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Unformatted text preview: UNIVERSITY OF CALIFORNIA IRVINE STABLE ELECTROSTATIC ACTUATION OF MEMS DOUBLEENDED TUNING FORK OSCILLATORS
THESIS submitted in partial satisfaction of the requirements for the degree of MASTER OF SCIENCE in Mechanical and Aerospace Engineering by Andreu FargasMarques Thesis Committee: Professor Andrei M. Shkel, Chair Professor Kenneth D. Mease Professor Athanassios Sideris 2001 c 2001 Andreu FargasMarques The thesis of Andreu FargasMarques is approved: Committee Chair University of California, Irvine 2001 ii Contents List of Figures List of Tables Acknowledgments Abstract of the Thesis 1 Introduction 1.1 1.2 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi xv xvi xvii 1 2 5 7 7 10 11 12 16 17 20 2 DoubleEnded Tuning Fork Design 2.1 2.2 Mechanical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . Dynamic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 2.3 Damping Forces . . . . . . . . . . . . . . . . . . . . . . . . . . Electrostatic Driving and Sensing . . . . . . . . . . . . . . . . . . . . 3 DETF Simulation and Stability Analysis 3.1 3.2 Dynamic Behavior of a DETF with Lateral Actuation . . . . . . . . . Numerical Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . iii 3.3 Pullin in Parallel Plate Actuators . . . . . . . . . . . . . . . . . . . . 3.3.1 3.3.2 3.3.3 Static Pullin Voltage . . . . . . . . . . . . . . . . . . . . . . . Dynamic Pullin Voltage . . . . . . . . . . . . . . . . . . . . . AC Dynamic Pullin Voltage . . . . . . . . . . . . . . . . . . . 25 27 31 38 49 51 51 52 55 56 59 62 65 65 70 70 71 75 80 83 83 85 3.4 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 DETF Prototypes 4.1 MUMPs Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 4.2 Process Description . . . . . . . . . . . . . . . . . . . . . . . . ........................ DETF Prototypes Design 4.2.1 4.2.2 Structural Design . . . . . . . . . . . . . . . . . . . . . . . . . Actuation and Sensing . . . . . . . . . . . . . . . . . . . . . . 4.3 4.4 Final Prototypes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Finite Elements Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Modal Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 5 DETF Testing 5.1 5.2 5.3 5.4 Experimental Evaluation Setup . . . . . . . . . . . . . . . . . . . . . Analysis of Prototypes . . . . . . . . . . . . . . . . . . . . . . . . . . Numerical Simulation Analysis . . . . . . . . . . . . . . . . . . . . . . AC Dynamic Pullin Analysis . . . . . . . . . . . . . . . . . . . . . . 6 Conclusions and Future Work 6.1 6.2 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv 6.2.1 6.2.2 6.2.3 AC Dynamic Pullin Characterizations . . . . . . . . . . . . . Oscillator Electronics . . . . . . . . . . . . . . . . . . . . . . . Frequency Shift Detection . . . . . . . . . . . . . . . . . . . . 6.2.3.1 6.2.3.2 Tunable DETF Oscillators . . . . . . . . . . . . . . . Angular Accelerometers . . . . . . . . . . . . . . . . 85 86 87 88 89 93 97 97 Bibliography A Matlab listings A.1 snap detf.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.2 eq snap detf.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 B Maple listings 102 B.1 tk lat bal 1b.mws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 B.2 AC dynamic pullin.mws . . . . . . . . . . . . . . . . . . . . . . . . . 104 C Ansys listings 107 C.1 lb1b beam4.inp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 C.2 lb1b beam4 sens.inp . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 v List of Figures
1.1 1.2 Schematic of a beam oscillating with electrostatic actuation and sensing Schematic of a DETF oscillator with electrostatic actuation and sensing 1.3 .................................. 3 2 Scheme of the dynamics of an electrostatic actuator. The force that appear between the plates depends on the voltage (V ) and the position of the plate (y ). ............................. ... 4 8 2.1 2.2 Vibrating beam oscillating under the inﬂuence of an axial force Force balance in a diﬀerential of a laterally oscillating beam. Adapted from [Rao, 1990] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 11 13 14 17 2.3 2.4 2.5 3.1 3.2 (a) Couette ﬂow damping; (b) Squeeze ﬁlm damping . . . . . . . . . (a) Lateral electrostatic actuator; (b) Parallel electrostatic actuator . (a) Lateral comb ﬁngers actuation; (b) Parallel plate actuation. . . . Beam with lateral electrostatic actuation and sensing . . . . . . . . . Frequency response of a DETF. The gain at the resonant pick is proportional to the quality factor. ................... 19 vi 3.3 Diﬀerent conﬁgurations for driving and sensing with electrostatic parallel plates. (a) Beam driven with lateral combs and motion sensed with parallel plates; (b) The driving and sensing is done with parallel plates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3.4 (a) Amplitude of oscillation of a beam driven with lateral combs; (b) Output current of a beam sensed with parallel plates . . . . . . . . . 21 3.5 Illustration of DETF response when the system is driven into oscillation using DC+AC signals combination. The DETF is driven using an AC signal at its mechanical natural frequency (fn =148 kHz, Q=100). Due to mismatch between driving frequency and fundamental frequency of the electromechanical system, increase of the DC bias does not always translate into increase of the amplitude. (a) Driven with 10 V DC and 15 V AC; (b) Driven with 15 V DC and 15 V AC; (c) Driven with 25 V DC and 15 V AC; (d) Driven with 50 V DC and 15 V AC. .... 22 3.6 (a) DETF Oscillating at its resonant frequency under the action of the control loop; (b) 5 V square wave signal that drives the DETF to resonance. ................................ 24 3.7 Amplitude and frequency of oscillation of the DETF (Q=30). (a) Relationship between the amplitude and the DC bias for a ﬁxed AC voltage; (b) Stiﬀness softening translates in shift of the frequency of oscillation with the increase of the DC bias. . . . . . . . . . . . . . . 25 26 3.8 Scheme of a parallel plate actuator . . . . . . . . . . . . . . . . . . . vii 3.9 Evolution of the potential energy along the gap between the electrodes. (a) Low voltages present stable and unstable static equilibrium; (b) The equilibrium points are points where the force of the capacitor is compensated by the restoring spring force. .............. 28 3.10 Evolution of the potential energy along the gap between the electrodes at Pullin Voltage. (a) The energy only presents an inﬂexion point; (b) The force of the capacitor is only compensated by the restoring spring force at one third of the gap, what corresponds to the inﬂexion point. 3.11 Evolution of the energy proﬁle in a DETF with a gap of 2 µm when changing the voltage between the plates. (a) Low voltages present stable and unstable static equilibrium; (b) At high voltages both equilibrium points get closer until they disappear at pullin voltage (61.9 V in the example). . . . . . . . . . . . . . . . . . . . . . . . . . 3.12 Evolution of the system when the voltage is applied as a step function. (a) Displacement of the moving electrode when the voltage is applied. It can be seen how the position of the electrode overshoots the stable equilibrium before it settles down; (b) Evolution of the energy for the same trajectory. The initial energy is dissipated due to the damping forces until it reaches the stable equilibrium. ............. 31 30 29 viii 3.13 Evolution of the system when the voltage applied leads the system to snap. (a) Displacement of the moving electrode when the voltage is applied. It can be seen how the position of the electrode reaches the unstable equilibrium. At that point, the evolution becomes unstable; (b) Evolution of the energy for the same trajectory. In the plot, the energy losses are small, so the energy is almost constant, reaching over the unstable equilibrium and leading the system to pullin. Energy levels above this line all lead over the unstable equilibrium. ..... 33 3.14 Pullin Voltage as a function of the Quality Factor. The Dynamic Pullin Voltage is 56.9 V for our example. . . . . . . . . . . . . . . . 3.15 (a) Evolution of the position of a DETF for a 56 V step input (Q=500); (b) Evolution of the same DETF showing snapping at 57 V. (Vpin = 61.9 V, Q=500) ............................. 36 35 3.16 Evolution of the energy for the examples in Figure 3.15. (a) Smooth energy evolution using a step of 56 V; (b) Unstable energy evolution using a step of 57 V. . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.17 The evolution of the energy including the dynamic component is compared to the static energy depending on the position in the gap for the examples in Figure 3.15. (a) The smooth evolution using a step of 56 V is bounded by the static energy; (b) The unstable evolution condition is achieved using a step of 57 V because the energy has reached over the unstable equilibrium. . . . . . . . . . . . . . . . . . 37 37 ix 3.18 Potential energy bounds when the system is oscillated with a DC bias voltage and an AC voltage. ....................... 39 3.19 Calculation of the maximum amplitude of oscillation for the simulated DETF depending on the combination of DC and AC voltages. (a) Plot of the amplitude for each voltage over which the DETF is at risk of snapping (Calculated for VD + VAC ); (b) Plot of the amplitude for each voltage over which the DETF is pulledin (Calculated for VD − VAC ). 3.20 Stable oscillation of a DETF. The potential energy curves deﬁne a steadystate oscillation or limit cycle. ................. 41 42 40 3.21 Oscillation of a DETF using 20 V DC and 5 V AC. . . . . . . . . . . 3.22 Evolution of the energy using 20 V DC and 5 V AC. (a) The energy of the system increases until it reaches the steady state oscillation; (b) Evolution of the energy in the gap. In stable oscillation, the energy is conﬁned between the static energy limits, without overshooting the unstable equilibrium. .......................... 42 3.23 Unstable oscillation. Beginning from the initial position, the amplitude of oscillation of the system increases until it reaches over the unstable equilibrium point for VDC + VAC . Then, the system snaps. . . . . . . 3.24 Unstable evolution of a DETF using 30 V DC and 5 V AC. (a) The amplitude of oscillation at the resonant frequency leads the DETF to the unstable region; (b) Closeup of the point where the unstable region is reached. The cross indicates the position where the unstable equilibrium point has been overshot. . . . . . . . . . . . . . . . . . . 44 43 x 3.25 Energy evolution when using 30 V DC and 5 V AC. (a) The energy increases without settling until pullin is reached; (b) The increase of the energy leads the position to overshoot the unstable equilibrium limit what leads to snapping. . . . . . . . . . . . . . . . . . . . . . . 3.26 AC Dynamic Pullin algorithm. The energy distribution is stable because amplitudes close to the unstable region decrease. ...... 46 44 3.27 AC Dynamic Pullin algorithm. The energy distribution is unstable because amplitudes close to the unstable region increase, leading the system to snapping. ........................... 46 3.28 AC Dynamic Pullin Voltage Curves. (a) Minimum DC bias and AC voltage that when applied produces snapping for a ﬁxed Quality Factor; (b) Evolution of the minimum bias that produces snapping for a ﬁxed AC signal in function of the Quality Factor. . . . . . . . . . . . . . . 4.1 4.2 4.3 MUMPs Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Section of a DETF designed in MUMPs ................ 49 52 54 The inclusion of the actuation mechanism increases the mass of the structure. (a) Lateral comb ﬁngers actuation; (b) Parallel plate 57 actuation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 (a) Doubleended tuning fork prototype with electrostatic lateral comb driving and sensing; (b) Closeup of the axial force actuator placed at one end of the DETF to test axial force sensitivity and frequency tunability. 4.5 ................................ ........ 64 65 Model using BEAM4 element for the prototype DL1B xi 4.6 First mode of oscillation for the prototype DL1B. It can be observed the outofplane motion. . . . . . . . . . . . . . . . . . . . . . . . . . 66 4.7 Second mode of oscillation for the prototype DL1B. Desired mode of inplane oscillation. ........................... 68 4.8 Third mode of oscillation for the prototype DL1B. Torsional mode of the actuation comb. ........................... 69 4.9 Fourth mode of oscillation for the prototype DL1B. Torsional outofplane oscillation of the actuation comb. ................ 69 5.1 Scanning Electron Microscope (SEM) picture of the 1cm × 1cm chip fabricated in the Cronos MUMPs run 42. In the image, the sector where the doubleended tuning forks are located. ........... 71 5.2 (a) Microscope probestation available in the Microsystems Lab at UCI; (b) Closeup of the precisionpositioners used to supply the power to the chip. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 72 5.3 5.4 Experimental setup used to test the prototypes. ........... (a) SEM picture of one of the doubleended tuning forks; (b) Picture of a doubleended tuning fork taken with the camera embedded in the microscope probestation. . . . . . . . . . . . . . . . . . . . . . . . . 73 5.5 (a) SEM of a beam testing structure with parallel plate actuation after snapping; (b) A laterallyactuated doubleended tuning fork burned after snapping. