Force Multiplier in a Microelectromechanical Silicon Oscillating Accelerometer. Nathan St. Nathan St. Mi-hel, MM. Allrights reserved.

Transcription

1 Force Multiplier in a Microelectromechanical Silicon Oscillating Accelerometer by Nathan St. Michel Submitted to the Department of Mechanical Engineering in partial fulfillment of the requirements for the degree of Masters of Science in Mechanical Engineering at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY -May Nathan St. Mi-hel, MM. Allrights reserved. The author hereby grants to MIT permission to reproduce and distribute publicly paper and electronic copies of this thesis document in whole or in part. A u th or Department of Mechanical Engineering a e, May 22, 2000 Certified by... David L. Trumper Associate Professor Thesis Supervisor C ertified by Marc $'. Weinberg Draper Laboratory e-,.-.2iiesis Supervisor Accepted by... Dr. Ain A. Sonin Chairman, Department Committee on Graduate Students MASSACHUSETTS ISTITUTE OF TECHNOLOGY SFP LIBRARIES

2 Force Multiplier in a Microelectromechanical Silicon Oscillating Accelerometer by Nathan St. Michel Submitted to the Department of Mechanical Engineering on May 22, 2000, in partial fulfillment of the requirements for the degree of Masters of Science in Mechanical Engineering Abstract This thesis describes the third generation design of a silicon oscillating accelerometer (SOA.) The device is a micromachined vibrating beam accelerometer consisting of two tuning fork oscillators attached to a large proof mass. When accelerated along the input axis, the proof mass exerts a force on the oscillators, causing their natural frequencies to shift in opposing directions. Taking the difference of these frequencies gives a measurement of the applied acceleration. A significant feature in the new design is a force multiplying lever arm which connects the oscillators to the proof mass. With this addition, the surface area of the proof mass is reduced by a factor of four, while the scale factor and axial resonant frequency of the previous design is retained. All vibration modes of the device are identified, and stiffness nonlinearity in the oscillators is investigated. A method to compensate for thermally induced acceleration errors is also introduced. Testing of the fabricated device is shown to agree with model predictions. Thesis Supervisor: David L. Trumper Title: Associate Professor Thesis Supervisor: Marc S. Weinberg Title: Draper Laboratory 2

3 Acknowledgments This thesis was prepared at The Charles Stark Draper Laboratory, Inc., under Draper Laboratory IR& D/CSR project Publication of this thesis does not constitute approval by Draper or the sponsoring agency of the findings or conclusions contained herein. It is published for the exchange and stimulation of ideas. Permission is hereby granted by the Author to the Massachusetts Institute of Technology to reproduce any or all of this thesis. A uthor Nathan St. Michel I am very grateful to all the people at Draper and MIT who have helped me to complete this project. In particular, I am thankful to Dr. Marc Weinberg and Prof. David Trumper for the enormous amount of guidance and instruction that they have given me. Without their influence, this thesis would have never happened. I am also thankful to all of the Draper staff, including Ralph Hopkins and the rest of the SOA team, who have helped bring this project to completion. Thanks to David Nokes and Greg Kirkos for their invaluable instruction and technical expertise. Thanks also to Bernie Antkowiak and Jim Bickford for their outstanding analysis work and help. I appreciate the help of Joe Miola and Rich Elliott, without whom there would be no test data to report. Jeff Borenstein, Lance Niles, Nicole Gerrish, and the rest of the fabrication team have been key to the success of the project, as have Paul Ward and the electronics team. Thanks to Rob White, Phil Rose, Dave Hom, Steve Daley, Paul Magnussen, and everyone else in the micromechanical test lab who were always willing to help out and always kept things interesting. Finally, I am very grateful for the support of my family and friends, who were always patient with me and full of encouragement. I feel blessed that God has placed you all in my life. 3

4 Contents 1 Introduction 10 2 Background Device Description Basic Operation Stiffness Nonlinearity General Formulation Error Predictions Thermal Sensitivity Capacitive Drive System Design and Analysis Previous Work Thermal Sensitivity Beam Models using ANSYS Detailed Analysis Modal Analysis Thermal Modeling Nonlinear Analysis Fabrication and Testing Fabrication Results Scale Factor and Drive Frequency Testing

5 4.3 Q Testing Temperature Sensitivity Axial Frequency Stiffness Nonlinearity Conclusions and Recommendations A Predicting Stiffness Nonlinearity B SOA Vibrational Modes

6 List of Figures 1-1 Completed SOA design Schematic diagram of SOA-2 (not drawn to scale) Simple model of flexure and oscillating mass Portion of capacitive comb drive Concept drawing of force multiplying lever arm connecting the proof m ass to the oscillator Plot of oscillator reaction forces vs. input force Spring model of SOA used in thermal analysis Concept drawing of oscillator and force multiplier with anchor beam repositioned for thermal compensation Beam model of SOA created in ANSYS Finite element model of one half of the SOA The contribution of base beam width to proof mass axial frequency, drive frequency, and total scale factor (frequency difference of two oscillators.) Lever arm width for this analysis is 125 pm The contribution of lever arm width to proof mass axial frequency and total scale factor (frequency difference of two oscillators.) Base beam width for this analysis is 45 pm Drive mode for SOA Out-of-plane mode for SOA Finite element model of oscillator and force multiplier used in nonlinear analysis

7 3-12 Predicted nonlinear stiffness curve for flexure displacement SOA with rotationally symmetric configuration SOA with mirror configuration Profile- of flexure beam Profile of comb finger Frequency response of an SOA Scale factor vs. drive frequency for twelve SOA units. Values calculated using FEA are also shown Oscillator position signal during ringdown Thermal sensitivity data for one SOA oscillator Thermal sensitivity data of the SOA. Here, the frequency difference has been taken to show the overall effect of temperature on 4-10 PSD of amplitude modulated drive signal showing the 3.4 the device. khz axial frequency of the proof mass Plot of measured and calculated nonlinear stiffness curves. A-i Modified spring model of SOA oscillator and force multiplier..... B-i Proof mass axial mode B-2 "Hula" mode in which flexures resonate in phase with one another.. B-3 Rotational mode in which the proof mass rotates about its center point B-4 Drive mode for SOA B-5 Cross-axis mode, similar to axial mode in Figure B-1, but along the axis perpendicular to the input axis B-6 Another in-plane mode that resembles a "tail wagging" motion of the oscillator B-7 First out-of-plane mode. This mode is symmetric to the other half of the device B-8 Second out-of-plane mode. This mode is antisymmetric to the other half of the device

