res- onant frequency with some of the simulations performed with the frequency above and below this value. The output of the simulation is converted into a voltage based on calculations performed on a sample ADXRS300, as shown in Equation 3.3. Vout = 3400(180 _ ) + 2:5 (3.3) The MATLAB run les documented in Appendix B run MATLAB function ode45 in three phases. In the rst and third phases, the amplitude is reduced to zero, while in the second phase the amplitude is varied for each test run. 3.2 Model Results Using the initial values found in Table 3.1, the model was simulated several times with di erences due only to the adjustment in the amount of simulated Noise Reduction. The test results are presented in Figures 3.1-3.5. These simulations were performed with no simulated rotation of the device to be similar to the tests conducted on the experimental subjects. Variable Value Timestep 0.1s SPL 130dB Freq 14000Hz Rotation 0 deg/s Table 3.1: Initial data for simulations 11 Test Noise Noise Reduction No Noise 0dB 0dB No Isolation 130dB 0dB 8dB Isolation 130dB 8dB 10dB Isolation 130dB 10dB 15dB Isolation 130dB 15dB Table 3.2: Simulations performed 12 Figure 3.1: Simulation results without noise and for 130dB with 0dB, 8dB, 10dB, and 15dB of isolation with noise 13 Figure 3.2: Comparison of simulation output without noise and with 130dB of simulated noise 14 Figure 3.3: Comparison of simulation output for tests at 130dB without isolation and with 8dB of isolation 15 Figure 3.4: Comparison of simulation output for tests at 130dB without isolation and with 10dB of isolation 16 Figure 3.5: Comparison of simulation output for tests at 130dB without isolation and with 15dB of isolation 17 Chapter 4 Experimental Setup This chapter describes the experimental test setup and discusses the tests and test procedures. 4.1 Experimental Test Setup The experimental tests were performed within an acoustic isolation chamber. The experimental con guration inside the chamber included a computer controlled single axis rate table from Aerotech. This rate table was attached to a massive aluminum base plate to prevent the device from rocking. In order to attach the gyroscopes to the table, a plate was designed and machined to allow the wires to have access to the mounting points that correspond to the individual gyroscope boards. This plate is adequate for mounting up to 5 individual gyroscope circuit boards, as seen in Figure 4.1. Figure 4.2 is a diagram of the plate(with the dimensions removed which allows the mounting positions to be observed). The rate table was surrounded by a frame which supported several acoustic drivers. The two drivers that provided the high frequency noise are Community Speakers VHF-100 drivers. These drivers are capable of outputs of up to 140dB at frequencies higher than 12kHz. Additional high range acoustic noise was provided by an EV DH1A all purpose driver. The other two drivers in this experimental con- guration are also made by Community Speakers. They are a M4 for the low range frequencies and an EM282 for the middle range. The frame provided adjustment of 18 the high and midrange speakers and held the low range speaker directly over the rate table. Figure 4.3 shows the basic layout of the test setup inside the acoustic cham- ber. The control area for the experiments was external to the acoustic chamber, as shown in Figure 4.4. The control area consisted of two computers, each containing a data acquisition card. The primary computer was responsible for generating the noise and recording the output from the gyroscopes. This computer provided the control for the rate table. The secondary computer was used only for monitoring and recording sound output levels via the piezo-electric microphone located inside the chamber. National Instruments Labview software was used to create virtual instrumentation to provide recordings of the data from both the gyroscopes and the microphone. The block diagram of this is shown in Figure 4.5. NCH Software?s Tone Generator was used to create the high frequency noise used in these experiments. This