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 xii 5.6 Comparison between the electrostatic stiﬀness softening predicted by the model when increasing the DC bias and the experimental results. (a) Spring softening for 3 V AC; (b) Spring softening for 5 V AC; (c) Spring softening for 7 V AC; (d) Spring softening for 10 V AC. ... 76 5.7 Comparison between the amplitude of oscillation predicted by the model when increasing the DC bias and the experimental results. (a) Amplitude for 3 V AC; (b) Amplitude for 5 V AC; (c) Amplitude for 7 V AC; (d) Amplitude for 10 V AC. . . . . . . . . . . . . . . . . . . 78 5.8 Evolution of the amplitude of oscillation when increasing the DC bias; (a) Beam at rest; (b) Beam oscillating with 15 V DC and 7 V AC; (c) Beam oscillating with 18 V DC and 7 V AC; (d) Beam after snapping with 20 V DC and 7 V AC. . . . . . . . . . . . . . . . . . . . . . . . 79 5.9 Comparison of the values for which the experimental DETF was pulledin. The trend of the experimental results agree with that of the 80 simulations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.10 Comparison between the experimental amplitude of oscillation and the maximum stable amplitude for the diﬀerent combination of voltages. (a) Evolution for 3 V AC. The amplitude does not reach the unstable region; (b) Evolution for 5 V AC. Snapping occurs when the amplitude reaches the unstable region at 29 V DC; (c) Evolution for 7 V AC. Snapping occurs for 19 V DC; (d) Evolution for 10 V AC. Snapping occurs for 15 V DC; 6.1 ........................... ........................ 82 86 Vacuum testing chamber. xiii 6.2 6.3 6.4 6.5 DETF Oscillator electronics . . . . . . . . . . . . . . . . . . . . . . . Vibrating beam frequency shift under the action of an axial force . . DETF with force tuning in one of the ends. . . . . . . . . . . . . . . (a) Schematic of a linear accelerometer using a DETF; (b) Schematic of an angular accelerometer using the same principle. . . . . . . . . . 87 88 89 90 6.6 (a) SEM of the Asymmetric Vibratory Angular Accelerometer; (b) Closeup of the proof mass of the device. ............... 91 6.7 (a) SEM of the Symmetric Vibratory Angular Accelerometer; (b) Closeup of a DETF attached to the proof mass. ........... 92 xiv List of Tables
4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 5.1 MUMPs layers and attributes [Koester et al., 2001] . . . . . . . . . . Polysilicon properties .......................... .............. 55 56 58 60 61 63 63 64 67 Example of a DETF structural parameters Estimated DETF lateral combs parameters in air . . . . . . . . . . . Estimated DETF lateral combs parameters in vacuum at 1 Torr . . . Design parameters for testing prototypes . . . . . . . . . . . . . . . . Testing prototypes structural parameters ............... ....... Testing prototypes actuation and sensing voltages in air Modes of oscillation for the Actuation Prototypes. .......... Frequencies and voltages of actuation of the fabricated prototypes in air 73 xv ACKNOWLEDGMENTS I want to thank the Balsells family, the Generalitat de Catalunya and the University of California for their support through the Balsells Fellowship. Without their help this thesis wouldn’t have been written. My gratitude also to Professor Roger H. Rangel, coordinator of the Balsells Fellowship program, for his invaluable help before and after the fellowship was awarded. I would also like to thank Professor Shkel for his support and guidance, and the rest of students in the Microsystems Laboratory for their companionship and help. Thanks to my Balsells fellows for their support and friendship. Their presence here in UCI made more pleasant this year of work. Finally, thanks to my girlfriend and parents for their patience while I have been far away. xvi ABSTRACT OF THE THESIS STABLE ELECTROSTATIC ACTUATION OF MEMS DOUBLEENDED TUNING FORK OSCILLATORS
by Andreu FargasMarques Master of Science in Mechanical and Aerospace Engineering University of California, Irvine, 2001 Professor Andrei M. Shkel, Chair In this thesis, the study of stable electrostatic actuation is presented for doubleended tuning fork oscillators. This thesis proposes to use the analysis of the energy of the electromechanical system to deduce the conditions of stable electrostatic actuation. Two cases are considered: (i) Static case, when balance of the energy stored in the elastic deformation and the energy stored in the capacitive drive are contemplated; (ii) Dynamic case, when in addition to the energy balance of the static case kinetic energy is accounted for. Step function as well as harmonic excitation are considered in the second case. Understanding of the energy evolution of the system leads to the implementation of the AC Dynamic Pullin Algorithm. This algorithm allows to determine the stability of the electrostatic actuation of a doubleended tuning fork at resonance. The conditions of stability are validated experimentally using designed surfacemicromachined doubleended tuning forks. The issues involved in the fabrication are discussed. Applications of the devices for onchip frequency references and angular acceleration sensing are introduced. xvii Chapter 1 Introduction
The use of oscillators as a frequency source is common in many applications ranging from communications systems to wristwatches. To build an oscillator, a frequency source must be provided. A common source for these frequencydependent devices is a mechanical resonator. Micro Electro Mechanical Systems (MEMS) technologies such as quartz micromachining, silicon bulk micromachining, or polysilicon surface micromachining, all have demonstrated their ability to fabricate mechanical resonators. The stability of the frequency output is one of the main challenges when designing mechanical oscillators. A resonator like the doubleended tuning fork consists of a beam that is driven into resonant oscillation (Figure 1.1). The characteristics of the output signal are deﬁned by the mechanical parameters of the beam and the dynamics of its deformation. Stable frequency output depends on the amplitude of oscillation. The beam must oscillate at a constant amplitude and frequency. The actuation mechanism determines this capability. There are several types of MEMS oscillators using diﬀerent actuation 1 Figure 1.1: Schematic of a beam oscillating with electrostatic actuation and sensing mechanisms: piezoelectric actuation [Cheshmehdoost et al., 1994], electrostatic actuation [Beeby et al., 2000b], optical actuation [Zook et al., 1995] and thermal actuation [Minfan and Tien, 1999]. In MEMS, electrostatic actuation is considered a highly attractive actuation mechanism. Electrostatic actuators are simple to design and can be easily implemented using standard micromachining technologies [Tang, 1990]. On the other hand, the characteristics of the electrostatic forces, and specially their nonlinearity, require careful analysis. This is particularly important in oscillators where high amplitudes are desired. 1.1 Motivation The goal of this research is to analyze and derive conditions for stable electrostatic actuation of doubleended tuning forks. A Doubleended tuning fork (DETF) is a mechanical resonator that consists of two beams oscillated laterally in antiphase (Figure 1.2). DETF oscillators are interesting 2 MEMS devices because they can be used as onchip reference oscillators or as resonant sensors. In the latter case, the frequency of oscillation is proportional to the tension generated in the beams by the input forces to be measured. Figure 1.2: Schematic of a DETF oscillator with electrostatic actuation and sensing Electrostatic parallel plate actuation of microbeams was the ﬁrst type of design implemented in MEMS. This type of actuator is easy to implement and the forces generated are large. However, since the ﬁrst beam oscillators [Newell, 1968], it is widely accepted that the amplitude of electrostatically driven oscillations is limited by the nonlinearity of the actuation force. This nonlinearity is a source of actuation instability. An electrostatic actuator consists of two plates where the force is generated when a voltage diﬀerence is applied between them (Figure 1.3). One plate is ﬁxed and the opposite one is attached to the moving structure. The voltage applied generates the attractive force between the plates. Due to nonlinearity of this force, disbalance between electrostatic and restoring linear spring forces result in the actuation instability, thus leading to collision of the actuator plates. This phenomena 3 is called Pullin. Figure 1.3: Scheme of the dynamics of an electrostatic actuator. The force that appear between the plates depends on the voltage (V ) and the position of the plate (y ). It is possible to derive the pullin conditions. In the static case when electrostatic and spring forces are considered, electrostatic deﬂections are limited to one third of the initial actuator gap (g0 ) due to the pullin phenomena (Figure 1.3). The control voltage providing this displacement is called pullin voltage and it is the maximum voltage that can be applied without producing snapping. In the dynamic case, the pullin conditions are diﬀerent. Oscillation is a dynamic process where the kinetic energy and damping forces have to be taken into account. The importance of the inertia of the mass and damping forces in the pullin phenomena has been demonstrated in micromachined microrelays [Gupta and Senturia, 1997]. They characterized the case when the beam is forced to snap in order to close an electrical connection. The dynamics is introduced because it aﬀects the time constant of the pullin process. To the best of our knowledge, there is no rigorous analysis in the literature that 4 provides insight into the limits of stable actuation when oscillation is the desired eﬀect. To ﬁll this gap, this thesis studies the following issues: • Starting from the static analysis of the pullin eﬀect based on the energy method, the formulation is extended to the dynamic case. • The study of the energy evolution is used to extract conditions of stable actuation. The maximum amplitude of oscillation for a given voltage is analyzed. The AC Dynamic Pullin Voltage is introduced. An algorithm for prediction of stable actuation is presented. • Numerical integration of the analytical model is used to simulate the behavior of the doubleended tuning fork oscillators. Simulation allows to predict the evolution of the position of a DETF. Numerically obtained pullin conditions validate the results of the energy analysis. • Fabrication issues in surface micromachined doubleended tuning fork are studied. The analysis leads to the design of a set of prototypes to experimentally validate the dynamic pullin conditions. • Experimental data from the fabricated prototypes is presented. The analytical models and the conditions of dynamic pullin are validated experimentally. 1.2 Thesis Outline The thesis includes three parts: modeling and numerical simulation of doubleended tuning fork resonators, surface micromachining fabrication of prototypes, and testing 5 of the prototypes. Background on structural analysis and electrostatic actuation is presented in Chapter 2. The study allows to obtain an analytical model of the dynamic behavior of microbeams. The DETF model is used in Chapter 3 to simulate the dynamic behavior of a DETF oscillator. Analytical solution is not possible for this system due to the nonlinearities of the electrostatic actuation force. Instead, the energy method is used to deduce the conditions that lead to stable actuation. Chapter 4 discusses the implementation of doubleended tuning forks in a surface micromachining process. The chapter presents a study of the fabrication issues involved in the design of the actual devices. A family of designs used to validate the actuation principles is developed. The devices are experimentally tested in Chapter 5. The results are used to conﬁrm the conditions of stable actuation developed in the thesis. Finally, Chapter 6 reviews the main accomplishments of this research, as well as, the work that remains to be done. 6 Chapter 2 DoubleEnded Tuning Fork Design
This chapter derives the oscillatory behavior of a microbeam. The derived model is used to analyze the stability of actuation of doubleended tuning forks. In the mechanical analysis of a beam, its natural frequency is derived, as well as its sensitivity to axial forces. These are two main parameters in the design of a doubleended tuning fork. The dynamic model is completed with the introduction of the damping forces. Finally, electrostatic actuation and sensing is analyzed. The two diﬀerent conﬁgurations of electrostatic actuation are presented. The analysis is complemented with the study of the electrostatic sensing. The formulas derived here will be used in the next chapter to simulate and analyze the behavior of DETFs. 2.1 Mechanical Analysis The properties and behavior of a DETF are derived using the basics of beam theory. Each of the tines conforming a DETF can be analyzed independently as a beam oscillating laterally under the inﬂuence of an axial load (Figure 2.1). 7 Figure 2.1: Vibrating beam oscillating under the inﬂuence of an axial force The deﬂection of an homogeneous beam can be studied under the EulerBernouilli theory of thin beams. In this case, the force balance in a diﬀerential section of the beam is governed by the following diﬀerential equation [Rao, 1990]: ∂2 ∂x2 ∂2w EI 2 ∂x ∂ + ∂x ∂w F ∂x ∂ 2w + ρA 2 = P (x) ∂t (2.1) where w is the oscillation amplitude, ρ is the density of the beam, A is the area of the section, E is the Young Modulus, I is the moment of inertia, F is the axial force and P (x) is the actuation force (Figure 2.2). The case under analysis introduces the axial force in the equation because the goal is to obtain a relationship between the axial force applied and the behavior of the oscillation of the beam. Figure 2.2: Force balance in a diﬀerential of a laterally oscillating beam. Adapted from [Rao, 1990] Solving this equation analytically is only possible if a constant force is assumed. 8 Under this assumption, separation of variables can be applied for decomposition of the amplitude of oscillation as follows: w(x, t) =