8 B-9 Third out-of-plane mode. This mode is symmetric to the other half of the device B-10 Fourth out-of-plane mode. This mode is symmetric to the other half of the device B-11 Fifth out-of-plane mode. This mode is antisymmetric to the other half of the device B-12 Sixth out-of-plane mode. This mode is symmetric to the other half of the device B-13 Seventh out-of-plane mode. This mode is symmetric to the other half of the device B-14 First out of plane mode. This mode is symmetric to the other half of the device B-15 First out of plane mode. This mode is antisymmetric to the other half of the device

9 List of Tables 3.1 Modal frequencies of SOA with force multiplier. All in-plane modes were calculated using the full FEA model, and out-of-plane modes were calculated using the half FEA model

10 Chapter 1 Introduction In recent years micromechanical inertial sensors have been gaining popularity in a number of applications. The advantage to using these devices is their compact size, low weight, low cost, and high reliability. These benefits are gained by the use of semiconductor fabrication technology, which has the potential to produce large quantities of micromachined devices that are cheaper and physically less complex than their conventional counterparts. With these benefits, micromechanical sensors have the potential to become wide-spread not only in military applications, but in a variety of commercial uses as well. In addition to military applications including "smart" munitions and missile guidance, these sensors have a virtually endless number of other uses including automobile systems, factory automation systems, bio-medical devices, and many other areas where inexpensive, robust sensors are required [3]. A variety of micromechanical inertial sensors have been designed and fabricated from such materials as quartz and silicon. These include gyroscopes using tuning fork resonators that are sensitive to coriolis acceleration [11], [10] and vibrating beam accelerometers [8], [1]. Draper Laboratory is currently developing a version of a vibrating beam accelerometer called the Silicon Oscillating Accelerometer (SOA). This device consists of a large proof mass and two mechanical oscillators which are attached to it. The proof mass and oscillators are attached via anchors to a Hoya glass substrate below. The oscillators are each formed out of two flexure beams which are driven at resonance using a capacitive comb drive system. When the proof 10

11 mass experiences acceleration along the input axis of the device it exerts opposing tensile and compressive forces on the oscillators, causing the frequency of one to increase and the frequency of the other to decrease. By taking the difference of these frequencies, a measure of acceleration is obtained. The change in differential frequency per unit acceleration is known as the device's scale factor, and the unloaded natural frequency of an oscillator is referred to as the drive or bias frequency. Chapter 2 gives a more in-depth description of this device, along with the basic equations governing its operation. Two previous designs of this device have accomplished much by first proving the feasibility of the device and then refining its operation by increasing its sensitivity and developing a means for eliminating errors caused by temperature sensitivity. The first design, SOA-1, which was developed in Kevin Gibbons' Master's thesis [7], produced a device measuring 2,400,um square, with a scale factor of 4.0 Hz/g, a temperature sensitivity of 1.03 Hz/ 0 C (41 ppm/sc), and an oscillator resonant frequency of 25.1 khz. The second design, SOA-2, created at Draper Laboratory, produced a device with a scale factor of 100 Hz/g, a temperature sensitivity of approximately 1.6 Hz/ 0 C (75 ppm/ 0 C), and an oscillator resonant frequency of 21 khz. This dramatic improvement in scale factor was accomplished by altering the configuration of the oscillators and increasing the size of the proof mass to 1 cm square. Unfortunately, the large size of the proof mass has also presented a number of manufacturing difficulties in the microfabrication process. In the present project, we have developed new SOA design with performance similar to the previous SOA-2 design, but with one quarter the surface area. In addition to maintaining the previous scale factor and temperature sensitivity, it is also specified that the resonant frequency of the proof mass along the input axis be above 4 khz. This requirement is set so that externally applied vibrations, which are specified to exist below this frequency, will not cause false acceleration measurements. To meet these objectives, a design was created that incorporated the use of a lever arm to produce a factor of four force magnification [15]. This mechanism, attached to a modified oscillator, allowed the required proof mass size to be reduced while leaving 11

12 LOW MAG PHOTO OF THE DEVICE. Figure 1-1: Completed SOA design the scale factor constant. Other requirements were met by altering beam dimensions and anchor placement. The completed device is shown in Figure 1-1. Work during a previous thesis [13] provided the groundwork for this design. This work, which is discussed in Section 3.1, involved the creation of a Matlab script that allowed a network of one-dimensional beams to be constructed and analyzed for displacements and reaction forces. With this script, a number of early designs were analyzed, and requirements on necessary beam dimensions were found for the device to function as desired. Following this work, a more complex finite element model, described in Section 3.3, was created that again used one-dimensional beams for the oscillator and force multiplier, but also included the proof mass. This model was used to both verify the results from the previous analysis and to perform a preliminary calculation of the SOA's expected thermal sensitivity. Using a simple beam and spring model, described in detail in Section 3.2, the 12

13 behavior of the new SOA under thermal expansion was analyzed, and an optimal placement for the lever anchor point was found. With this information, a finite element half model of the proposed instrument was created, and modal analysis was performed to find the device's scale factor and natural modes. This analysis was later verified with a full FEA model. After final optimization of the beam dimensions, a thermal analysis was performed. A nonlinear analysis on the oscillator was also carried out to predict the cubic stiffness nonlinearity of the flexures during oscillation. These analyses are described in section 3.4. Unique contributions compared to previous work [13] include creation of a full finite element device model and meeting a number of design objectives such as temperature sensitivity, scale factor, drive frequency, and mode placement. The predicted scale factor of the new SOA design was 91 Hz/g, and the expected drive frequency was 20.9 khz. The axial frequency was calculated to be 4.1 khz, and the closest natural mode to the drive mode was at 89% of the drive frequency. Thermal modeling predicted that the device would have a temperature sensitivity of 0.87 ppm/ 0 C, and nonlinear analysis predicted a cubic stiffness nonlinearity that was four times greater than the previous SOA design. Devices which were fabricated in the Draper Laboratory micro-fabrication facility possessed thinner flexures and comb teeth than designed. Test results from these devices, given in Chapter 4, show that this decrease in mass and stiffness of the beams caused the average natural frequency of the devices to drop to 16.6 khz and the average scale factor to increase to 197 Hz/g. Cubic stiffness nonlinearity measurements were found to be close to predicted values. Temperature tests revealed a 196 ppm/ 0 C sensitivity of each oscillator, but common mode effects reduced the sensitivity of the overall instrument to 6.6 ppm/ 0 C. Further modeling of the device shows that this difference between expected and measured values of temperature sensitivity occurred because materials underneath the glass substrate were neglected in the initial modeling stage. 13