software has the ability to create multiple tones, sweeps over ranges of tones, and several types of noise generation, while allowing control over the dB level cre- ated in the software. The tones generated by the software were then passed through the computer?s sound card into two Crown XTi-1000 model ampli ers. The ability of the computer tone generating software to control the sound output level to the ampli ers simpli ed some tests by allowing more precise control over the pressure level in the chamber rather than controlling the analog inputs for output power level on the ampli ers. A script was written using AutoHotKey to make testing more e cient and to create less potential for human error. 19 4.2 Test Results 4.2.1 Baseline Testing The MEMS gyroscopes used during the experiments are model ADXRS300 from Analog Devices. Testing began with one gyroscope used as the reference. First, the device was subjected to high frequency sound sweeps between 6kHz and 18kHz to determine the frequency that had the most detrimental e ect on the output signal. A sample is shown in Figure 4.6. The high frequency sound sweep range was narrowed after calculating which frequencies were closest from visual inspection of the output signal. After narrowing the sweep to what was perceived to be the correct frequency, the tests were changed from continuous noise to 30 second tests with no sound for 10 seconds, followed by 10 seconds of sound, and completed with 10 seconds of silence. This format was used throughout the remaining tests. When the resonant frequency of the device was found, errors were generated in the device?s output signal as shown in Figure 4.7. After it was shown that high power acoustic noise had such a detrimental e ect, the other gyros were tested to see if they were responsive to such e ects. Figures 4.8-4.33 show the output of the gyros during these tests. It was found that the frequency of noise for the reference gyroscope was di erent from the rest of the gyroscopes tested. Table 4.1 shows the di erent resonant frequencies for each of the gyroscopes. The output for Gyro 2 in Figure 4.8 demonstrates the drastic e ect on the output signal due to the exposure to high power noise. The signal spikes downward to about 1.75V then promptly rises to almost 3.5V remaining at that level until the noise is removed. The output signal from this device during noise exposure is similar to that of a gyroscope rotating at an approximate rate of 120deg/s. Examining the output of Gyro 3 in Figure 4.9 reveals an immediate drop due to the introduction 20 Gyro Number Frequency (Hz) Reference (1) 13903.77 2 15386 3 16234 4 16340 5 15796 7 14255 8 16071 9 13836 12 15433.51 13 15576 14 15689.6 15 16009.62 16 15983.83 18 13766 19 13640.75 20 14041 21 15923.59 22 15732 23 15943.1 24 15933.5 Table 4.1: List of resonant frequencies of gyroscopes of the high power noise. This noise creates a false output signal at just below 2V, which would be the equivalent of a rotation at 60deg/s. The output from Gyro 4 in Figure 4.10 is similar to that of Gyro 3 but the signal has a sinusoidal pattern during exposure to the noise. This can cause detrimental e ect on a system as it is generating output signals similar to a Gyroscope that has a varying rotational rate. The e ects had on the output signal from Gyro 5 can be seen in Figure 4.11. The signal quickly spikes down to under 1.75V and then climbs back to 3.25V. Gyro 6 was non-functional during testing. Figure 4.12 shows a downward spike followed by an gradual increase to the baseline voltage followed by a small spike when the acoustic noise was removed from the environment. As shown in Figure 4.13 Gyro 8 experienced a drop in voltage as the noise was introduced. However, 21 as the noise continued the signal began to take on a sinusoidal pattern. Figure 4.14 shows the response of Gyro 9 to its resonant frequency having a sharp drop under 2V then immediate spike to almost 4V followed by a leveling o at 3V. This would be similar to a rapid change from no rotation to -60 deg/s then to 180 deg/s and changing to 60 