i qi (t)φi (x) (2.2) where qi (t) is the timedependent modal displacement for the oscillation mode i and φi (x) is the positiondependent modal shape. Substituting and rearranging terms, the oscillation of each of the inﬁnite modes is governed by Mef f · qi + Kef f · qi = 0 ¨ where Kef f EI = L3
0 1 1 (2.3) d2 φi dε2 φ2 dε + i 2 F dε + L
0 1 dφi dε 2 dε (2.4) Mef f = ρAL
0 mj (φi (εj ))2
j (2.5) The equation has been normalized deﬁning ε = x/L as the normalized position along the beam ranging from 0 to 1. It has also been added the possibility of having masses, mj , placed along the beam at a distance εj . This could be needed to account for masses added by the drive and sense electrodes. With these equations, one can solve for the natural frequency of the beam for a particular oscillating mode: ωn = Kef f Mef f (2.6) and taking the derivative with respect to the axial force, obtain the frequency sensitivity to axial loads [Roessig, 1998]: 1 ∂fn = ∂F 4πωn Mef f L 9
1 dφ i dε 2 dε (2.7) 0 The sensitivity is the main parameter in any sensor. In this case, the sensitivity provides the amount of shift in the oscillation frequency due to the applied axial force. Based on this expression, it can be pointed out that in order to increase the sensitivity it is important to reduce both mass and stiﬀness while increasing the longitude of the beam 1 ∂ωn ∝ ∂F L Kef f Mef f (2.8) or, analyzing the geometrical content, reducing the width and thickness of the beam ∂ωn 1 ∝ 2√ ∂F b t Eρ (2.9) A thorough analysis of the main factors in the sensitivity equation can be found in [Roessig, 1998]. These conclusions will be used later in the design of the DETF prototypes. 2.2 Dynamic Model The mechanical analysis provides insight in the inherent characteristics of the doubleended tuning fork. However, as the beams are driven into resonance, the complete set of equations involving the dynamic behavior must be obtained. As have been presented previously, the behavior of a beam oscillated laterally is governed by a secondorder diﬀerential equation. More precisely, the behavior of each oscillation mode can be seen as a massspring system (Eq. 2.3). But this equation lacks of velocitydependent forces, also called damping eﬀects. Once the damping forces are introduced, the complete system can be modeled as a massspringdamper system, and for each of the oscillation modes, the equation can 10 be written as follows Mef f · qi + Bef f · qi + Kef f · qi = Pef f ¨ ˙ (2.10) what allows to study transient behavior and steadystate response once the actuating force Pef f is deﬁned. 2.2.1 Damping Forces In Micro Electro Mechanical Systems, there are two basic sources of damping forces: structural damping and viscous damping. The structural damping is generated by the molecular interaction in the material due to deformations. As should be expected, the values of these forces in the polysilicon are negligible compared to the viscous damping eﬀects. (a) (b) Figure 2.3: (a) Couette ﬂow damping; (b) Squeeze ﬁlm damping Two diﬀerent types of viscous damping can be identiﬁed in micromachined structures: couette ﬂow damping and squeeze ﬁlm damping. The former occurs when two plates move parallel one to the other and are separated by a Newtonian ﬂuid 11 (Figure 2.3a), and can be modeled as bcouette = µA g (2.11) While the later occurs when the plates move one against the other, trapping the ﬂuid between them (Figure 2.3b). In this case, the damping factor, using the HagenPoiseuille law is modeled by bsqueeze = 7 µAt2 g3 (2.12) where µ = 17.9e−6 P a · s is the viscosity of air at room pressure and µ = 3.7e−4 · p · g in low pressure (p << 50 torr) [Clark, 1997]. Note that g is the gap between surfaces, A the overlapping area and t the lateral dimension of the plates. With these deﬁnitions, the damping factor in the dynamic equation (2.10) is Bef f = bcouette + bsqueeze (2.13) 2.3 Electrostatic Driving and Sensing If electrostatic forces are used, an alternate voltage at the natural frequency of the beam provides the necessary excitation. Under this condition, the beam tends to resonance, oscillating freely with large amplitudes, what allows to generate oscillation without the need of a large power supply [Beeby et al., 2000a]. In resonance, the driving force is only needed to compensate for the losses produced by the damping forces that limit the free oscillation of the beam. Electrostatic forces appear when two conducting surfaces are held close together while charged with diﬀerent potential. Using this property, two types of actuators can be designed: a lateral actuator (Figure 2.4a) or a parallel actuator (Figure 2.4b). 12 (a) (b) Figure 2.4: (a) Lateral electrostatic actuator; (b) Parallel electrostatic actuator Given the capacitance between two parallel electrodes C = ε0 A g (2.14) the potential energy (U ) stored in a capacitor when a voltage (V ) is applied is U= 1 Q2 1 = CV 2 2C 2 (2.15) where Q is the accumulated electrical charge. Then, the force is deﬁned as the gradient of the potential energy F =− U = 1 ∂C 2 1 ∂C 2 V x+ Vy 2 ∂x 2 ∂y (2.16) where ε0 is the dielectric constant, A is the overlapping area, g is the gap between the electrodes, C is the capacitance, V is the voltage diﬀerence between the plates and x and y are the directions deﬁned in Figure 2.4. With this deﬁnition, if the displacement of the plates of the capacitor is in parallel in the xdirection, it is called lateral actuation (Figure 2.4a). In this case, the resulting force is independent of the displacement 1t F = ε0 V 2 2g (2.17) If the displacement of the plates is one against the other, in the ydirection, it is called parallel actuation (Figure 2.4b). In this case, the force varies nonlinearly with 13 the change of the gap between plates 1A F = − ε0 2 V 2 2g More rigorous studies can be found in the literature [Clark, 1997]. (2.18) (a) (b) Figure 2.5: (a) Lateral comb ﬁngers actuation; (b) Parallel plate actuation. Consequently, in order to drive the beam into resonance, its center is connected to an electrostatic actuator (Figure 2.5). With this conﬁguration, the force applied to the beam (Pef f ) is provided by lateral comb ﬁngers Pef f = or by parallel plate actuation Pef f = ∂C 1 ∂Cd 2 ε0 nf A Vd ; = 2 ∂q ∂q g2 (2.20) 1 ∂C 2 ∂C 2 ε0 nf t Vd ; = 2 ∂q ∂q g (2.19) where ε0 = 8.85e−12 F/m is the dielectric constant in free space, nf is the number of 14 comb ﬁngers, g is the gap between plates, Vd is the voltage between plates, t is the thickness of the plates and A is the overlapping area in the plates. To determine the frequency of oscillation when the structure is resonating, electrostatic combs can also be used. With a constant voltage applied between the capacitor plates, Vs , the relative movement between electrodes generates, by deﬁnition, a ﬂow of current i(t) = ∂C Vs ∂t (2.21) In this formula, the movement of the beam gives variations in the capacitance. Then, the relationship between the detected current and the displacement of the beam is as follows i(t) = Vs where value of
∂C ∂q ∂C ∂q ∂q ∂t (2.22) for lateral combs is given by equation (2.19), and the value for parallel plate sensing is given by equation (2.20). 15 Chapter 3 DETF Simulation and Stability Analysis
This chapter analyzes the dynamic behavior of the system based on the fundamental equations derived in the previous section. When lateral comb ﬁngers are used to drive and sense the oscillation of a beam, an analytical solution for the evolution of the beam exists. This simpliﬁes the selection of design parameters. When using parallel plates to actuate or sense, the nonlinearity of the drive force complicates the equations of motion and a closedsolution of the dynamic equation cannot be obtained. Consequently, numeric simulation is used to predict the behavior of the electrostatically driven beam. This chapter analyzes the dynamic behavior of the beam using both actuation mechanisms. Special emphasis is taken in delimiting the conditions under which parallel plates can be used to actuate a beam without reaching the instability region. The analysis provides insight in the characteristics of the pullin phenomena. 16 3.1 Dynamic Behavior of a DETF with Lateral Actuation An example of a beam where actuation and sensing is done using electrostatic lateral comb ﬁngers is presented in Figure 3.1. Figure 3.1: Beam with lateral electrostatic actuation and sensing From Chapter 2, the dynamic model is known to have the following form Mef f · q + Bef f · q + Kef f · q = ¨ ˙ 1 ∂Cd 2 1 ∂Cs 2 V− V 2 ∂q d 2 ∂q s (3.1) where Mef f , Bef f , Kef f are the dynamic parameters of the beam, q is the amplitude of oscillation for a given mode, Cd and Vd are the capacitance and voltage applied to the driving combs, and Cs and Vs are the capacitance and voltage applied to the sensing combs, respectively. The output current is obtained according to the equation i(t) = Vs ∂Cs ∂q ∂q ∂t (3.2) 17 In the case of lateral combs, equation (3.1) can be solved analytically. A beam driven with a sinusoidal voltage and a DC bias Vd = vD + vd · sin(ω · t) and with a bias voltage at the sensing combs of Vs , has the following steadystate response q (t) = 12 1 ∂Cs 2 1 ∂Cd 2 vD + vd − V 2Kef f ∂q 2 2Kef f ∂q s (∂Cd /∂q )vD vd + sin(ω · t + ψ1 ) 2 (Kef f − Mef f ω 2 )2 + Bef f ω 2
2 (∂Cd /∂q )vd 2 4 (Kef f − 4Mef f ω 2 )2 + 4Bef f ω 2 (3.3) − cos(2ω · t − ψ2 ) The equation shows that using lateral combs the ﬁrst term is constant while other terms are small if Vd = Vs ,
∂Cd ∂q = ∂Cs . ∂q Moreover, if vd << vD , the last term can be considered negligible because it includes a second order term. Consequently, under the above assumptions, the amplitude of oscillation is approximately q (t) ≈ (∂Cd /∂q )vD vd
2 (Kef f − Mef f ω 2 )2 + Bef f ω 2 sin(ω · t + ψ1 ) (3.4) It can be observed that the amplitude is maximized when the frequency of the input voltage (ω ) is equal to the natural frequency of the system (ωn ), ω = ωn . This is the condition of resonance (Figure 3.2) qres ≈ (∂Cd /∂q )vD vd (∂Cd /∂q )vD vd =Q 2 Bef f ωn ωn Mef f (3.5) The equation is useful to introduce a constant, Q, which is used to quantify the value of the resonant peak. Its magnitude is proportional to the amplitude at resonance and is called Quality Factor. 18 The Quality Factor is directly related to the damping constant (ξ ) of the structure Q = ωn Mef f 1 = Bef f 2ξ (3.6) This can be The quality factor of an oscillator characterizes the energy losses. translated to amplitude of oscillation at resonance. Higher amplitude is achieved when the quality factor is large because there is less dissipation. Usually, resonant oscillation is desired in order to achieve maximum amplitude with the minimum driving force. Thus, the Quality Factor is a parameter that determines the gain achieved when the system is at resonance. Large value of the Quality Factor is a desired property of resonators.
Bode Diagram −180 −200 Magnitude (dB) −220 −240 −260 −280 0 −45 Phase (deg) −90 −135 −180 10
5 10 6 10 7 Frequency (rad/sec) Figure 3.2: Frequency response of a DETF. The gain at the resonant pick is proportional to the quality factor. Finally, knowing the behavior of the beam, the value of the output current can be 19 estimated. Using equation (3.4), the sensed current is i(t) = Vs ∂Cs ∂q ∂q ∂t Vs ω ∂Cs ∂q (∂Cd /∂q )vD vd cos(ω · t + ψ1 ) (Kef f − Mef f ω 2 )2 + B 2 ω 2 (3.7) 3.2 Numerical Simulation An important step in any design process is the prediction of the expected result. Given the equation of motion (3.1), the diﬀerential equation can be solved numerically. The equations derived in the previous section do not apply if parallel plates are used for actuation. The nonlinearity of
∂Cs ∂q prevents from solving the equation analytically. Figure 3.3 shows two conﬁgurations that can only be analyzed using numerical methods. (a) (b) Figure 3.3: Diﬀerent conﬁgurations for driving and sensing with electrostatic parallel plates. (a) Beam driven with lateral combs and motion sensed with parallel plates; (b) The driving and sensing is done with parallel plates. Using simulations, the response of the DETF to the applied voltages can be 20 predicted. This allows to estimate the expected evolution, output voltage and stability of the actuation scheme. These parameters are extensively used during the design process. The simulations have been carried out using one of the ODE solvers (ODE45) of MATLAB 6.0, MathWorks (Appendix A). A typical output from the simulation is shown in Figure 3.4. The amplitude of oscillation of the DETF (Figure 3.4a) and the current detected at the output (Figure 3.4b) are presented for one of the simulated models where lateral combs are used for driving and parallel plates for sensing (vD = 15 V, vd = 5 V, Vs = 15V ).