14 Chapter 2 Background 2.1 Device Description As discussed in [7], the SOA is a micromachined device formed from single crystal silicon using techniques common to semiconductor fabrication. It is a monolithic structure that consists of two oscillators attached to a large proof mass. A schematic representation is shown in Figure 2-1. Each of the oscillators is made up of two smaller masses, suspended by flexures, which are driven at resonance through an electrostatic comb-drive system. Acceleration is sensed by reading the change in resonant frequencies of the two oscillators due to loads placed on them by the proof mass. The proof mass is attached to the glass substrate via anchor beams, which are machined out of the proof mass. These structures are flexible in the direction of the input axis (which is the Y-direction in Figure 2-1,) but are rigid along the other axes. In this way only motion along the input axis is allowed. The oscillators each consist of two oscillating masses, which are placed in the center of a flexure. The flexures are attached at either end to base beams, which run between the flexures. The base beams are then attached to either an anchor at one end of the oscillator, or to the large proof mass at the other end. This configuration is sometimes referred to as a "tuning fork" design. The SOA is operated by driving each of the small masses in an oscillator at 14

15 X-axis -axis Y-axis (Input Axis) Anchor Point Silicon Proof Mass Base Beam A / A / '1 4AV( VI F~jih Jim i- - W C1+_ 0 go [-.L-7 aff6/i I - LL -- I ii 7 Oscillating Mass I / Line of Oscillating Flexure Mass Line Symmetry of Resonator II '/7 I Glass Substrate Glass Substrate

16 resonance (referred to as the drive or bias frequency,) 180' out of phase with the other, in the direction of the X-axis of Figure 2-1. Driving the oscillating masses out of phase with one another serves to cancel out reaction forces caused by each individual mass' motion. Excitation of the oscillators is accomplished using a capacitive comb-drive system, which is described in Section 2.5. When the SOA is accelerated along the input axis, the proof mass exerts an equal force on each of the two oscillators. Since the oscillators are arranged in an opposing orientation, one of the oscillators is placed in tension, while the other is placed in compression. These forces, which are transmitted equally to the flexures in the oscillators, cause the resonant frequency of the oscillator in tension to increase and that of the oscillator in compression to decrease proportional to the amount of acceleration applied to the device. Taking the difference of these frequencies provides a measure of the applied acceleration. The amount of differential change in oscillator frequency due to acceleration is known as the scale factor of the device. Since other factors, such as variations in device temperature or oscillator drive amplitude, affect each oscillator's frequency in the same way, most of the errors caused by such sources are rejected by taking the frequency difference. Slight manufacturing differences between the oscillators, however, cause small differences in each oscillator's sensitivity to these inputs. This situation prevents the common mode error rejection from being total. For this reason, a great deal of effort must be placed into reducing these types of error sources. 2.2 Basic Operation This section describes some of the basic equations governing the operation of the SOA. The flexures and masses that comprise each oscillator can be modeled as shown in Figure 2-2. In this model, the base beams are assumed to be much stiffer than the flexures. Each flexure can then be modeled as two cantilever beams with guided ends, with each attached to the mass, m, at the center of the flexure. Detailed finite element analysis in Section 3.4 confirms the validity of this model. 16

17 P Flexure Base Beam Oscillation ase Beam Figure 2-2: Simple model of flexure and oscillating mass. The transverse deflection of a cantilever beam with guided ends is given in Table 10 of [19] as X = 2 tan U) -- ki (2.1) kp 1 2 where W is the transverse load, P is the axial compressive load (applied to the flexures by the proof mass), 1 is the length of the beam, E is the Young's Modulus of Elasticity, I is the area moment of inertia of the flexure, and k = (P/EI) 1. The stiffness, K, of the beam is obtained by W _ kp K k i (2.2) X 2tan ()- k A linear approximation of the stiffness as a function of P can be made by substituting for k and expanding about P = 0. Dropping all second and higher order terms yields 12EI 6P K ~5 (2.3) 17

18 Since each flexure is made up of two of these beams, placed end-to-end, the total linearized stiffness for one flexure is then 24E1 12P (2.4) K(oscillator = The oscillating mass and flexure system of figure 2-2 can now be modeled as a second order mass-spring system. The natural frequency is then K(oscilator m 24EI = 12P 3 =5 (2.5) m Equation 2.5 demonstrates how the stiffness of the flexure, and thus the natural frequency of the oscillator decreases when the flexure is placed in compression and increases when it is placed in tension. This is the main effect we wish to utilize in order to measure the acceleration applied to the device. Other effects, however, also contribute to shifts in natural frequency. These error sources, which include cubic stiffness effects and temperature sensitivity, are introduced in the following two sections. 2.3 Stiffness Nonlinearity General Formulation An important property of the SOA flexures that affects the oscillators' resonant frequencies is stiffness nonlinearity. This condition describes spring softening or stiffening as displacement amplitude increases. This change in stiffness acts to increase or decrease the resonant frequency of the oscillator as the amplitude of oscillation increases. In this case, the springs stiffen with amplitude, and thus the resonant frequency is increased. To observe these effects, we first assume that the spring forces caused by a flexure's 18

19 transverse motion can be described by F = k 1 x + k 2 x 2 + k 3 x 3. (2.6) The undamped equation of motion for such an oscillator is then m, + k 1 x + k 2 x 2 + k 3 x 3 = 0. (2.7) If we assume a sinusoidal motion of the flexure of x(t) = X sin (wt) (2.8) then substituting into 2.7 yields -w2x sin(wt) + ki X sin(wt) + k2 X2 sin2(wt) m m + k3 X 3 sin 3 (Wt) = 0 m (2.9) k i 2 M- w in ' X sin(wt) + 1 k 2 2m -X 2 [m- cos(2wt)] + X3 - sin(wt) - I sin(3wt) 0. M 4 4 (2.10) Since this motion is occurring at resonance, the higher frequency terms involving 2w and 3w can be neglected, as can the bias term. This leaves the equation ki m _)2 + 3 k3x2 4 m )X sin(wt) = 0. (2.11) Solving for w 2 gives 2 = ki 3 k 3 X2 m 4 m (2.12) Expanding 2.12 about X = 0 gives an expression for the natural frequency as a function of amplitude wa = -+ m 3 ink 3 8 ki m (2.13) 19