deg/s. Gyro 10 was also non-functional for the tests performed. Gyro 11 was non- functional through out the testing process. Gyro 12 in Figure 4.15 presents an increase in the noise oor that can be seen in Figure 4.16. The detail in Figure 4.17 seems to not show change in output signal for Gyro 13. However in Figure 4.18 it also shows an increase in voltage and an increase in the noise in the output signal. Figure 4.19 shows an almost sinusoidal output signal that swings from -60 deg/s to almost 60 deg/s. Gyro 15 in Figure 4.20 demonstrates a minute reaction to the noise similar to those of Gyro 12 and Gyro 13. Figure 4.21 reveals that Gyro 15 has a small increase in voltage that gradually begins to decrease until the removal of the noise. Gyro 16 in Figure 4.22 reveals reactions similar to that of Gyro 14; however, the voltage swing is much less than that of Gyro 14. Gyro 17 was non-functional throughout testing. In Figure 4.23 Gyro 18 demonstrates a sudden drop o followed by erratic output signal changes until the removal of noise. Figures 4.24-4.25 show the output of Gyro 19. Figure 4.25 displays the details of an increase of voltage followed by sinusoidal drop and peak before the noise is removed. Gyro 20 shows a initial spike followed by a drop in voltage as seen in Figure 4.27. Gyro 21 in Figure 4.29 experiences a drop followed by a half sinusoidal motion that is not perceived in Figure 4.28. Figure 4.30 shows that Gyro 22 has inverted reaction to the noise that is similar to that of Gyro 5. Gyro 23 in Figure 4.32 demonstrates a small voltage change that 22 begins to drop through the duration of the noise portion of the test. The output of Gyro 24 in Figure 4.33 shows a voltage drop while exposed to the high frequency noise. While each gyro was a ected by the acoustic noise it was determined that the e ects of the acoustic noise did not have an established pattern. 4.2.2 Foam Testing Several di erent types of foams were used in the testing - a half inch thick gray closed cell foam, half inch thick black closed cell foam, and three di erent types of open cell foam (pink, blue, and green) which were all 1 inch thick. The black and gray foams were doubled in thickness for the tests to create one inch cubes of foam. The insertion loss for each of these foams is listed in Table 4.2. These tests were performed by inserting the microphone into the foam and measuring the sound level recorded. This shows that the black foam has an advantage at reducing the amount of noise in the baseline measured tests with the microphone. Foam Sound Level (dB) No Foam 130 Gray 122 Black 119 Pink 120 Blue 122 Green 121 Table 4.2: Sound levels The tests consisted of placing the shaped foam over the gyroscope then subject- ing the device again to its resonant frequency in order to determine what e ect the foam had as a mitigation technique. Figures 4.35-4.39 present a comparison of the e ects of the noise with and without each type of foam on the reference gyroscope. The average increase in output voltage due to the noise over the baselines at the 23 beginning and end of each test, where there was no noise for each type of foam, can be seen in Table 4.3. Foam Absolute Di erence (V) Gray 0.010681 Black 0.005188 Pink 0.006256 Blue 0.011750 Green 0.005646 Table 4.3: The absolute di erence in output voltage between baseline of no noise to that of noise during tests with foams for the reference gyro This shows that the black foam was the most e ective at mitigating the e ects of the noise for the reference gyro, as indicated by the smallest average voltage change. Figure 4.40 shows a comparison of the mitigation e ects of the foams. During testing of the mitigation techniques, several of the gyroscopes began to perform sporadically, perhaps due to the exposure to extended periods of high power, high frequency noise. There were only two other gyroscopes that would perform reliably during the testing. These gyroscopes were numbers 14 and 16. A comparison of the e ects of noise with and without each type of foam on Gyro 14 is seen in Figures 4.41-4.45. The data