3 x 10
−7 3 x 10 −9 2 2 1 amplitude of oscillation (m) 1 0 output current (A) 0 0.5 1 time (sec) 1.5 2 x 10 2.5
−3 0 −1 −1 −2 −3 −2 −4 −3 0 0.5 1 time (sec) 1.5 2 x 10 2.5
−3 (a) (b) Figure 3.4: (a) Amplitude of oscillation of a beam driven with lateral combs; (b) Output current of a beam sensed with parallel plates Simulation results highlight the nonlinearity of the response of the DETF. Figure 3.5 shows how the actuation of a DETF can become unstable at low DC voltages, while higher voltages can generate stable oscillation. However, it can be seen that an appropriate choice of the system parameters allows oscillation in almost the entire gap region. 21 Dynamic evolution using 10 V DC and 15 V AC signal 0.8 Dynamic evolution using 15 V DC and 15 V AC signal 0.8
0.6 0.6
0.4 0.4
normalized amplitude (y/g0) 0.2 normalized amplitude (y/g0) 0.2 0 0 −0.2 −0.2 −0.4
−0.4 −0.6
−0.6 −0.8
−0.8 0 0.1 0.2 0.3 0.4 0.5 time (sec) 0.6 0.7 0.8 0.9 x 10 1
−3 0 0.2 0.4 0.6 0.8 time (sec) 1 1.2 1.4 x 10
−4 (a)
Dynamic evolution using 25 V DC and 15 V AC signal 0.8 (b)
Dynamic evolution using 50 V DC and 15 V AC signal 0.8
0.6 0.4 normalized amplitude (y/g0) 0.6 normalized amplitude (y/g0) 0.2 0.4 0 0.2 −0.2 −0.4 0 −0.6 −0.2 0 0.1 0.2 0.3 0.4 0.5 time (sec) 0.6 0.7 0.8 0.9 x 10 1
−3 0 0.2 0.4 0.6 time (sec) 0.8 1 1.2 x 10
−5 (c) (d) Figure 3.5: Illustration of DETF response when the system is driven into oscillation using DC+AC signals combination. The DETF is driven using an AC signal at its mechanical natural frequency (fn =148 kHz, Q=100). Due to mismatch between driving frequency and fundamental frequency of the electromechanical system, increase of the DC bias does not always translate into increase of the amplitude. (a) Driven with 10 V DC and 15 V AC; (b) Driven with 15 V DC and 15 V AC; (c) Driven with 25 V DC and 15 V AC; (d) Driven with 50 V DC and 15 V AC. 22 The regions of stability and instability in the example appear because of the mismatch between the input frequency and the frequency of the electromechanical system. To analyze the system correctly, the driving force must be at the natural frequency of the system. The main contribution to this frequency variation is The nonlinear the electrostatic stiﬀness softening [Ananthasuresh et al., 1996]. electrostatic force can be presented as F= 1 ε0 A 1 ε0 A V 2 ε0 A V 2 V2 = + y + O(2) 2 3 2 (g0 − y )2 2 g0 g0 (3.8) using Taylor expansion in y = 0. Then, the eﬀect of the electrostatic force shifts the natural frequency of the system M y + B y + (K − ¨ ˙ ε0 A V 2 1 ε0 A V 2 )y= + O(2) 3 2 g0 2 g0 (3.9) ωn = K− ε0 A V 2 3 g0 M (3.10) Thus, the electrostatic force acts like a negative spring. It has been reported that the amplitude of the oscillation also generates a shift of the frequency due to the nonlinearity of the force [Ananthasuresh et al., 1996]. Consequently, unless this fact is taken into account, the frequency of the system and the frequency of the forcing force do not match. Then, the eﬀect of the variation of the driving voltage cannot be predicted as shown in Figure 3.5. To ensure that the DETF oscillates at its natural frequency certain control must be introduced. The simplest way to drive the oscillation of a beam to its resonant frequency is to build a positive feedback loop. At resonant frequency, the position of the beam and the force are phaseshifted 90 degrees. That means that the force and 23 Driving voltage
Dynamic evolution using 10 V DC and 5 V AC signal 0.15 5 4 0.1 3 2
normalized amplitude (y/g0) Driving voltage (V) 0.05 1 0 0 −1 −0.05 −2 −3
−0.1 −4 0 0.1 0.2 0.3 0.4 0.5 time (sec) 0.6 0.7 0.8 0.9 x 10 1
−3 −5 0 0.1 0.2 0.3 0.4 0.5 time (sec) 0.6 0.7 0.8 0.9 x 10 1
−4 (a) (b) Figure 3.6: (a) DETF Oscillating at its resonant frequency under the action of the control loop; (b) 5 V square wave signal that drives the DETF to resonance. the velocity are in phase. Then, if the DETF is driven with a force in phase with the velocity of the beam, the feedback loop will drive the oscillation into resonance. To simplify the design, the forcing function is generated as a square wave signal of constant amplitude at the fundamental frequency of the system (Figure 3.6). Once oscillation at resonant frequency is guaranteed, the proportionality between the voltage applied and amplitude of oscillation is regained. Figure 3.7a shows the evolution of the amplitude of oscillation depending on the DC bias for ﬁxed AC signal. As it can be seen, for higher AC voltages the amplitude increases quicker. To show the necessity of the control loop, Figure 3.7b plots the shift of the frequency against the DC bias voltage. The shift of frequency increases faster when getting close to the instability region. As has been presented in this section, numerical simulation allows to study amplitude and frequency of oscillation of the system for any applied voltages. 24 Amplitude of oscillation 0.5 0.45 Electrostatic stiffness softening 148 146
Normalized amplitude of oscillation (y/g0) 0.4 10 V 7V 10 V
0.35 0.3 7V 5V
Frequency (kHz) 144 5V 3V 1V
0.25 0.2 142 140
0.15 3V
0.1 0.05 AC=1 V AC=3 V AC=5 V AC=7 V AC=10 V 5 10 15 20 25 30 DC voltage (V) 35 40 45 50 138 136 AC=1 V AC=3 V AC=5 V AC=7 V AC=10 V 5 10 15 20 DC voltage (V) 25 30 35 1V
40 (a) (b) (a) Figure 3.7: Amplitude and frequency of oscillation of the DETF (Q=30). Relationship between the amplitude and the DC bias for a ﬁxed AC voltage; (b) Stiﬀness softening translates in shift of the frequency of oscillation with the increase of the DC bias. Moreover, the introduction of the oscillation control loop allows to simulate the characteristics of the DETF oscillator. 3.3 Pullin in Parallel Plate Actuators The use of electrostatic parallel plate actuators and sensors introduces important limitations to the range of voltages and amplitudes of oscillation that can be used for actuation. An example of the problem has been shown in Figure 3.5, where incorrect selection of the driving voltages produce snapping between electrodes. Figure 3.8 shows the simpliﬁed lumped massspring system modeling of a parallel plate actuator M y+B y+K y =F ¨ ˙ (3.11) where y is the displacement of the moving electrode from its initial equilibrium. In 25 y g0 B M K + V _ Figure 3.8: Scheme of a parallel plate actuator the model, the force generated between the parallel plates is F= 1 ε0 A V2 2 (g0 − y )2 (3.12) where ε0 is the dielectric constant, g0 is the initial gap between the plates, A is the area of the plates and V is the applied voltage between the electrodes. As it can be observed, the force is inversely proportional to the gap of the actuator. As the gap decreases, the generated attractive force increases quadratically, and at some point the mechanical forces deﬁned by the linear spring cannot balance it. Once reached this state, the electrodes will snap one against the other, and in most cases, the system could be permanently disabled. To understand the phenomena, one must turn to the energy of the electromechanical system 1 1 1 ε0 A E = Ek + Ep + U = M y 2 + K y 2 − ˙ V2 2 2 2 (g0 − y ) (3.13) that is composed of the kinetic energy (Ek ), the potential energy stored in the spring (Ep ), and the potential energy stored in the parallel plate capacitor (U ). 26 The study of the evolution of the energy can be used to determine the equilibrium positions of the system, as well as the regions of instability. The following sections analyze this instability phenomena in the three main cases of electrostatic actuation. First of all, static electrostatic deﬂection is introduced and reviewed from the point of view of the energy evolution. This initial analysis allows to understand the concept of stable and unstable equilibrium relative to the position of the electrodes. Following this discussion, the dynamics of the system is introduced. The role of the equilibrium points is explained in the dynamic case. Finally, the discussion is extended to oscillating systems. Again, the importance of equilibrium points in determining the stability of the system is highlighted. This allows to introduce the concept of AC Dynamic Pullin Voltage. An algorithm to predict stable actuation is presented and validated using simulation of the system for diﬀerent dynamic conﬁgurations. 3.3.1 Static Pullin Voltage Statement 1 In the static case, the distribution of the potential energy in the gap between the electrodes deﬁnes the possible equilibrium points for each voltage. For voltages larger than the Static Pullin Voltage no equilibrium positions exist and the parallel plate snaps. In static equilibrium, y = y = 0, the energy of the system (3.13) consists only of ¨ ˙ the potential terms 1 1 ε0 A E = K y2 − V2 2 2 (g0 − y ) (3.14) The distribution of the system energy along the gap between the electrodes is constant 27 and unique for each voltage applied. (a) (b) Figure 3.9: Evolution of the potential energy along the gap between the electrodes. (a) Low voltages present stable and unstable static equilibrium; (b) The equilibrium points are points where the force of the capacitor is compensated by the restoring spring force. For low voltages, the energy proﬁle along the gap exhibits two equilibrium points, stable and unstable. Figure 3.9a shows a typical proﬁle with the equilibrium points indicated. Analytically, the static equilibrium corresponds to points where the variation of energy is zero; this translates to equilibrium between the electrostatic force and the restoring spring force of the structure (Figure 3.9b) 1 ε0 A dE =K ·y− V2 =0 dy 2 (g0 − y )2 (3.15) The type of equilibrium is deﬁned by the second derivative. If the value is positive, indicating a minimum of the energy, the system is stable, i.e. ε0 A d2 E =K− V2 >0 2 dy (g0 − y )3 28 (3.16) The maximum and minimum of the energy are sketched in Figure 3.9a and Figure 3.9b. The plots show the correspondence between the energy of the system and the relative value of restoring spring force and electrostatic force. (a) (b) Figure 3.10: Evolution of the potential energy along the gap between the electrodes at Pullin Voltage. (a) The energy only presents an inﬂexion point; (b) The force of the capacitor is only compensated by the restoring spring force at one third of the gap, what corresponds to the inﬂexion point. The limit condition for existence of stable equilibrium is d2 E dy 2 = 0. If this condition is satisﬁed, an inﬂection point exists (Figure 3.9a). In the inﬂection point the maximum and minimum merge into one point. The electrostatic and restoring spring forces compensate each other at this point. This condition provides the analytical value for the maximum static stable displacement from the initial equilibrium and the voltage needed to reach this position ypin = g0 ; Vpin = 3 3 8 K g0 27 ε0 A (3.17) This limit voltage is called Static Pullin Voltage. In parallel plates, snapping between 29 the electrodes occurs if the voltage exceeds this value [Senturia, 2001].