20 Retaining the first two terms yields = I 1+ 3 X). (2.14) The change in frequency with amplitude is then -w - 3k3X2 wn 8 ki (2.15) Nayfeh and Mook [14] provide a similar derivation of this result, as well as other methods for finding a more accurate solution to this problem Error Predictions Since the SOA uses shifts in the oscillators' resonant frequencies to measure acceleration, any changes that occur because of the flexure stiffness nonlinearity will become sources of error. If the oscillators were able to be driven at a constant amplitude, both oscillator frequencies would shift by the same amount, and this error would be canceled by taking the difference of the two frequencies. However, since small variations in the drive amplitude exist, the stiffness nonlinearity will contribute to bias uncertainty. If we assume that instead of 2.8, the periodic motion of the flexures includes small amplitude variations X(t) = (X + AX) sin(wt) (2.16) then equation 2.15 becomes /dw 3k 3 Aw - -(X+ AX) 2. (2.17) Wn 8 ki Bias uncertainty is caused by frequency variations about a nominal, stiffnessshifted frequency. This variation is the difference between 2.17 and 2.15 (ZAw) 0 3 k3 k k(x + AX)2 X2 (2.18) Wn 8 k, 8ki 20

21 (Aw)o = 3 k Wn [2X (AX) + AX2 (2.19) 8 ki where (Aw). is the change in frequency (due to AX) about a nominal shifted frequency (due to X). A = If we now assume that the amplitude of oscillation can be controlled to some level -,and if we allow for the common mode effect to provide some error reduction, C, then equation 2.19 becomes 3 k 3 (Aw) 0 = 3k*nX 2 AC (2.20) where the higher order terms have been ignored, and C < 1. Equation 2.20 shows that, while improving amplitude control of the oscillators helps to decrease bias uncertainty, the largest factor in reducing this error is lowering the amplitude at which the oscillators are driven. 2.4 Thermal Sensitivity Temperature sensitivity in the SOA is caused by two factors which act against each other. The first of these factors involves the bonding of the silicon device to the Hoya glass substrate. The coefficient of thermal expansion (CTE) for Hoya glass (a,,= x 10-6 /oc) [9] is less than that of silicon (a, = 2.75 x 10-6 /oc) [6]. Because of the orientation of the oscillators and the placement of the oscillators' anchor points, an increase in device temperature causes the flexures to be placed in tension. This tension causes an increase in their natural frequency, according to equation 2.5. Section 3.2 provides an analysis of this effect. The second way in which temperature affects the natural frequency of the oscillators is by lowering Young's modulus of silicon as temperature increases. Equation 2.5 shows that the unloaded natural frequency of an oscillator depends on the square root of E. Wf Un 24EI 3. (2.21) m13 21

22 The change in the oscillators' natural frequencies as a result of the change in the Young's modulus of Silicon can be found by differentiating this expression. This gives Aw, 61 3 m I AE = EI 1 M11 (2.22) The fractional change in resonant frequency with the Young's modulus is then obtained by dividing equation 2.22 by equation 2.21 =Aw I w,, 2 (E E (2.23) As with the stiffness nonlinearity effect, if both oscillators responded in exactly the same way to temperature, differencing their frequencies would cancel out this effect. In actual devices, however, small manufacturing differences between the two oscillators cause slight variations in each oscillator's temperature sensitivity. To correct for this condition, the SOA can be designed so that the two temperature effects cancel each other in each oscillator. The Young's modulus of silicon decreases with increasing temperature by AEE AT- -50 ppm/ 0 C (-50 x 10-6 /oc) [12]. This change corresponds to a downward shift in the SOA's resonant frequency of -25 ppm/ 0 C. In section 3.2, an SOA design with close to zero temperature sensitivity is created by positioning the oscillator anchor points so that the CTE mismatch effect raises the natural frequency by this amount. 2.5 Capacitive Drive System The flexures in the SOA are excited at resonance through the use of intermeshed comb capacitor drives. Figure 2-3 shows a view of a portion of the comb drive. Each oscillating mass has two of these capacitors - one to drive the oscillation of the flexure, and the other to sense the flexure's displacement. A discussion of this type of drive capacitive system can be found in [16]. 22

23 Stator Comb Motion Oscillating Mass Figure 2-3: Portion of capacitive comb drive The capacitance between two charged parallel plates is where C = Av (2.24) 9 co =permittivity of free space A = area of comb finger overlap g = gap between comb fingers v = fringing coefficient (v ~ 1.5 for SOA combs). Each comb capacitor consists of N intermeshed fingers, which create 2N parallel plate capacitors. Referring to Figure 2-3, the total capacitance is then C=2N WV (2.25) 9 23

24 where I is the comb engagement, and w is the depth of the finger. During operation, engagement changes with the motion of the rotor comb, x, so that 1 = 1. - x, where lis the equilibrium value of comb overlap. These capacitors drive the motion of the flexure by using electrostatic forcing. For a capacitor, the force acting between the two sides is F = -V2 0 0 (2.26) 2 Ox where V is the voltage across the capacitor, and 2 is the change in capacitance with the transverse motion of the oscillating mass. For the capacitors described above, DC Ox cowii = 2N. (2.27) g Note that since the force depends on the voltage squared, applying a zero-bias sinusoidal voltage at the resonant frequency of the flexures will not create any force at that frequency. To remedy this problem, a bias voltage is added to the voltage drive signal [18]. While the flexure is driven at resonance by one comb capacitor on the oscillating mass, the other comb measures the flexure's displacement. This is accomplished by again utilizing the change in capacitance that corresponds to the motion of the comb fingers. The current caused by a change in capacitance of the combs is I = -(CV) dt DV DC C +V.(2.28) at at By setting the voltage across the capacitor to a constant level, half of this expression is set to zero so that I = VD C (2.29) at A voltage readout of position can then be obtained by passing this current into a preamplifier configured as an integrator [18]. 24

25 Chapter 3 Design and Analysis The first SOA design, SOA-1, created by Kevin Gibbons in his Master's thesis [7], had a slightly different configuration from that depicted in Figure 2-1, with two flexures attached to each oscillating mass. The proof mass was approximately 2.6 mm square and was 12 pm thick. This design had an oscillator frequency of 27 khz, a scale factor of 4 Hz/g, and a temperature sensitivity of 1.03 Hz/ 0 C (38 ppm/sc). To improve scale factor and decrease temperature sensitivity, David Nokes, of Draper Laboratory, created a second design, the SOA-2, with a layout similar to Figure 2-1. In this design, the scale factor was increased to 100 Hz/g by increasing the size of the proof mass to 1 cm square. This increase in mass provided a larger force per unit acceleration to be transmitted to the flexures, thereby increasing the oscillators' g-sensitivity. The proof mass thickness was also increased to 50 Pm. Nokes' design also featured an oscillator anchor point that was placed between the two flexures. This placement allowed the two thermal effects discussed in Section 2.4 to cancel each other, and gave the instrument very low thermal sensitivity. The oscillator resonant frequency of this device was 21 khz. This project's main goal was to overcome the manufacturing difficulties caused by the SOA-2's large size by designing a new device with the same performance as the current device, but with a proof mass that was at most one quarter its area. To accomplish this goal, six objectives were established [17]: 25