provided from Gyro 14 generated inconclusive results as the foams appear to have had no e ect on mitigating the acoustic noise at its respective resonant frequency. Gyro 16, however, as shown in Figures 4.46-4.50, provided useful data with the black foam having the greatest e ect at reducing the e ects of noise in Gyro 16. The results of these tests seem to indicate that the black foam has a better overall ability to mitigate the e ects. The results from Gyro 16 also con rm that the black foam has the highest noise reduction of all the foams. This seemed to be predicted as the insertion loss of the black foam was greater than that of the rest of the other types of foam. 24 Figure 4.1: Gyroscopes mounted to the rate table Figure 4.2: Gyroscope mounting plate 25 Figure 4.3: Basic con guration of the test setup 26 Figure 4.4: Control area directly external to the acoustic chamber 27 Figure 4.5: Block diagram of con guration for monitoring sound levels Figure 4.6: Reference gyro frequency sweep from 6kHz to 18kHz 28 Figure 4.7: Output response from the reference gyroscope at resonant frequency of 13907.77Hz 29 Figure 4.8: Output response from gyro 2 at resonant frequency 30 Figure 4.9: Output response from gyro 3 at resonant frequency 31 Figure 4.10: Output response from gyro 4 at resonant frequency 32 Figure 4.11: Output response from gyro 5 at resonant frequency 33 Figure 4.12: Output response from gyro 7 at resonant frequency 34 Figure 4.13: Output response from gyro 8 at resonant frequency 35 Figure 4.14: Output response from gyro 9 at resonant frequency 36 Figure 4.15: Output response from gyro 12 at resonant frequency 37 Figure 4.16: Output response from gyro 12 at resonant frequency with y-scale en- larged 38 Figure 4.17: Output response from gyro 13 at resonant frequency 39 Figure 4.18: Output response from gyro 13 at resonant frequency with y-scale en- larged 40 Figure 4.19: Output response from gyro 14 at resonant frequency 41 Figure 4.20: Output response from gyro 15 at resonant frequency 42 Figure 4.21: Output response from gyro 15 at resonant frequency with y-scale en- larged 43 Figure 4.22: Output response from gyro 16 at resonant frequency 44 Figure 4.23: Output response from gyro 18 at resonant frequency 45 Figure 4.24: Output response from gyro 19 at resonant frequency 46 Figure 4.25: Output response from gyro 19 at resonant frequency with y-scale en- larged 47 Figure 4.26: Output response from gyro 20 at resonant frequency 48 Figure 4.27: Output response from gyro 20 at resonant frequency with y-scale en- larged 49 Figure 4.28: Output response from gyro 21 at resonant frequency 50 Figure 4.29: Output response from gyro 21 at resonant frequency with y-scale en- larged 51 Figure 4.30: Output response from gyro 22 at resonant frequency 52 Figure 4.31: Output response from gyro 23 at resonant frequency 53 Figure 4.32: Output response from gyro 23 at resonant frequency with y-scale en- larged 54 Figure 4.33: Output response from gyro 24 at resonant frequency 55 Figure 4.34: Photograph of foam samples used during the experimental testing 56 Figure 4.35: Output responses from the reference gyro without foam and with gray foam 57 Figure 4.36: Output responses from the reference gyro without foam and with black foam 58 Figure 4.37: Output responses from the reference gyro without foam and with pink foam 59 Figure 4.38: Output responses from the reference gyro without foam and with blue foam 60 Figure 4.39: Output responses from the reference gyro without foam and with green foam 61 Figure 4.40: A comparison of the reference gyro?s output signal for all foams 62 Figure 4.41: Output responses from gyro 14 without foam and with gray foam 63 Figure 4.42: Output responses from gyro 14 without foam and with black foam 64 Figure 4.43: Output responses from gyro 14 without foam and with pink foam 65 Figure 4.44: Output responses from gyro 14 without foam and with blue foam 66 Figure 4.45: Output responses from gyro 14 without foam and with green foam 67 Figure 4.46: Output responses from gyro 16 without foam and with gray foam 68 Figure 