x 10 4
12 Evolution energy profile with increasing voltage x 10 1 12 Evolution energy profile with increasing voltage 10 V 20 V
2 50 V 56 V 61.9 V 2 30 V
0 3 40 V
Energy 4 Energy 65 V 2 5 4 6 6 7 8 8
10 V 20 V 30 V 40 V 0 0.1 0.2 0.3 0.4 0.5 0.6 normalized displacement (y/g0) 0.7 0.8 0.9 9 0 10 50 V 56 V 61.9 V 65 V 0.1 0.2 0.3 0.4 0.5 normalized displacement (y/g0) 0.6 0.7 0.8 (a) (b) Figure 3.11: Evolution of the energy proﬁle in a DETF with a gap of 2 µm when changing the voltage between the plates. (a) Low voltages present stable and unstable static equilibrium; (b) At high voltages both equilibrium points get closer until they disappear at pullin voltage (61.9 V in the example). Figure 3.11a and Figure 3.11b show the evolution of the energy in the gap between the parallel plates for a DETF. The parameters used for simulation along this chapter are K = 3.1 N/m, K = 3.76e−12 Kg, g0 = 2 µm and A = 228 µm2 . As can be observed, for low voltages the stable equilibrium is placed near the initial position and the unstable equilibrium close to the ﬁxed electrode. When increasing the voltage, the stable and unstable equilibrium move closer until they merge at Static Pullin Voltage (Vpin = 61.9V in our case). No equilibrium points exist for voltages higher than Static Pullin Voltage. 30 3.3.2 Dynamic Pullin Voltage Statement 2 When the kinetic energy of the system and damping forces are taken into account, higher amplitudes than the
g0 3 can be reached without snapping. The above derivation only takes into account the static equilibrium and does not consider the dynamics of the system response when the voltage is applied. (a) (b) Figure 3.12: Evolution of the system when the voltage is applied as a step function. (a) Displacement of the moving electrode when the voltage is applied. It can be seen how the position of the electrode overshoots the stable equilibrium before it settles down; (b) Evolution of the energy for the same trajectory. The initial energy is dissipated due to the damping forces until it reaches the stable equilibrium. In the dynamic case, it can occur that due to the gained kinetic energy, the position of the plate overshoots that of the stable equilibrium position (Figure 3.12a). After the transient, the plate would return to the stable position. In Figure 3.12b, the evolution of the energy shows how initially all the energy is provided by the potential energy, but the kinetic term allows to overshoot the energy equilibrium. The energy oscillates between the potential energy bounds, until it settles down at the stable 31 equilibrium. The behavior can be explained analytically. If the evolution of the energy (3.13) is analyzed dE 1 ε0 AV 2 =M y y+K y y− ˙¨ ˙ y= ˙ dt 2 (g0 − y )2 M y+K y− ¨ 1 ε0 AV 2 2 (g0 − y )2 y ˙ (3.18) and the dynamics of the system is introduced (3.11) M y+K y− ¨ 1 ε0 AV 2 = −B y ˙ 2 (g0 − y )2 (3.19) the evolution of the energy is deﬁned by dE = −B y 2 ˙ dt (3.20) This equation indicates that the energy decreases continuously due to the damping (B ) of the system, until it reaches an equilibrium at y = 0. ˙ Using the same reasoning, the evolution of the energy along the gap is dE 1 ε0 A =K ·y− V2 2 dy 2 (g0 − y ) And if the dynamic model is introduced, the evolution is deﬁned by dE = −M y − B y ¨ ˙ dy (3.22) (3.21) where the inertia and damping forces deﬁne the slope of the loss of energy in Figure 3.12b. Statement 3 In the dynamic case, snapping at lower voltages than Vpin can occur. The Dynamic Pullin Voltage is the minimum constant voltage that when applied in the form of a step function can produce snapping. Voltages lower than the Dynamic Pullin Voltage never produce snapping. 32 The evolution of the system depends on the value of the voltage applied between the electrodes. If the voltage is increased, at some value the overshoot could lead the plate past the critical point, producing snapping of the plates. Figure 3.13a shows how the position of the plate increases until it reaches the unstable region. Then the evolution becomes unstable and the electrode moves until it collides with the opposite electrode. (a) (b) Figure 3.13: Evolution of the system when the voltage applied leads the system to snap. (a) Displacement of the moving electrode when the voltage is applied. It can be seen how the position of the electrode reaches the unstable equilibrium. At that point, the evolution becomes unstable; (b) Evolution of the energy for the same trajectory. In the plot, the energy losses are small, so the energy is almost constant, reaching over the unstable equilibrium and leading the system to pullin. Energy levels above this line all lead over the unstable equilibrium. Based on the energy equation (3.13), it has been observed, Figure 3.11, that increase of the voltage results in migration of the stable and unstable equilibrium points. Both peaks displace to a closer position, and at the same time, the value of the energy at the unstable peak decreases. At the limit voltage, the initial energy of the system and the energy of the unstable 33 peak have the same magnitude (Figure 3.13b). Assuming that the system has no damping forces, the energy of the system remains constant, what implies that applying this voltage, the system will move until it overshoots the unstable equilibrium, and the electrodes will collide. This limit voltage is called Dynamic Pullin Voltage. Any voltage lower than its magnitude cannot produce snapping. Analytically, the voltage can be calculated imposing the condition that the value of the potential energy at the initial position (E0 ) equals that of the potential energy at the unstable peak (Euns ): E0 = − 1 ε0 A 2 V 2 g0 dpin (3.23) 1 1 ε0 A 2 2 Euns = K yuns − Vdpin 2 2 (g0 − yuns ) where the unstable equilibrium (yuns ) satisﬁes 1 ε0 A 1 K yuns − V2 =0 2 dpin 2 2 (g0 − yuns ) (3.24) (3.25) Solving the system of equations, the limit voltage has the unstable equilibrium at the center of the gap yuns = g0 2 (3.26) and the Dynamic Pullin Voltage has the following expression Vdpin =
3 1 K g0 4 ε0 A (3.27) When damping forces are introduced, the voltage that leads to pullin increases. The energy losses due to the damping prevent from overshooting the unstable peak. Figure 3.14 plots the evolution of the pullin voltage against the value of 34 Evolution dynamic pull−in voltage 62 61 Dynamic Pull−in Voltage (V) 60 59 58 57 56 −1 10 10 0 10 1 10 Quality Factor 2 10 3 10 4 10 5 Figure 3.14: Pullin Voltage as a function of the Quality Factor. The Dynamic Pullin Voltage is 56.9 V for our example. the Quality Factor obtained via simulation. This result was already presented in [Gupta et al., 1996], where the numerical simulation was conﬁrmed with experimental results. It can be observed that the pullin voltage decreases from the Static Pullin Voltage while increasing the quality factor, Q, until it settles for values of Q higher than 1000, at the Dynamic Pullin Voltage. The fact that the system becomes unstable when it reaches the unstable equilibrium can be shown analytically. The evolution of the energy, dE = dt M y+K y− ¨ 1 ε0 AV 2 2 (g0 − y )2 y ˙ (3.28) at the unstable equilibrium point 1 1 ε0 A K yuns − V2 =0 2 2 (g0 − yuns )2 dpin (3.29) 35 follows the expression dE =M y y=0 ˙¨ dt (3.30) ... Consequently, if the point is not static equilibrium y = y = y = . . . = 0, that ˙ ¨ point is a minimum of the energy: ... d2 E d3 E d4 E = 0 ; 3 = 0 ; 4 = 3M y2 > 0 2 dt dt dt (3.31) The smooth decrease of the energy is broken at that point, because the unstable region has been reached. The unstable trajectory of the moving electrode is indicted by the inﬂexion point in the trajectory (Figure 3.13b).
DETF evolution 56 V DC 0.4 DETF evolution 57 V DC 0.9 0.35 0.8 0.3 normalized amplitude (y/g0) 0.7 normalized amplitude (y/g0) 0.6 0.25 0.5 0.2 0.4 0.15 0.3 0.1 0.2
0.05 0.1 0 0 0.5 1 1.5 2 2.5 time (sec) 3 3.5 4 4.5 x 10 5
−3 0 0 1 2 3 4 5 time (sec) 6 7 8 9 x 10
−6 (a) (b) Figure 3.15: (a) Evolution of the position of a DETF for a 56 V step input (Q=500); (b) Evolution of the same DETF showing snapping at 57 V. (Vpin = 61.9 V, Q=500) Numerical simulation allows to observe all these phenomena. Using the nonlinear equation (3.11), the example in Figure 3.15b shows snapping of a DETF when applying a step function of 57 V, while the calculated Static Pullin Voltage is 61.9 V. The snapping is produced with a voltage 8% lower than the static calculation, Vpin [Ananthasuresh et al., 1996]. 36 x 10 −12 Energy evolution for 56 V DC
−1.639 x 10 −12 Energy evolution for 57 V DC −1.6
−1.6395 −1.62 −1.64 −1.64 energy
energy −1.6405 −1.66 −1.641 −1.6415 −1.68
−1.642 −1.7
−1.6425 0 0.5 1 1.5 2 2.5 time (sec) 3 3.5 4 4.5 x 10 5
−3 0 1 2 3 4 5 time (sec) 6 7 8 9 x 10
−6 (a) (b) Figure 3.16: Evolution of the energy for the examples in Figure 3.15. (a) Smooth energy evolution using a step of 56 V; (b) Unstable energy evolution using a step of 57 V. −1.58 x 10 −12 Energy evolution in the gap for 56 V DC −1.6 x 10 −12 Energy evolution in the gap for 57 V DC −1.65
−1.6 −1.7
−1.62 −1.75 −1.8
−1.64 energy energy
−1.66 −1.68 −1.7 −1.85 −1.9 −1.95 −2 −2.05
total energy static energy −1.72 0 0.05 0.1 0.15 0.2 0.25 0.3 normalized displacement (y/g0) 0.35 0.4 0.45 total energy static energy −2.1 0 0.1 0.2 0.3 0.4 normalized displacement (y/g0) 0.5 0.6 0.7 (a) (b) Figure 3.17: The evolution of the energy including the dynamic component is compared to the static energy depending on the position in the gap for the examples in Figure 3.15. (a) The smooth evolution using a step of 56 V is bounded by the static energy; (b) The unstable evolution condition is achieved using a step of 57 V because the energy has reached over the unstable equilibrium. 37 When the evolution of the system is stable (Figure 3.15a), the energy decreases smoothly (Figure 3.16a) until the stable equilibrium is reached. The total energy is bounded by the potential energy as shown in Figure 3.17a. However, when the voltage is higher than the Dynamic Pullin Voltage, the system can become unstable (Figure 3.15b). Figure 3.16b shows how the energy evolution losses its smoothness when the unstable peak is reached (Figure 3.17b), leading the system to snapping. 3.3.3 AC Dynamic Pullin Voltage
When the system is driven into resonant oscillation, snapping can Statement 4 occur at very low voltages. The system oscillation is stable if the resonant amplitude is lower than the maximum stable amplitude for the applied voltage. The results of the previous section on dynamic pullin raise the question of what are the conditions of stable actuation when driving a DETF to oscillation. The study of the energy can be done for the case of forced oscillation, where V (t) = VDC + VAC (t). However, the voltage varies with time, what implies that the equilibrium points given by
dE dy are changing continuously (3.32) 1 ε0 A dE (t) = K · y − V (t)2 = 0 dy 2 (g0 − y )2 and that the stable equilibrium depends on the pattern of the AC voltage 1 ε0 A ε0 A dE ˙ =M y y+K y y− ˙¨ ˙ V (t)2 − V (t)V (t) y ˙ 2 dt 2 (g0 − y ) (g0 − y ) (3.33) Using the same reasoning explained in the previous section, the potential energy curves allow to determine the maximum amplitude of oscillation that can be achieved 38 Figure 3.18: Potential energy bounds when the system is oscillated with a DC bias voltage and an AC voltage. without reaching the pullin zone. As shown in Figure 3.18, the combination of DC and AC voltages delimits two potential energy curves. Consequently, the unstable equilibrium positions of both energy proﬁles deﬁne the amplitude of oscillation. For amplitudes smaller than the peak of the VD + VAC curve, no pullin occurs (Figure 3.18). In the region between both peaks, the pullin depends on the AC signal. For amplitudes larger than the peak of the VD − VAC curve the electrode is pulledin. Solving the equation (3.32)