26 Anchor Point Proof Mass Anchor Beam Connecting Lever Arm Oscillating Mass Flexure Base Beam Anchor Point Figure 3-1: Concept drawing of force multiplying lever arm connecting the proof mass to the oscillator " The new device must have a scale factor of about 100 Hz/g. " The resonant frequency of the proof mass along the input axis must stay above 4 khz. This limit is set to prevent measurement errors that would be produced if external shocks and vibrations excited this mode. " The temperature sensitivity of the device should be as low as possible. This objective can be met by balancing the change in Young's modulus with stresses induced by the thermal expansion mismatch between the silicon device and the glass substrate. " The stiffness nonlinearity of the flexures must be as small as possible. * The oscillator resonant frequencies must be between 20 and 25 khz. In order to decrease the size of the proof mass, while retaining the sensitivity of the device, the concept of a connecting the oscillator to the proof mass with a lever arm was conceived [15]. A schematic of this idea is presented in Figure

27 As depicted in this figure, the force multiplier consists of four beams: a lever arm that provides the force multiplication, two vertical beams that connect the lever arm to the oscillator and the proof mass, and an anchor beam, which serves as the pivot point for the lever. By tuning the force multiplier to provide a 4-to-1 force amplification from the proof mass to the oscillator, the size of the proof mass can be cut by a factor of four. In this way, the mass would apply the same amount of force on the oscillator as in the previous design, and the scale factor would remain unchanged. 3.1 Previous Work Much of the groundwork for this project was established in my Bachelor's thesis, [13]. During this project, a Matlab script was developed that allowed structures to be modeled using one-dimensional beam elements. By assigning material properties to these elements, such as stiffness and area moment of inertia, the structure's behavior under loading could be studied. In the Bachelor's thesis, the focus of the design work was tuning the force multiplier to fit the existing oscillator. In this way, many of the characteristics of this oscillator, such as a resonant frequency of 21 khz and the placement of other vibrational modes away from the primary drive mode were retained. Since the Matlab model used linear beam equations, and since it was also not able to calculate thermal expansion, only scale factor and axial resonant frequency were investigated. To use the Matlab script, a beam model of the oscillator and force multiplier was created with a configuration similar to Figure 3-1. Forces modeling the action of the proof mass were added to one end of the force multiplier, while the anchor points were held fixed. Resonant frequencies of the oscillator drive mode were calculated by displacing the oscillating masses to mimic drive motion. By estimating the size of the masses and calculating the force required to displace them, a measure of resonant frequency was obtained. Scale factor was calculated by simultaneously applying a force mimicking the action of the proof mass to the lever arm, and finding the change 27

28 Reaction Forces vs. Vertical Connecting Beam Width R. eaction Force at Lever Anchor Point 0.2- S -- Input Force From Proof Mass ~0. u. Reaction Force at Oscillator Base Beam Desired Force Level Vertical Connecting Beam Width (non-dimensionalized) Figure 3-2: Plot of oscillator reaction forces vs. input force in resonant frequency. Axial frequency of the proof mass was calculated in a similar way by applying a displacement to the lever arm and calculating the resultant forces. The results of this analysis showed that because the beams in the force multiplier were built into each other, rather than connected with pin joints (as would be the case for an ideal lever) the dimensions of the lever arm must be greater than four to one. The extra forces created by this lever are taken up in the bending of the connecting beams and the lever arm itself. For this reason, the connecting beams were made much thinner than the lever arm. Figure 3-2 shows the reaction forces created at both the lever anchor point and the oscillator base beam (this is the force that is transmitted to the flexures.) Beam widths in this plot have been normalized against the lever arm width so that a value of 1 on the horizontal axis indicates that the connecting beams have the same width as the lever arm. This plot clearly shows that the lever arm must be much wider than the connecting beams to achieve sufficient force multiplication. In the finished design for this project, the lever arm was 4.5 times wider than the connecting beams. With this thickness and with a lever ratio of 4.2 to 1, the structure produced a force 28

29 Proof Mass Input Axis Attaches Here t P A Anchork X2 X4 X3 Lever Ann 2 Oscillator 1 Anchor Line of Symmetry Figure 3-3: Spring model of SOA used in thermal analysis multiplication of 3.8. The axial resonant frequency of the proof mass, which is also dependent on the size and thickness of the lever arm, remained at 4 khz at these dimensions. 3.2 Thermal Sensitivity In the SOA-2 design, pictured in Figure 2-1, the temperature sensitivity was controlled through the placement of the oscillator anchor point so that the thermal stresses induced in the flexure beams would counteract the decrease in Young's modulus with temperature. In the new SOA-3 design, the presence of two anchors, each moving with the expansion of the Hoya glass underneath, alters the forces that are transmitted to the flexures. In order to study the temperature sensitivity of the SOA-3 oscillator with a force multiplier, a simple model of the SOA was constructed using springs to represent the compliance of the various components involved. This model 29

30 was then used to find the forces transmitted to the flexures due to the thermally induced displacements of both the silicon device and the Hoya substrate. The following analysis follows the work done by Marc Weinberg at Draper Laboratory [17]. Figure 3-3 shows a spring model of the SOA oscillator that includes the force multiplier. Only one side of the SOA is depicted, and displacements, x, are given relative to the line of symmetry separating the two oscillators. In this model, the compliance of the flexures and base beams shown in Figure 3-1 are modeled as one spring, k 2. Since these beams, which are not pictured in Figure 3-3, can be modeled as springs connected in parallel, the total stiffnes of the oscillator is then k kbbkf (3.1) kbb + k 1 31 where kbb = stiffness of the base beams kf = stiffness of the flexures This spring is attached to an anchor point on one end and a beam representing the lever arm on the other. The lever arm is assumed to be rigid, compared to the other elements of the model. Also included in the model are the beams attaching the lever arm to the the proof mass and the second anchor point. The load applied externally by the proof mass is denoted P,, and internal loads of the springs are denoted Fe, where n corresponds to the spring number. These internal spring forces are defined as positive when the spring is in tension and negative when the spring is in compression. All elements of this model are assumed to be connected with pin joints so that there are no moments transmitted. The validity of this assumption was addressed in Section 3.1. We are interested in finding the force F 2 applied to the flexures due to the thermal expansion of the Hoya glass and silicon. This force can be found by first identifying the displacement relationships within the model and solving the resulting equations. The displacement of the oscillator anchor, x 1 due to the thermal expansion of the 30