4.47: Output responses from gyro 16 without foam and with black foam 69 Figure 4.48: Output responses from gyro 16 without foam and with pink foam 70 Figure 4.49: Output responses from gyro 16 without foam and with blue foam 71 Figure 4.50: Output responses from gyro 16 without foam and with green foam 72 Chapter 5 Conclusion Applications for MEMS gyroscopes have increased in recent years due to the small size and low cost of MEMS fabrication. Many of these environments include harsh conditions such as extreme temperatures, mechanical shock, vibration, and high frequency, high power acoustic noise. Studies have been performed by Weinberg on basic design error sources and by Pryputniewicz on the e ects of temperatures on the packaging and devices. Studies have been performed by Dean, et. al., on the e ects of vibrations and the means of isolation through the use of lowpass lters controlled with electrostatic actuation and signal processing [7] and by the use of mechanical lters made of either silicon or a polymer membrane[6]. The focus of this thesis was to demonstrate that MEMS Gyroscopes are susceptible to high power, high frequency acoustic noise and to evaluate ways of mitigating these e ects. The main task accomplished in this work was the creation of a model to simulate the e ects of high frequency acoustic noise on a MEMS Gyroscope. The model was created using the base equations of motion of a vibratory MEMS Gyroscope and introducing a vibration into the equations to simulate the e ects that acoustic noise would have on the gyroscope. This model was used to determine what type and amount of isolation were required to mitigate the adverse e ects on the gyroscope?s rate output due to acoustic noise. This data was then used to nd similar foams with characteristics that corresponded to the simulated isolation. Experimental testing began without isolation to determine the e ects on each of the gyroscopes that were used during the testing. These experiments demonstrated that there is 73 a wide range of e ects that this noise can have on gyroscopes. After investigation, foams were found with characteristics similar to those used in the simulation. Tests were run with these foams in order to determine the ability of each foam to mitigate the e ects of the acoustic noise. It was found that the denser black foam provided the best isolation from acoustic noise. The results from the gyro isolation tests also corresponded to the performed insertion loss tests. During the experimental testing several of the gyroscopes became unresponsive or intermittently responsive this could be due to prolonged exposure to the acoustic noise and warrants further investigation. While the mitigation techniques used in this work only used several di erent types of foams, the study has provided a good background for work into the isolation of MEMS devices from high frequency, high power acoustic noise. This study pro- vided a few means of mitigation of high frequency acoustic noise, the area remains open for further research into the eld of mitigation techniques for acoustic noise isolation of MEMS devices. 74 Bibliography [1] Gad-el-Hak, Mohamed, ed., The MEMS Handbook, CRC Press 2001. [2] Weinberg, Marc S.; Kourepenis, Anthony, \Error Sources in In-Plane Silicon Tuning Fork MEMS Gyroscopes," Journal of Microelectromechanical Systems, vol. 15, no. 3, June 2006, p 479-491. [3] Pryputniewicz, R J.; Marinis, Thomas F.; Soucy, Joseph W.; Furlong, Cosme, \Development of Packaging for MEMS Inertial Sensors," Record - IEEE PLANS, Position Location and Navigation Symposium, PLANS - 2004 Po- sition Location and Navigation Symposium, 2004, p 56-62. [4] Dean, Robert N. ; Flowers, George T.; Hodel, A. Scotte; Roth, Grant; Castro, Simon; Zhou, Ran; Moreira, Alfonso; Ahmed, Anwar; Rifki, Rifki; Grantham, Brian E.; Bittle, David; Brunsch, J, \On the degradation of MEMS gyroscope performance in the presence of high power acoustic noise,"IEEE International Symposium on Industrial Electronics, 2007 IEEE International Symposium on Industrial Electronics, ISIE 2007, Proceedings, 