2 2 3 K yuns − 2g0 K yuns + Kg0 yuns − e0 A 2 V =0 2 (3.34) for the unstable equilibrium allows to determine the maximum amplitude of oscillation depending on the actuation voltage, V . Figure 3.19a and Figure 3.19b show both bounds on the amplitude of oscillation for a DETF depending on the actuation DC and AC voltages. Statement 5 In forced oscillation, the system is stable if the supplied energy 39 Calculation of maximum safe amplitude 1 0.95 0.95 0.9 0.9 0.85 Normalized amplitude (y/g0) Normalized amplitude (y/g0) 0.85 Calculation of maximum amplitude 0.8 0.8 0.75 0.75 0.7 0.7 0.65 0.65 0.6 AC=1 V AC=3 V AC=5 V AC=7 V AC=10 V 5 10 15 20 25 30 DC voltage (V) 35 40 45 50 55 AC=1 V AC=3 V AC=5 V AC=7 V AC=10 V 5 10 15 20 25 30 DC voltage (V) 35 40 45 50 55 0.6 0.55 0.55 (a) (b) Figure 3.19: Calculation of the maximum amplitude of oscillation for the simulated DETF depending on the combination of DC and AC voltages. (a) Plot of the amplitude for each voltage over which the DETF is at risk of snapping (Calculated for VD + VAC ); (b) Plot of the amplitude for each voltage over which the DETF is pulledin (Calculated for VD − VAC ). does not allow the system to overshoot the unstable equilibrium point of the potential energy deﬁned by the applied voltage. Stable actuation is deﬁned by the existence of a stable limit cycle. The AC Dynamic Pullin Voltage is the combination of minimum DC and AC voltages that when applied produce snapping. Once known the maximum amplitude, the next step is to study if the unstable region will be reached. Forced resonant oscillation is an intrinsically unstable process. Energy is supplied to the system, trying to drive the system into resonance. To simplify the understanding of the phenomena, the analysis is developed using an AC square wave signal. However, the explanation can be extended to any kind of AC signal. Figure 3.20 shows the evolution of the energy when the system is oscillated at 40 Figure 3.20: Stable oscillation of a DETF. The potential energy curves deﬁne a steadystate oscillation or limit cycle. resonance using a square wave voltage signal. Beginning from the initial position, the amplitude of oscillation increases bounded by the potential energy until reaches the steadystate oscillation. The use of a square wave signal means that only two potential energy curves exist, one for the maximum voltage and one for the minimum one. The existence of a steadystate oscillation deﬁnes the stability of the actuation voltages. This sustained oscillation is called a limit cycle [Senturia, 2001]. An example obtained via simulation is shown in Figure 3.21, where the DETF oscillates using 20 V DC and 5 V AC. The evolution of the energy is presented in Figure 3.22a and Figure 3.22b. The energy increases with time until the limit cycle is reached, then, oscillates in a closed loop. The oscillation is conﬁned between the potential energy curves. The existence of a limit cycle prevents from reaching the unstable equilibrium peak, and the oscillation is stable. If no limit cycle exists for the selected actuation voltages, the amplitude of 41 Dynamic evolution using 20 V DC and 5 V AC 0.3 0.2 gap percentage (g/g0) 0.1 0 −0.1 −0.2 −0.3 0 0.1 0.2 0.3 0.4 0.5 time (sec) 0.6 0.7 0.8 0.9 x 10 1
−3 Figure 3.21: Oscillation of a DETF using 20 V DC and 5 V AC. x 10 7 6 5 4 3 energy 2 1 0 −1 −2 −3 0 −13 Energy evolution in the gap for 20 V DC and 5 V AC x 10 7 6 5 4 3 energy 2 1 0 −1 −2 −3 −13 Energy evolution in the gap for 20 V DC and 5 V AC total energy static energy −0.3 −0.2 −0.1 0 0.1 normalized gap (g/g0) 0.2 0.3 1 2 3 4 5 time (sec) 6 7 8 9 x 10
−4 (a) (b) Figure 3.22: Evolution of the energy using 20 V DC and 5 V AC. (a) The energy of the system increases until it reaches the steady state oscillation; (b) Evolution of the energy in the gap. In stable oscillation, the energy is conﬁned between the static energy limits, without overshooting the unstable equilibrium. 42 oscillation increases continuously until the system is pulledin. This happens because at some point the energy accumulated in the system drives the amplitude of oscillation over the unstable peak, at which point the electrode snaps. Figure 3.23 shows the evolution of the energy when it leads to snapping. It can be seen how the relative position of the potential curves do not deﬁne a limit cycle. Thus, the energy builds up until the system is pulledin. Figure 3.23: Unstable oscillation. Beginning from the initial position, the amplitude of oscillation of the system increases until it reaches over the unstable equilibrium point for VDC + VAC . Then, the system snaps. Again, numerical simulation shows more accurately how the phenomena occurs. Figure 3.24 plots the unstable evolution of the amplitude of oscillation of a DETF. Figure 3.25a shows how the energy in the unstable case increases without settling. This increase in the energy leads the system over the unstable equilibrium point as can be observed in Figure 3.25b. And the parallel plate actuator snaps. To predict the existence of a stable limit cycle, the relationship between the potential energy curves and the damping of the system must be studied. The evolution of the energy (3.13) at each of the positive and negative cycles 43 Dynamic evolution using 30 V DC and 5 V AC
Close−up instability point 0.95 0.8
0.9 0.6
0.85 gap percentage (g/g0) 0.4
gap percentage (g/g0) 0.8 0.75 0.2 0.7 0 0.65 −0.2
0.6 −0.4 0.55 0 0.2 0.4 0.6 time (sec) 0.8 1 1.2 x 10
−4 0.5 1.324 1.326 1.328 1.33 1.332 1.334 time (sec) 1.336 1.338 1.34 1.342 x 10
−4 (a) (b) Figure 3.24: Unstable evolution of a DETF using 30 V DC and 5 V AC. (a) The amplitude of oscillation at the resonant frequency leads the DETF to the unstable region; (b) Closeup of the point where the unstable region is reached. The cross indicates the position where the unstable equilibrium point has been overshot. x 10 20 −13 Energy evolution for 30 V DC and 5 V AC 20 x 10 −13 Energy evolution in the gap for 30 V DC and 5 V AC 15 15 10 energy energy 5 10 5 0 0 −5 0 0.2 0.4 0.6 time (sec) 0.8 1 1.2 x 10
−4 −5 total energy static energy −0.4 −0.2 0 0.2 normalized gap (g/g0) 0.4 0.6 (a) Figure 3.25: (b) Energy evolution when using 30 V DC and 5 V AC. (a) The energy increases without settling until pullin is reached; (b) The increase of the energy leads the position to overshoot the unstable equilibrium limit what leads to snapping. 44 decreases again depending on the damping (B ) of the system dE = −B y 2 ˙ dt And the evolution of the energy along the gap is dE = −M y − B y ¨ ˙ dy (3.36) (3.35) Knowing the slope of the decrease of energy in the gap (3.36), the existence of a limit cycle can be analyzed. If stable oscillation exists for an actuation voltage, an oscillation loop in the energy of the system close to the unstable amplitude should decrease the amplitude of oscillation. That means the system energy diminishes until reaching the limit cycle. This reasoning is explained graphically in Figure 3.26. However, if the voltage applied generates unstable oscillation, when an oscillation loop is generated in the energy close to the unstable peak, the amplitude of oscillation increases, indicating that the system will become unstable and snapping will occur. Figure 3.27 shows an unstable energy distribution. Although the slope of the energy evolution depends on the velocity and acceleration of the trajectory, its magnitude can be estimated, what leads to an analytic algorithm that allows to predict the stability of oscillation of the system. Assuming sinusoidal oscillation of the system of amplitude A and frequency ω y = A sin(ω t) ; y = A ω cos(ω t) ; y = −A ω 2 sin(ω t) ˙ ¨ the slope of the energy has the following value dE = M A ω 2 sin(ω t) − B A ω cos(ω t) dy 45 (3.38) (3.37) Figure 3.26: AC Dynamic Pullin algorithm. The energy distribution is stable because amplitudes close to the unstable region decrease. Figure 3.27: AC Dynamic Pullin algorithm. The energy distribution is unstable because amplitudes close to the unstable region increase, leading the system to snapping. 46 Using the slope at the center of the trajectory as approximation over the entire gap, dE ≈ −B A ω dy (3.39) the stability can be successfully estimated. Figure 3.26 and Figure 3.27 show the procedure followed to determine the stability of the actuation voltages using the AC Dynamic Pullin algorithm. AC Dynamic Pullin Algorithm: 1. First of all, the unstable equilibrium peak, yuns , for VD + VAC is calculated:
3 2 2 K yuns − 2g0 K yuns + Kg0 yuns − e0 A (VD + VAC )2 = 0 2 (3.40) 2. Then, beginning from a point close to the unstable peak (e.g. yP1 = 0.9 · yuns ), an oscillation loop is built. From P1 at VD + VAC , the system evolves to P2 at VD − VAC . The new energy level can be calculated from: 1 1 ε0 A 2 EP2 = K yP2 − (VD − VAC )2 2 2 (g0 − yP2 ) (3.41) 3. From this point, the energy decreases until it reaches the the potential energy curve again. The position of P3 is determined from the intersection of the decreasing energy and the potential energy at VD − VAC : 1 1 ε0 A 2 EP3 = K yP3 − (VD − VAC )2 = EP2 + B A ω (yP3 − yP2 ) 2 2 (g0 − yP3 ) (3.42) where B is the damping factor, ω the frequency of oscillation and A is the amplitude of oscillation, calculated as A = yP1 − y0 , being y0 the equilibrium position for VD and center of oscillation. 47 4. Next step is calculating the energy for P4 , knowing that yP3 = yP4 : 1 1 ε0 A 2 EP4 = K yP4 − (VD + VAC )2 2 2 (g0 − yP4 ) (3.43) 5. Finally, the loop is closed calculating the position of P5 as intersection of the decrease of energy and potential energy: 1 1 ε0 A 2 EP5 = K yP5 − (VD + VAC )2 = EP4 − B A ω (yP5 − yP4 ) 2 2 (g0 − yP5 ) 6. Stability discussion: • If the ﬁnal amplitude is smaller than the initial amplitude, the system is stable (Figure 3.26): yP5 < yP1 ⇒ Stable oscillation (3.45) (3.44) • If the ﬁnal amplitude is larger than the initial amplitude, the system is unstable (Figure 3.27): yP5 > yP1 ⇒ U nstable oscillation (3.46) The algorithm can be implemented in any software and provides a quick check of the stability of the actuation (e.g. Maple 6.0, Waterloo Maple Inc. in Appendix B). The results of the algorithm have been compared to the numerical values of the AC Dynamic Pullin Voltage. Using numerical simulation, the curves of the combination of DC and AC voltage that produce snapping of the system can be calculated for diﬀerent values of the Quality Factor. The algorithm shows accuracy in predicting the stability of the system. It is a powerful tool to provides a good approximation of the stability without need of simulating the whole evolution. 48 Figure 3.28a compares the combination of minimum bias and AC signal that once applied produces snapping depending on the Quality Factor. Figure 3.28b highlights the drop in the maximum bias that can be applied for a ﬁxed AC signal. The relationship between the DC bias and the Quality Factor is logarithmic. These curves can be used as a guide to choose the actuation voltages when oscillating the fabricated prototypes.