31 glass substrate is where 1 = 1 1 aoat (3.2) 11 = the distance from the line of symmetry to the oscillator anchor point ag = the coefficient of thermal expansion for the Hoya glass substrate AT = temperature change For the oscillator, the internal spring forces are F 2 -- XI - l 2 aat. k2 (3.3) From the lever to the anchor point, F 3 = (l ) agat - X3-13 aat, k3 (3.4) and from the lever to the proof mass, F 4 - X5 - - l 4 cxat (3.5) where as is the coefficient of thermal expansion for silicon. that F3 + F4 = F2, Force balance requires (3.6) and torque balance gives nf 4 = F 2 (3.7) where n is the force multiplication (in this case n = 4.) Finally, for the lever arm, the nodal displacements have the relationship (X4 - x 3 ) = n(x 2 - X 3 ). (3.8) The stiffness of the structure along the input axis, kax, can be found by setting 31

32 AT = 0 and applying a proof mass load Pp = F 4. Solving equations 3.3 through 3.8 yields 1 x 5 n 2 (n -1) (3.9) kax Pp k 2 k3 k4 k 2 k 3 k 4 ka =.(.0 ax k 2 k 3 + (n - 1) 2 k 2 k 4 + n 2 ksk 4 Since the proof mass is much more rigid than either the oscillator or the force multiplier, we can assume that it moves independently from them under thermal expansion. With this in mind, the displacement of the proof mass at the point of connection with the force multiplier is then X5 = ( ) asat. (3.11) The force transmitted to the flexures due to both acceleration and thermal expansion can be found by solving this equation and equations 3.2 through 3.8 for F 2 Fflexure = F 2 = npp + nkax [( ) (n - 1) - 11] (ag - a,) AT. (3.12) In the SOA-2 design shown in Figure 2-1, the lack of a force multiplier (n = 1) causes the above expression to revert to the expected form, where the thermal component is described by -k (ag - as) AT. The axial stiffness term, kax, also reflects the added compliance produced by the lever arm. Since the area of the proof mass in the SOA-3 is reduced by a factor of four, the axial stiffness must decrease by the same amount for the axial resonant frequency to remain the same. With this in mind, we see that the term nkax remains unchanged between the SOA-2 design and the SOA-3 design (n = 4). Since the change in Young's modulus with temperature reduces the natural frequency of the oscillator, the thermal component of equation 3.12 must be positive to counteract this effect. The coefficient of thermal expansion for Hoya glass is less than that of silicon, so the term [( ) (n - 1) - 1i] must then be negative. In the previous SOA-2 design, this term was equal to -li, since there was no force multiplier. 32

33 Lever Arm Proof Mass Oscillating Mass Base Beam Flexure =--Anchor Point Figure 3-4: Concept drawing of oscillator and force multiplier with anchor beam repositioned for thermal compensation In the new design, the force multiplier anchor must be flipped, as in Figure 3-4, so that this term has the same value as before. 3.3 Beam Models using ANSYS As a first step in creating a full model of the SOA-3 using the concepts defined in the previous section, a second beam model, similar to the Matlab model of Section 3.1, but analyzed using ANSYS 5.5 finite element software, was created. This model, in which the oscillator was again modeled with one-dimensional beam elements, was more detailed than the previous model in that it included both the proof mass and the proof mass anchor beams. Including all of the device's components in this model allowed easy modeling of the full device later by simply laying the actual geometry over the beam elements. All geometry was created in Pro/Engineer and exported into ANSYS for analysis. This model is shown in Figure 3-5. To create the model, a three-dimensional shape representing the proof mass (the 33

34 Figure 3-5: Beam model of SOA created in ANSYS 34

35 shaded, meshed portion in figure 3-5) was first drawn in Pro/Engineer. Datum points were then drawn and connected with datum curves to form the shape of the oscillator and proof mass anchor points. Using the Pro/Engineer finite element package, onedimensional beam elements were then laid over the datum curves, and the proof mass geometry was meshed using 8-node two-dimensional shell elements (shell93 elements in ANSYS.) The beam elements were then assigned material properties and an area moment of inertia according to the actual dimensions of the beam that the element represented. The proof mass elements were also assigned material properties and a thickness of 50 pm. For this analysis, as well as future analyses, the following material properties were used: " E (Young's Modulus) = 1.68 x pn/pm 2 " v (Poisson's Ratio) = 0.26 * G (Shear Modulus) = 6.67 x pn/pm 2 " p (Density) = 2.33 x 10-/g/M2 For these material properties, a Young's modulus corresponding to beams in the (110) orientation was used, and isotropic properties were assumed for the Poisson's ratio and shear modulus. This simplification has little effect on the model since a slender beam approximation for bending, which does not include these properties, can be used for the flexure beams. Further finite element models using orthotropic properties have been used to validate this assumption. Once this model had been created, it was then exported into ANSYS. Oscillator drive frequency was calculated by first applying fixed boundary boundary conditions to all anchor points and then using ANSYS' Block-Lanczos solver to find the device's natural modes and frequencies. The proof mass axial frequency was also found in this manner. Scale factor was calculated by applying a ig acceleration to the model in the direction of the input axis. A static analysis was then performed to calculate the resulting stresses and displacements within the device. This prestress data was then 35

36 included in the modal analysis to find the device's new oscillator frequencies. Scale factor was then calculated by taking the difference of the shifted frequencies. This model predicted an oscillator drive frequency of 25.1 khz and a proof mass axial frequency of 5.1 khz. Because of the added stiffness of the proof mass anchor beams, the scale factor of this model dropped to 86 Hz/g. This number was increased by lengthening the lever arm by 30 pm so that the lever ratio was 5:1. This change resulted in a scale factor of 96 Hz/g. A thermal analysis was also performed on this model by including a representation of the Hoya glass substrate. This was done in Pro/Engineer by creating a large block of Hoya glass directly underneath the proof mass. The oscillator and proof mass anchor points were then connected to the glass using rigid beam elements, and the glass substrate was then meshed using the same type of elements as the proof mass. A coefficient of thermal expansion of ppm/ 0 C was then assigned to the Hoya glass, and one of 2.75 ppm/ 0 C was assigned to the silicon. Since the Hoya substrate is much thicker than the silicon, the elements representing it were given a much greater stiffness than silicon. In this way, the movement of the anchor points due to temperature changes would be driven solely by the Hoya glass. To find the thermal sensitivity of the model, a temperature increase of 1 'C was applied. A static analysis was then performed to find the stresses induced in the oscillator by the mismatch in CTE between the Hoya glass and silicon. A modal analysis was then performed with the prestress data included, and the resulting oscillator frequency was compared to the unloaded bias frequency. With an overlap between lever anchor and the oscillator anchor in Figure 3-4 of 6 = 43pum the thermal sensitivity due to CTE mismatch was calculated at +25ppm/ C. Since Section 2.4 shows that the change in the Young's modulus with temperature for silicon creates a -25 ppm/ 0 C change in oscillator frequency, this design was shown to have zero temperature sensitivity, in theory. With the proven concept of positioning the anchor to attain the desired thermal sensitivity, a detailed model of the device could then be constructed and analyzed. 36