2007, p 1435-1440. [5] Castro, Simon; Roth, Grant; Dean, Robert; Flowers, George T.; Grantham, Brian, \In uence of acoustic noise on the dynamic performance of MEMS gyro- scopes," ASME International Mechanical Engineering Congress and Exposition, Proceedings, v 9 PART C,Proceedings of the ASME International Mechanical Engineering Congress and Exposition, IMECE 2007, 2008, p 1825-1831. [6] Dean, Robert ; Flowers, George; Hodel, Scotte; MacAllister, Ken; Horvath, Roland; Matras, Alex; Glover, Rob, \Vibration isolation of MEMS sensors for aerospace applications," Proceedings of SPIE - The International Society for Optical Engineering, v 4828, 2002, p 166-170. [7] Dean, Robert; Flowers, George; Sanders, Nicole; MacAllister, Ken; Horvath, Roland; Hodel, A. Scotteward; Johnson, Wayne; Kranz, Michael; Whitley, Michael, \Damping control of micromachined lowpass mechanical vibration isolation lters using electrostatic actuation with electronic signal process- ing,"Proceedings of SPIE - The International Society for Optical Engineering, v 5760, Smart Structures and Materials 2005 - Damping and Isolation, 2005, p 11-22. 75 [8] Franden, Jacob, AIP Handbook of Modern Sensors Physics, Designs and Ap- plications American Institute of Physics, 1993. [9] Beeby, Stephen; Ensell, Graham; Kraft, Michael; White, Neil, MEMS Mechan- ical Sensors, Artech House, Inc., 2004. [10] Kovacas, Gregory T.A., Micromachined Transducers Sourcebook, McGraw-Hill, 1998. [11] Patil, Nishad, \Design and Analysis of MEMS Angular Rate Sensors," Master?s Thesis, Indian Institute of Science Bangalore, 2006. [12] Harris, Cyril M., Handbook of Noise Control, McGraw-Hill, 1957. [13] Saukoski, Mikko; Aaltonen, Lasse; Halonen, Kari A. I., \Zero-Rate Output and Quadrature Compensation in Vibratory MEMS Gyroscopes," IEEE Sensors Journal, v 7, no. 12, December 2007, p 1639-1652. 76 Appendices 77 Appendix A Derivation of Equations of Motion The derivation of the Equations of Motion for a vibratory gyroscope using Lagrange?s Equation. First nd the kinetic energy T = 12m(~Vg ~Vg) + 12~! ~Hg (A.1) Given the following statements: ~ro = x^i+y^j (A.2) ~Vo = _x^i+ _y^j (A.3) ~! = _ ^k (A.4) ~Vg = ~Vo +~! ~ro (A.5) ~Hg = 1 12m(l 2 x +l 2 y) _ (A.6) This leaves us with: T = 12m( _x2 + _y2 2 _ y_x+ 2 _ x_y + _ 2x2 + _ 2y2) + 12( 112m(l2x +l2y)) _ 2 (A.7) Next the Rayliegh dissipation function D and the potential energy: D = 12cx _x2 + 12cy _y2 (A.8) 78 V = 12kxx2 + 12kyy2 (A.9) Then nd the Lagrange equations of motion by plugging A.7, A.8, and A.9 into: d dt @T @ _qi ! @T@ _q i + @D@ _q i + @V@q i = Qi (A.10) Where i is x;y; and Qx = Fd;Qy = 0;Q = 0 The equations of motion are: m x+cx _x+ (kx m _ 2)x 2m _ _y m _y = Fd (A.11) m y +cy _y + (ky m _ 2)y + 2m _ _x+m _x = 0 (A.12) m (x2 +y2 + 112(l2x +l2y)) + 2m _ (x_x+y_y) my x+mx y = 0 (A.13) 79 Appendix B Simulation Code B.1 Plotting % This program is designed to plot a % simulation of a 30 second test run % of the gyroscopes in the acoustic chamber % while not in motion % It is divided into 3 phases %************************************************************ %Phase 1 Ten seconds of Silence %************************************************************ tic clc clear delete *.mat run('sim run') clear run('init Data') format long int=Tfinal Timeglobal; n=Timeglobal/Tstep; Time1=Timeglobal; 80 if n==0; N=1; else N=n; end nfinal=(Tfinal)/Tstep; fileinit=int2str(N); load (fileinit, 't','z') i=1; X=z(1,1); X dot=z(1,4); Y=z(1,2); Y dot=z(1,5); Theta=z(1,3); Theta dot=z(1,6); for i=N:nfinal; file=int2str(i); load ( file, 't','z') ttemp=t; x=z(:,1); x dot=z(:,4); y=z(:,2); y dot=z(:,5); theta=z(:,3); theta dot=z(:,6); clear t z 81 Time1 = cat(1,Time1,ttemp); X dot=cat(1,X dot,x dot); X=cat(1,X,x); Y dot=cat(1,Y dot,y dot); Y=cat(1,Y,y); Theta dot=cat(1,Theta dot,theta dot); Theta=cat(1,Theta,theta); i=i+1; end index Time=length(Time1); index=length(Theta dot); if index Time==index voltage out1 = .0075*(180/pi*Theta dot)+2.5; end save('phase1','Time1','voltage out1') %************************************************************ %Phase 2 Ten seconds of