AC Pullin Voltage 60 Q=30 Q=70 Q=100 Q=500 Q=1000
50 AC Pullin Voltage AC=1 V AC=3 V AC=5 V AC=7 V AC=10 V AC=20 V AC=30 V AC=50 V 45 1V 50 40 35 40 3V
DC voltage (V) 30
30 DC voltage (V) 30 25 5V
20 70
20 7V 100 500 15 10 V
10 10 20 V
5 1000
0 0 5 10 15 20 25 30 AC voltage (V) 35 40 45 50 30 V 50 V
10
2 10 Quality Factor 3 (a) (b) Figure 3.28: AC Dynamic Pullin Voltage Curves. (a) Minimum DC bias and AC voltage that when applied produces snapping for a ﬁxed Quality Factor; (b) Evolution of the minimum bias that produces snapping for a ﬁxed AC signal in function of the Quality Factor. 3.4 Remarks The chapter provides insight in the conditions that lead to unstable actuation, as well as, supply with the guidelines to choose the actuation voltages. Moreover, knowing the maximum position in the gap that can be achieved without becoming unstable for each voltage, a stable control algorithm can be build that controls the amplitude 49 of oscillation and avoids the unstable region. The analysis of the pullin eﬀect has also its inﬂuence on the design of electrostatic lateral combs. Although in a ﬁrst approximation the behavior of lateral actuators is considered independent of the position, this model do not take into account the fringing ﬁelds and the inﬂuence of the tip of the ﬁngers. The tips can be functionally modeled as a parallel plate actuator. Electrostatic lateral combs show pullin eﬀects when the amplitude of oscillation is large and the opposing ﬁngers get close. Introducing the force generated by the ﬁngers tips in the equation of motion allows to calculate the maximum amplitude of oscillation before pullin occurs. The results of this chapter give the tools to choose the appropriate voltage needed to drive the DETF prototypes. Chapter 4 uses the numerical simulation and the stability conditions to optimize the values of the prototypes to be fabricated. 50 Chapter 4 DETF Prototypes
This chapter presents the fabrication process used for designing and manufacturing of the DETF prototypes. Characteristics of the proposed designs are discussed. The prototypes have been designed to support theoretical results on stable actuation of doubleended tuning forks. Lateral and parallel actuation and sensing designs are presented. Parametric variations in the designs allow to study the characteristics of surface micromachined doubleended tuning forks. The goal of the study is to analyze the range of frequencies and sensitivity to axial loads that can be achieved. This analysis allows to delimit the possible applications of micromachined DETFs. The design process is supported by Finite Elements Analysis (FEA), which allows to extract the oscillation modes of the ﬁnal designs and provide detailed inside into the eﬀect of loading of the devices. 4.1 MUMPs Process The prototypes have been designed and fabricated in the MultiUser MEMS Process (MUMPs). This is a commercially available threelayer surface micromachining 51 process provided by Cronos (JDS Uniphase). MUMPs is a quick and costeﬀective method to fabricate prototypes. It is a good choice as a ﬁrst step to validate the working principles of a new design. As a standard surfacemicromachining process, the devices are formed using polysilicon as a structural material, silicon dioxide is used as a sacriﬁcial material, lower electrical interconnects are deﬁned by heavilydoped polysilicon and upper interconnects are deﬁned by metal. All features are deﬁned using lithographic techniques commonly used in the semiconductor industry. 4.1.1 Process Description The process consists of a series of steps including material deposition, photolithographic masking and material etching that allow to create 3D planar structures freestanding over the substrate. Figure 4.1: MUMPs Process 52 In the ﬁrst step, a 600 nm silicon nitride layer is deposited using lowpressure chemical vapor deposition (LPCVD) over the 100 mm ntype (100) silicon wafer. This layer is used as an electrical isolation layer. Then, a ﬁrst 500 nm polysilicon layer is deposited (Poly 0). This layer is used to run the electrical connections between structures. To pattern the polysilicon, photoresist is coated, exposed with the appropriate mask (POLY0), and developed. This procedure leaves a hard photoresist mask that is used to etch away the Poly 0 using Reactive Ion Etch (RIE). The remaining photoresist is then etched away. This step is then followed by deposition of the First Oxide layer (Oxide 1), a 2 µm phosphosilicate glass (PSG) sacriﬁcial layer. This material will be removed at the end of the process in order to free the rest of the freestanding structure. This layer is photolithographically patterned with the DIMPLES mask and the ANCHOR mask. The anchors provide holes that connect Poly 0 with the second polysilicon layer. After patterning the Oxide 1, the ﬁrst polysilicon structural layer is deposited (Poly 1). This layer is 2 µm thick. Again the polysilicon is coated with a PSG masking layer, which is patterned with the POLY1 mask. This provides a hard mask for the ﬁnal RIE etching of the Poly 1. The process continues with the .75 µm Second Oxide PSG sacriﬁcial layer (Oxide 2). The POLY1 POLY2 VIA and ANCHOR2 masks are applied and the last 1.5 µm polysilicon layer is deposited (Poly 2). Patterns are provided with the POLY2 mask. The ﬁnal step includes a metal layer that is deposited and patterned using a liftoﬀ process [Kovacs, 1998]. Once ﬁnished, the wafer is diced and the structures are release using an HF 53 Figure 4.2: Section of a DETF designed in MUMPs solution. This last step removes all the PSG sacriﬁcial layers, leaving freestanding polysilicon structures. When designing structures it is not necessary to use all the available layers. Usually, in inertial devices, the Poly 0 layer is used for electrical connections and the Poly 1 layer is used as the structural layer, deﬁning the structure of the devices. When the sequence of steps is completed, one is left with a polysilicon structure suspended 2 µm over the substrate. In the Poly 1 layer one can implement capacitors to actuate and sense motion. The MUMPs process has a number of design rules that must be followed in order to guarantee an errorfree fabrication. These rules include minimum size features, minimum gaps between layers and the use of etching holes to ensure release of the structure [Koester et al., 2001]. Table 4.1 provide a quick overview of the layers provided by the MUMPs process. 54 Material Layer Nitride Poly 0 Oxide 1 Poly 1 Oxide 2 Poly 2 Metal Thickness Mask name Minimum feature Minimum space 0.6 µm 0.5 µm 2 µm 2 µm 0.75 µm 1.5 µm 0.5 µm POLY0 HOLE0 DIMPLE ANCHOR1 POLY1 HOLE1 POLY1 POLY2 VIA ANCHOR2 POLY2 HOLE2 METAL HOLEM 2 µm 2 µm 2 µm 3 µm 2 µm 3 µm 2 µm 3 µm 2 µm 3 µm 3 µm 4 µm 2 µm 2 µm 3 µm 2 µm 2 µm 3 µm 2 µm 2 µm 2 µm 3 µm 3 µm 4 µm Table 4.1: MUMPs layers and attributes [Koester et al., 2001] It has been added the minimum features that can be achieved due to lithographic precision in each of the layers and masks. Table 4.2 provides the values of the material properties that can be used for the polysilicon layers used in the fabrication process. The variability in the polysilicon properties makes their use only an estimation [W.N. Sharpe et al., 1997]. 4.2 DETF Prototypes Design The design of a doubleended tuning fork has two steps. First of all, the structural characteristics have to be chosen, what deﬁnes the range of natural frequencies and the achievable sensitivities to axial loads. Secondly, the actuation and sensing mechanisms 55 Property Density Young Modulus (E) Poison’s ratio (ν ) Resistivity Linear Expansion Coeﬃcient Thermal Conductivity Speciﬁc heat Value 2330 Kg/m3 150 − 180 GP a 0.22 1 − 2e−3 Ω cm 2.33e− 6 /K 45 W/m K 920 J/Kg K Table 4.2: Polysilicon properties have to be selected. That includes the analysis of the behavior of the DETF and its stability. A discussion of the design possibilities when using surface micromachining is presented in this section. 4.2.1 Structural Design If a doubleended tuning fork is used as a force sensor, the sensitivity to the applied axial forces is the main parameter. And the natural frequency deﬁnes the sensor bandwidth. If a doubleended tuning fork is used as a frequency source, the natural frequency sets its working range and the sensitivity to axial forces deﬁnes its tunability.
1 1 fn = 2π 1 Kef f ∂fn ; = Mef f ∂F 4πωn Mef f L dφ i dε 2 dε (4.1) 0 As is derived in Chapter 2, the frequency sensitivity of a DETF is deﬁned by equation (4.1). Stiﬀness, mass and length of the beam must be reduced to increase sensitivity (2.8). Translated to geometrical values, equation (2.9) indicates that the width and 56 thickness of the beam are the parameters to be minimized. Process requirements deﬁne the thickness of the beams and the minimum width to 2 µm (Table 4.1). Other parameters that inﬂuence the sensitivity are the Young Modulus and the density of polysilicon. However, choosing a commercial surfacemicromachining process, like MUMPs, does not allow to change these values (Table 4.2). Residual stresses in the material (which act as a preloaded axial force) and the mode of oscillation, according to [Roessig, 1998], should increase sensitivity. Residual compression deriving from the fabrication could increase the sensitivity of the DETF but it is not a desired eﬀect because it can cause instability and buckling of the beam. Higher modes of oscillation show improved sensitivity at the expense of a more diﬃcult excitation scheme. (a) (b) Figure 4.3: The inclusion of the actuation mechanism increases the mass of the structure. (a) Lateral comb ﬁngers actuation; (b) Parallel plate actuation. In the designs, the actual mass is speciﬁed by the mass of actuation/sensing 57 combs (Figure 4.3), which is several times larger than the mass of the beams (e.g., Mcombs = 5.3 · 10−12 , Mbeam = 1.8 · 10−12 ) . In this case, once the width of the beam is set to the minimum feature size, the length of the beam is the design parameter. Increasing the longitude decreases the stiﬀness as well as increases the sensitivity: ∂ωn ∝ ∂F L1/2 E IMef f (4.2) Using the process speciﬁcations [Koester et al., 2001], long 2 µm beams are not recommended because they could compromise the lithography quality. Previous fabrication runs show that lengths up to 200 µm are achievable. Using this length as a maximum feature, an upper bound of the sensitivity of a doubleended tuning fork that can be achieved using MUMPs Process can be calculated. The maximum sensitivity using 2µmwide 200 µmlong polysilicon beams is 0.4728 Hz/nN. Beam design Length (L) Width (b) Thickness (t) Mass (Mef f ) Stiﬀness (Kef f ) Natural frequency (fn ) Sensitivity 200 µm 2 µm 2 µm 6.04 · 10−12 Kg 5.63 N/m 153 kHz 0.165 Hz/nN Table 4.3: Example of a DETF structural parameters Choice of the geometry also deﬁnes limits imposed on the natural frequency of the oscillator. Provided that width is chosen to its minimum value, frequency can only be changed using the length of the beam or the additional mass included due to the electrostatic 58 combs. The lower frequencies achievable with the current technology are around 100 kHz, unless longitude is pushed to the limit (L = 400 µm → f ∼ 50kHz ). Higher frequencies are easier to achieve, but this results in reduction of sensitivity. For this reason, the preferred designs include a longitude of about 200 µm producing frequencies in the range 100 to 200 kHz and sensitivities of 0.15 to 0.25 Hz/nN. Table 4.3 presents the values of the doubleended tuning fork with lateral comb drive and sense in Figure 4.4a. It is important to mention that the sensitivity of a doubleended tuning fork is half of the sensitivity of a beam (2.7). This happens because the applied force is equally distributed to both beams. 4.2.2 Actuation and Sensing The choice of design parameters for electrostatic actuation depends on the available power and the desired amplitude of oscillation. The choice of the sensing mechanism is dictated by the maximum desired value of the output signal. The selection of the acting and sensing mechanism determines the Quality Factor, Q, of the structure (3.6). Most of the ﬂuid damping in the device occurs in the electrostatic actuation/sensing combs, where the electrodes are very close moving against each other (squeeze ﬁlm damping) or sliding in parallel one along the other (couette ﬂow damping). Surfacemicromachined doubleended tuning forks can be designed with large Quality Factor. The proposed prototypes show estimated Q values in air for lateral combs around 290, while for parallel plates the quality factor 59 is near 140. In vacuum, regardless of the mechanism of actuation, the quality factor is over 106 . Electrostatic comb design Finger length Finger width Finger thickness (t) Gap (g) Number ﬁngers driving (nd ) Number ﬁngers sensing (ns ) Bias voltage driving (vD ) AC voltage driving (vd ) Bias voltage sensing (Vs ) 12 µm 2 µm 2 µm 2 µm 12 15 25 V 3V 25 V Dynamic parameters Mef f Kef f Qair = ω · Mef f /Bef f Oscillation amplitude Current amplitude 6.04 · 10−12 Kg 5.63 N/m 290 0.84 µm 5.28 nA Oscillation frequency (ωn ) 965160 rad/sec Table 4.4: Estimated DETF lateral combs parameters in air When choosing the ﬁnal speciﬁcations for the actuation and sensing combs, the results concerning AC Dynamic Pullin Voltage must be taken into account. For a needed amplitude of oscillation and a desired output signal, the force and motion induced sensing current depend on the design. It must be checked that the speciﬁcations do not lead the structure to snapping because the amplitudes needed cannot be achieved with the chosen design. The design parameters for the lateral combs are the gap size and the number of ﬁngers (Equation 2.19). Minimum gap is set by design rules. Then, the number of ﬁngers deﬁnes the maximum achievable capacitance, what translates in a bound to the maximum achievable values for the force and current. Stable electrostatic actuation deﬁnes the maximum amplitude that can be achieved, what sets the length of the comb ﬁngers. 