37 Anchor Points Proof Mass Lever Arm Flexures Figure 3-6: Finite element model of one half of the SOA 3.4 Detailed Analysis With the knowledge gained from the preliminary analyses a detailed finite element half model of the SOA with force multiplier was created. This model is pictured in figure 3-6. A half model of the device was used in the analysis to reduce both modeling effort and computational time. This simplification is possible because of the symmetry of the device along the X-axis. This model was used for all design iterations, and a full model was then created to verify the performance of the final design. To create this model, Pro/Engineer was used to lay the actual beam geometries of the oscillator components directly over the datum curves of the model created in section 3.3. The thicknesses of each of the beams were determined initially from the area moments of inertia of the beams in the final one-dimensional model. Attaching the beam geometries to the datum curves from the previous model allowed for easy manipulation of dimensions during design iterations. 37

38 Once this model had been drawn in three dimensions in Pro/Engineer, its geometry was exported to ANSYS and converted into a two-dimensional model. ANSYS Plane82 elements were used to mesh the parts. These elements each have eight nodes and are used to model plane stresses. They were used in all in-plane modal and thermal analyses for this project. The same silicon material properties used in previous models were used except for the density of the proof mass silicon. Since, in production, holes are placed in the proof mass, its density was reduced by 10% in the model to reflect the reduced weight. The primary purpose of this analysis was to accurately characterize the device's performance. This characterization included finding all natural frequencies and mode shapes of the device, determining the device's thermal sensitivity, and finding the nonlinear characteristics of the flexures. Modal analysis was performed in the same fashion as the analysis in section 3.3. Fixed boundary conditions were placed on the anchor points, and, depending on the analysis being performed, either symmetric or anti-symmetric boundary conditions were placed on the proof mass symmetry line. An anti-symmetric boundary condition allowed free movement of the proof mass parallel to the input axis, but restricted all other movement. This boundary condition was used in finding scale factor and proof mass axial frequency, as well as several out-ofplane modes found in later analyses. Symmetric boundary conditions allowed only movement perpendicular to the input axis. This boundary condition was used to find the "cross-axis" frequency of the proof mass and also to find out-of-plane modes Modal Analysis The first goal in developing a final design was to fine tune the beam dimensions so that the design objectives outlined at the beginning of this chapter were met. The two beam dimensions studied in this exercise were the base beam width and the lever arm width. Both of these dimensions play a large role in the determination of scale factor and proof mass axial frequency. The Figures 3-7 and 3-8 show the results of design iterations performed with these parameters. The final SOA-3 design was chosen with a base beam width of 45 pm and a lever arm width of 125 pm. With 38

39 4500 Effect of Base Beam Width on Axial Freq., Drive Freq, and Scale Factor N Base Beam Width (microns) 0 U) C) Base Beam Width (microns) Base Beam Width (microns) Figure 3-7: The contribution of base beam width to proof mass axial frequency, drive frequency, and total scale factor (frequency difference of two oscillators.) Lever arm width for this analysis is 125 pm. 39

40 4500 Effect of Lever Arm Width on Proof Mass Axial Freq. and Scale Factor I a, U- x ) Lever Arm Width (microns) a) 0 Ci) ) I I I I I I I I I Lever Arm Width (microns) 160 Figure 3-8: The contribution of lever arm width to proof mass axial frequency and total scale factor (frequency difference of two oscillators.) Base beam width for this analysis is 45 /Lm. 40

41 these dimensions, the device had a predicted drive frequency of 20.3 khz, an axial proof mass frequency of 4.1 khz, and a scale factor of 91 Hz/g. These figures show that both parameters have equally important effects on the axial resonant frequency and scale factor of the device. For each parameter, a 50% decrease from the final dimension results in a 20% decrease in scale factor and a 35% decrease in axial frequency. It is observed that the scale factor nearly reaches a steady state with the chosen dimensions, so any further increase must be accomplished by increasing the lever ratio. Axial frequency would also reach a steady state if the thickness of the beams were increased further, but this amount of rigidity was not needed in the current design. The effect of base beam width on drive frequency is not as dramatic as the effects on scale factor and axial frequency, but it is still important in retaining the 20 khz drive frequency of the previous design. With these key parameters successfully defined, further modal analysis was conducted to check the placement of other vibrational modes of the device. In-plane modes were found using the same model as in the above analysis, and out-of-plane modes were found by replacing the elements with ANSYS Shell93 elements. These two-dimensional elements allow the modeling of thin structures in three-dimensions, as long as no torsional forces are present. For the SOA, these elements were an ideal choice to reduce model complexity, while ensuring accuracy. Two full models of the SOA-3 were also created to verify the placement of the in-plane modes. In these models, two oscillator configurations were used. The first configuration, pictured in a completed unit in Figure 1-1, arranged the oscillators in a mirrored orientation. The other configuration, shown in Figure 4-1, positioned the oscillators to be rotationally symmetric about the center of the device. Out-of-plane modes of the full models were not calculated because the use of shell93 elements produced very computationally intensive models. Table 3.1 gives a list of all the relevant modes of the device calculated with the half and full models. All in-plane modes are given using the results from the full models, and the out-of-plane modes are given using the half-model results. In this table, "Hula" refers to the mode in which the flexures oscillate in phase with one another. 41