Noise %************************************************************ run('sim run2') clear run('init Data2') format long int=Tfinal Timeglobal; n=Timeglobal/Tstep; Time2=Timeglobal; 82 if n==0; N=1; else N=n; end nfinal=(Tfinal)/Tstep; fileinit=int2str(N); load (fileinit, 't','z') i=1; X=z(1,1); X dot=z(1,4); Y=z(1,2); Y dot=z(1,5); Theta=z(1,3); Theta dot=z(1,6); for i=N:nfinal; file=int2str(i); load ( file, 't','z') ttemp=t; x=z(:,1); x dot=z(:,4); y=z(:,2); y dot=z(:,5); theta=z(:,3); theta dot=z(:,6); clear t z Time2 = cat(1,Time2,ttemp); 83 X dot=cat(1,X dot,x dot); X=cat(1,X,x); Y dot=cat(1,Y dot,y dot); Y=cat(1,Y,y); Theta dot=cat(1,Theta dot,theta dot); Theta=cat(1,Theta,theta); i=i+1; end index Time=length(Time2); index=length(Theta dot); if index Time==index voltage out2 = .0075*(180/pi*Theta dot)+2.5; end save('phase2','Time2','voltage out2') %************************************************************ %Phase 3 Ten seconds of silence %************************************************************ run('sim run3') clear run('init Data3') format long int=Tfinal Timeglobal; n=Timeglobal/Tstep; Time3=Timeglobal; 84 if n==0; N=1; else N=n; end nfinal=(Tfinal)/Tstep; fileinit=int2str(N); load (fileinit, 't','z') i=1; X=z(1,1); X dot=z(1,4); Y=z(1,2); Y dot=z(1,5); Theta=z(1,3); Theta dot=z(1,6); for i=N:nfinal; file=int2str(i); load ( file, 't','z') ttemp=t; x=z(:,1); x dot=z(:,4); y=z(:,2); y dot=z(:,5); theta=z(:,3); theta dot=z(:,6); clear t z Time3 = cat(1,Time3,ttemp); 85 X dot=cat(1,X dot,x dot); X=cat(1,X,x); Y dot=cat(1,Y dot,y dot); Y=cat(1,Y,y); Theta dot=cat(1,Theta dot,theta dot); Theta=cat(1,Theta,theta); i=i+1; end index Time=length(Time3); index=length(Theta dot); if index Time==index voltage out3 = .0075*(180/pi*Theta dot)+2.5; end load('phase1','Time1','voltage out1') load('phase2','Time2','voltage out2') Tot Time = decimate(cat(1,Time1,Time2,Time3),decval); Tot voltage=decimate(cat(1,voltage out1,voltage out2,voltage out3), decval); plot(Tot Time,Tot voltage),title('Voltage Output vs Time'), xlabel('Time (s)'),ylabel('Voltage Out (V)'),ylim([0,5]) RunTime = toc save('good/no noise','Tot Time','Tot voltage') 86 B.2 Initialization Data B.2.1 init Data.m eo=8.854e 12; er=1.00054; V0=12; %Voltage from charge pump Vac=6; %AC drive voltage num=2500; %Number of comb fingers on thick=4e 6;%Thickness of gyro structure widthleg=1.7e 6;%Width of legs m=8e 9; %Mass of Proof mass driveamp=7e 6;%amplitude of drive V bias= 12; %Volts Vn=10e 8; CapSum=12e 12; Capc=1e 13; Vc=.2; g=1.7e 6; phase shift=0; lc=1.7e 6;%50e 6; x0=7e 6;%amplitude of drive alpha=1e 6; l1=1e 3; l2=5e 4; fn=14000; %rate ADXRS300 proofmass is supposed to vibrate ratio= 1.2;%the ratio between drive and sense usually sense is %approximately 20% higher than drive Q=45;%QUALITY FACTOR 87 diffchange=2e 12/3e 6; feedback =2e 12; %Frequencies and Spring Rates wn1= 2*pi*fn; wn2= 2*pi*fn*ratio; k1= wn1*wn1*m; k2= wn2*wn2*m; c1= m*wn1/Q; c2= m*wn2/Q; decval = 200; %Rotational Rate Information rotationDegrees=0; % Initial Rotational Rate of the Gyroscope omega in=rotationDegrees*(pi/180); %Acoustic Noise Parameters noiseamount=0; %sound level in dB noisefreq=0; %noise isolation=0; Tstep=.1; Timeglobal= 0; Tfinal=10; 88 B.2.2 init Data2.m run('init Data') %Acoustic Noise Parameters soundlevel=30;%sound level in dB minus 100 noise isolation=0;%reduction in dB noiseamount=(soundlevel noise isolation)*.75; noisefreq=wn2; Timeglobal= 10; Tfinal=20; 89 B.2.3 init Data3.m run('init Data') Timeglobal= 20; Tfinal=30; 90 B.3 Files That Run ode45 B.3.1 sim run.m clc clear all close all run('init Data'); x0=0; xd0=0; xdd0=0; y0=0; yd0=0; ydd0=0; theta0=0; thetad0=omega in; thetadd0=0; z0=[x0,y0,theta0,xd0,yd0,thetad0]; options = odeset('RelTol',1e 3,'InitialStep', 1e 16); t=0; int=Tfinal Timeglobal; n=Timeglobal/Tstep; Time=Timeglobal; if n==0; 91 N=1; else N=n; fileinitial = int2str(N); load(fileinital,'z'); Lin=size(z); xinit=z(Lin(1),1); yinit=z(Lin(1),2); thetainit=z(Lin(1),3); xdinit=z(Lin(1),4); ydinit=z(Lin(1),5); thetadinit=z(Lin(1),6); clear z end n=n+1; while Timeglobal