42 Fundamental Modes (Hz) Drive 20, ,281.8 Axial 4,141 Other In-Plane Modes (Hz) "Hula" 16,702 16,721 Rotational 18,251 (89% of drive) Perp. to Flexures. 24,482 Out-of-Plane Modes (Hz) Low Freq. 5,223 5,900 6,161 11,107 11,510 15,544 Close to Drive 17,933 (88% of drive) High Freq. 26,436 28,915 Table 3.1: Modal frequencies of SOA with force multiplier. All in-plane modes were calculated using the full FEA model, and out-of-plane modes were calculated using the half FEA model. The "Rotational" mode, which at 18.3 khz is the closest to the drive mode, refers to the rotation of the proof mass about its center. Appendix B contains half-model diagrams of all of these modes. In each of the full models, two identical oscillators showed a drive frequency separation of 0.3 Hz, which is due to the coupling existing between them through the proof mass. This coupling prevents the two oscillators from operating independently from one another, and it creates the possibility that the two frequencies will "lock" together, making it impossible to take differential frequency measurements. Since this phenomenon, which is being studied further at Draper Laboratory [5], occurs when the two drive frequencies are close to one another, the final design submitted for fabrication contained oscillators with drive frequencies separated by 1000 Hz. This separation, which was created by thinning the flexures of one oscillator by 0.2 pm, avoids mode locking by reducing the possibility that the oscillator frequencies will cross one another during operation Thermal Modeling A thermal analysis similar to the one described in section 3.3 was performed on the finite element half model, and an optimal length for the force multiplier anchor beam 42

43 ANSYS MAY :12:41 DISPLACEMENT STEP=1 SUB =4 FREQ=20282 PowerGraphics EFACET=1 AVRES=Mat DMX =1 *DSCA= ZV =1 *DIST=2750 *XF =1250 Z-BUFFER Drive Mode Figure 3-9: Drive mode for SOA 43

44 ANSYS My :18:29 NODAL SOLUTION STEP-1 SUB -2 FREQ-6105 UZ (AVG) RSYS-0 PowerGraphics. --- EFACET-1 AVRES-Mat DM _ SM smx A c D a E 'F V2 I =1.475 Out of Plane Mode (Symmetric) Figure 3-10: Out-of-plane mode for SOA 44

45 was chosen. Rather than create a computationally intensive shell model containing both the silicon part and the glass substrate, the plane stress model created for modal analysis was used. To model the effects of both the glass and silicon expansion, the anchor points were given fixed boundary conditions, and the silicon part was given a CTE equal to the difference of silicon and Hoya glass. By doing so, the silicon expanded the same amount relative to the fixed boundary conditions as it would relative to the expansion of the actual glass substrate. This simplified analysis was verified by comparing its results with another analysis performed by displacing each node along the anchor point by an amount equal to the expansion of the glass substrate and then assigning the CTE of silicon to its true value. In each of these tests, the same temperature sensitivity was derived. Final analysis predicted a thermal sensitivity of ppm/oc with an anchor point overlap of 6 = 13pm. Combined with the change in Young's modulus with temperature, the sensitivity was predicted to be 0.87 ppm/ 0 C. A change in thermal sensitivity of ±0.14 (ppm/ C)/pm was predicted for movement of the anchor point around this position Nonlinear Analysis A nonlinear analysis was performed on the oscillator to predict the cubic stiffness terms introduced in section 2.3. To perform this analysis, a piece of the model from the above analyses was taken and modified slightly. Figure 3-11 shows the model with its various components labeled. In this model plane82 elements were again used, and fixed boundary conditions were placed on both anchor points. Since the proof mass is much stiffer than the oscillator, the end of the force multiplier connecting to the proof mass was also modeled as fixed. To find the force vs. displacement curve for the flexures, opposing loads were applied to the two small proof masses. A nonlinear analysis including stress-stiffening effects was then performed in ANSYS. During this calculation, the software breaks the force applied into a number of load steps. Within each step, the displacement of the flexure resulting from a portion of the total load is calculated, and the internal stresses resulting from this displacement are also found. The next load step then uses 45

46 Anchor points FNF7 11AMAN Flexure Base F Beam Small Proof Proof mass Masses Force connects Multiplier surface. igs Figure 3-11: Finite element model of oscillator and force multiplier used analysis in nonlinear 46

47 SX 108 Nonlinear Stiffness Curve for SOA Oscillator Flexure Displacement (um) Figure 3-12: Predicted nonlinear stiffness curve for flexure displacement these stresses in its calculation of force and displacement. This until the entire load has been applied. The result of this analysis process continues is shown in figure Equation 2.6 gives an expression for force vs. displacement for a cubic spring F = kix + k 2 x 2 + k 3 X 3. (3.13) The calculated coefficients for this expression are given below. Note that the presence of a quadratic term means that the flexures experience spring softening for inward displacements and spring stiffening for outward displacements. k, = x 106 pn/p-m k2= x 103 pn/pum 2 47

48 k3 = x 103 pn/pm 3 These coefficients have a ratio g-k- that is 4 times greater than those calculated for the previous SOA design, which had a similar oscillator, but no force multiplier. This difference can be explained by the change in axial stiffness that occurred between designs. Since the proof mass was reduced by a factor of four in the new design, the axial stiffness of the oscillator was also reduced by the same amount so that the proof mass axial frequency would remain constant. Appendix A gives a brief explanation of how this decrease in stiffness acts to increase the stiffness nonlinearity of the flexures. 48

49 Chapter 4 Fabrication and Testing The final design for the new version of the SOA was fabricated at Draper Laboratory using a silicon-on-glass bulk dissolved wafer process similar to one described in [7]. Figures 4-1 and 4-2 show typical devices mounted on the glass substrate. Two of the bond pads, which provide electrical connections for the device, can be seen on the middle right and left-hand side of the image. Two configurations of oscillators were fabricated so that both symmetric versions could be tested. Figure 4-1 shows one configuration in which the oscillators are rotationally symmetric with one another, and Figure 4-2 shows the other configuration in which the oscillators mirror each other. Despite finite element models of each of these configurations showing agreement with the half model predictions from chapter 3, it was decided that both configurations should be produced to ensure that any significant, unmodeled properties of the device had not been overlooked. In addition to the two oscillator configurations, three anchor beam lengths were also fabricated to test the thermal sensitivity predictions from section These three configurations had anchor point overlaps of 6 = -7, 13 (original dimension), and 33 ytm. Overlaps of this size were predicted to give the device a temperature sensitivity of 0.9 ± 2.8 ppm/ 0 C (Young's modulus thermal effect included.) In order to prepare the devices for testing, all manufactured units were first inspected visually. Units that were free from debris and had all beams, flexures, and comb teeth intact were tested using a probe station to ensure that all vibrational 49

50 Figure 4-1: SOA with rotationally symmetric configuration LOW MAG PHOTO OF THE DEVICE. Figure 4-2: SOA with mirror configuration 50