60 A surfacemicromachined 200 µmlong DETF can have 10 to 15 ﬁngers, setting the capacitance to femtofarads. The corresponding forces for a 5 V biasing are in nanonewtons and the output current is in nanoamperes. The design parameters of a parallel plate are the gap size and overlapping area. Design rules deﬁne the minimum gap, leaving the area as the design parameter. To increase actuation force and sensing current the length of the plates has to be maximized. However, this reduces the maximum amplitude of stable oscillation. A surfacemicromachined 200 µmlong DETF can have around 150 µmlong parallel plates, limiting the capacitance again to femtofarads. The corresponding forces for a 5 V biasing are in nanonewtons and the output current in nanoamperes, but the values are three times those of the lateral combs. Electrostatic comb design Bias voltage driving (vD ) AC voltage driving (vd ) Bias voltage sensing (Vs ) 4V 1 mV 4V Dynamic behavior Q1T orr Oscillation amplitude Current amplitude 6 · 106 1.03 µm 1.06 nA Table 4.5: Estimated DETF lateral combs parameters in vacuum at 1 Torr As has been introduced in the previous section, the decisions in the design of the electrostatic actuation can alter the ﬁnal speciﬁcations due to the added mass needed to build the combs. The mass of the comb ﬁngers can be used to reduce the natural frequency of the structure, but that will also reduce the sensitivity to axial forces (2.8). Table 4.4 shows the values used to design an electrostatic lateral comb for driving and sensing. The table provides the values for the equation of the dynamics and 61 presents the voltages needed to drive the structure in air, as well as, the expected amplitude of oscillation. Table 4.5 shows the same values for the case of vacuum actuation at 1 Torr. 4.3 Final Prototypes A set of prototypes has been designed, which is aimed to analyze stable DETF actuation. Parametric changes in the actuation and sensing mechanisms allow to test the analytical results, as well as, the range of frequencies that can be achieved. The parametric changes in frequency and sensitivity of the prototypes allow to study the eﬀect of these changes in the frequency shift produced by axial forces. The design values have been calculated using Maple 6.0, Waterloo Maple Inc. (Appendix B). The set consist of ﬁve diﬀerent doubleended tuning forks with diﬀerent types of actuation and sensing mechanisms. Table 4.6 shows the speciﬁcations of the designs. A picture of one of the designs produced by the layout editor LEdit of Tanner Inc. is shown in Figure 4.4a. This layout editor allows to design masks for fabrication of MEMS devices. The masks are sent directly to the fabrication facility. The diﬀerent colors correspond to the diﬀerent material layers. Table 4.7 presents the summary of the main structural values. All the DETFs have been designed with a 200 µm beam of 2 × 2 µm crosssection. The values show the correlation between the total mass and the sensitivity and frequency of the designs. As has been pointed out, this is a consequence of the fact that the electrostatic comb counts for most of the mass of the structure. 62 DETF Name DL1 DL1B DM1 DM1B DPC1S Actuation Combs lateral lateral lateral parallel lateral parallel parallel No. ﬁngers/ Plate area 10 12 10 228 µm2 12 228 µm2 200 µm2 Gap Size 2 µm 2 µm 2 µm 2 µm 2 µm 2 µm 2 µm Sensing Combs lateral lateral parallel lateral parallel lateral parallel No. ﬁngers/ Plate area 12 15 228 µm2 10 228 µm2 12 240 µm2 Gap Size 2 µm 2 µm 2 µm 2 µm 2 µm 2 µm 2 µm Table 4.6: Design parameters for testing prototypes Name DL1 DL1B DM1 DM1B DPC1S Meﬀ (Kg) 7.25 · 10−12 6.04 · 10−12 4.69 · 10−12 3.84 · 10−12 2.70 · 10−12 Keﬀ (N/m) 5.63 5.63 5.63 5.63 5.63 fn (kHz) Sensitivity (Hz/nN) 140.197 153.610 174.336 192.612 229.931 0.151 0.165 0.188 0.207 0.248 Table 4.7: Testing prototypes structural parameters If the diﬀerent methods of actuation are compared, parallel plate driving and sensing shows the best results. Table 4.8 summarizes the values needed to actuate the diﬀerent designs for the same 1 µm amplitude of oscillation in air. It can be seen that parallel plates need less voltage to drive a DETF and produce more current output. However, amplitude larger than half of the gap easily leads to snapping. For this reason, being able to understand the conditions of stable actuation is really important. If higher voltages are not a limitation, lateral comb ﬁngers can be the choice for actuation of the resonator. They also allow higher amplitudes of oscillation with less 63 (a) (b) Figure 4.4: (a) Doubleended tuning fork prototype with electrostatic lateral comb driving and sensing; (b) Closeup of the axial force actuator placed at one end of the DETF to test axial force sensitivity and frequency tunability. risk of snapping. Mixed designs can be interesting if the parallel plate is used as a sensor, but then the amplitude of oscillation is limited again by the pullin conditions. Name DL1 DL1B DM1 DM1B DPC1S Qair 298 290 183 183 176 176 174 vD (V) 35 30 35 12 30 12 11 vd (V) 3 3 4 3 4 3 4 Vs (V) Current (A) 35 30 10 12 10 12 11 7 · 10−9 8 · 10−9 10.0 · 10−9 2.5 · 10−9 11 · 10−9 3 · 10−9 15 · 10−9 Table 4.8: Testing prototypes actuation and sensing voltages in air The designs have a parallel plate capacitor in one of the ends in order to apply axial forces electrostatically to the DETF (Figure 4.4b). This conﬁguration allows 64 to simulate a force applied to the DETF and would be used for force sensitivity characterizations. 4.4 Finite Elements Analysis The last step in the design of the prototypes has been the validation through Finite Elements Analysis (FEA). The analysis has been done using the FEA software ANSYS of ANSYS, Inc (Appendix C). This analysis provides insight on to the fundamental modes and force frequencyshift sensitivity of the ﬁnal designs. 4.4.1 Modal Analysis The modal analysis is important when designing compliant resonant MEMS devices. Any undesirable oscillation mode must be avoided or signiﬁcantly separated from the actuation frequency range. Figure 4.5: Model using BEAM4 element for the prototype DL1B 65 The analysis has been carried out using the element BEAM4 (Figure 4.5) and covers the range of frequencies from 10 kHz to 400 kHz. The modal analysis highlights the existence of a ﬁrst outofplane mode of oscillation that is not desired (Figure 4.6). This is a known problem when using a surfacemicromachined technology, because the reduced thickness of the polysilicon layer does not allow to have enough stiﬀness in the outofplane mode of oscillation. As this problem is inherent of the technology, it is important to place the actuation mode (Figure 4.7) at least a 10% apart from this mode. Figure 4.6: First mode of oscillation for the prototype DL1B. It can be observed the outofplane motion. Table 4.9 shows the ﬁrst four modes of the prototypes. It can be seen that the actuation frequencies (2nd Mode) match those of the analytical model (2% discrepancy). Moreover, the table provides the sensitivity of the modes to an axial force applied to the beam. The actuation mode is highly sensitive to the axial load, while the other modes are less aﬀected. For this reason, DETF oscillators are used 66 Name DL1 Unstressed Modes (Hz) 86592 136250 157742 174043 Stressed Modes (Hz) 87269 139133 158660 174682 91024 152866 170563 195672 130212 173719 230686 231683 134250 192763 251934 272239 210678 230885 364997 383509 Mode Sensitivity 10 µN (Hz) 677 2883 918 639 629 3182 993 694 1470 3618 1350 894 1279 4042 1503 1032 3653 4852 1729 2394 DETF Sensitivity 10 µN (Hz) 1441.5 DL1B 90395 149684 169570 194978 1591 DM1 128742 170101 229336 230789 1809 DM1B 132971 188721 250431 271207 2021 DPC1S 207025 226033 363268 381115 2426 Table 4.9: Modes of oscillation for the Actuation Prototypes. 67 Figure 4.7: Second mode of oscillation for the prototype DL1B. Desired mode of inplane oscillation. as resonant force sensors. The frequency shifts predicted by ANSYS are equal to the analytical estimations obtained in Table 4.7. The results of the modal analysis point out the importance of the design of the actuationsensing combs, because they are the source of the third (Figure 4.8) and fourth mode (Figure 4.9). In the prototypes this undesired modes are at least 20% away from the main mode. 68 Figure 4.8: Third mode of oscillation for the prototype DL1B. Torsional mode of the actuation comb. Figure 4.9: Fourth mode of oscillation for the prototype DL1B. Torsional outofplane oscillation of the actuation comb. 69 Chapter 5 DETF Testing
This chapter presents the experimental data obtained from the prototypes fabricated using a commercial surfacemicromachining process. Figure 5.1 shows a Scanning Electron Microscope (SEM) picture of one of the chips fabricated in the Cronos MUMPS run 42. A brief description of the experimental setup will lead to the evaluation of the outcome. The analysis of the experimental results is presented. The tests validate the dynamic DETF model and the conclusions on the pullin conditions presented in Chapter 3. 5.1 Experimental Evaluation Setup In order to test the prototypes, the microscope probestation in the Microsystems Lab at UCI has been used (Figure 5.2a). This probestation is equipped with three objectives (x10, x25, x50) and video output. To supply the power to the devices precision positioners are used. Figure 5.2b shows the fabricated chip ready to be tested on the probestation. 70 Figure 5.1: Scanning Electron Microscope (SEM) picture of the 1cm × 1cm chip fabricated in the Cronos MUMPs run 42. In the image, the sector where the doubleended tuning forks are located. The experimentation setup is shown in Figure 5.3. The DC power supply and the function generator are used to provide the needed excitation to drive the devices into resonance. The response is detected optically through the video camera. Using an ATI image acquisition card, images are sent to the computer where they are processed using the image processing software ImagePro Plus 4.1 by Media Cybernetics. Image processing allows to determine the resonant frequency and the amplitude of oscillation of the beams. 5.2 Analysis of Prototypes The properties of the fabricated prototypes have been analyzed. The goal is to extract conclusions about the suitability of surface micromachining to build doubleended tuning forks. Figure 5.4a and Figure 5.4b show a visual inspection of the prototypes. All 71 (a) (b) Figure 5.2: (a) Microscope probestation available in the Microsystems Lab at UCI; (b) Closeup of the precisionpositioners used to supply the power to the chip. Figure 5.3: Experimental setup used to test the prototypes. 72 prototypes were completely released and worked properly. (a) (b) Figure 5.4: (a) SEM picture of one of the doubleended tuning forks; (b) Picture of a doubleended tuning fork taken with the camera embedded in the microscope probestation. Table 5.1 summarizes the natural frequencies of the diﬀerent designs and the needed voltages to drive them to visible oscillations in air. Name DL1 DL1B DM1 DM1B DPC1S Frequency (kHz) 109115 118125 134140 143152 172180 DC Voltage (V) 20 20 15 15 15 AC Voltage (V) 10 10 5 5 5 Table 5.1: Frequencies and voltages of actuation of the fabricated prototypes in air The following conclusions have been reached in the analysis of the prototypes: • The devices present a frequency diﬀerence of 20% with respect to the analytical values of Table 4.7. This mismatch could be related to the overetching observed in the designs. Moreover, an average of 9 MPa residual compression stresses has 73 been acknowledged in the Poly 1 layer of the MUMPs run 42. A combination of these factors could account for the lower frequencies. • A mismatch of 5% has been found between the frequency of the beams of the doubleended tuning forks. It seems to be related to the orientation of the devices on the chip. As frequency matching is a desired goal between DETF tines, more insight is needed on this matter. • Analysis of the amplitudes of oscillation and voltages used to drive the structures allows to estimate the Quality Factor of the structures. The Quality Factor is proportional to the amplitude of oscillation at the resonant frequency. The designs present lower Quality Factor than those calculated in Chapter 4. The average value of the Quality Factor in air of the devices is 50. However, the value of the damping in the test seems to vary depending on the chip used to experiment. • Experimentation highlights the importance of understanding the snapping phenomena. Figure 5.5a shows snapping in a testing structure using parallel plates. Snapping occurred at voltages lower than the static pullin voltage (VDC = 20V, VAC = 10V, Vpin = 63V ). Moreover, snapping can be dangerous when actuating devices in air. The voltages needed to oscillate the structures in some cases imply that if the snapping happened, the device will melt (Figure 5.5b, VDC = 30V, VAC = 10V ). • Finally, the results of the testing conﬁrm the existence of some of the undesired modes predicted in the Finite Elements Analysis study (Table 4.9). 74 Under experimentation no outofplane mode was excited with the electrostatic actuation, but torsional planar modes were observed when actuating with high enough voltages. More analysis should be devoted to reduce the probability of exciting these modes. (a) (b) Figure 5.5: (a) SEM of a beam testing structure with parallel plate actuation after snapping; (b) A laterallyactuated doubleended tuning fork burned after snapping. Experimentation indicates that surface micromachining could be used to fabricate the doubleended tuning forks, but more consistent analysis is needed in order to delimit the extent of the variability in the fabrication and the parameters that are aﬀected. 5.3 Numerical Simulation Analysis This section presents the comparison between the experimental data and the numerical simulation results. This allows to validate the model used to analyze the DETF behavior. The parameters of the model were tuned based on the preliminary results. This 75 Electrostatic stiffness softening AC = 3 V 1 1 Electrostatic stiffness softening AC = 5 V 0.995 0.99 0.995 Normalized Frequency (kHz) Normalized Frequency (kHz) Model Experimental 12 14 16 18 20 22 DC voltage (V) 24 26 28 30 0.985 0.98 0.99 0.975 0.97 0.985 0.965 0.98 0.96 0.955 10 Model Experimental 10 12 14 16 DC voltage (V) 18 20 22 (a)
Electrostatic stiffness softening AC = 7 V 1 0.998
0.998 1 (b)
Electrostatic stiffness softening AC = 10 V 0.996 0.994 Normalized Frequency (kHz) 0.992 0.99 0.988 0.986 0.984 0.982 0.98 Model Experimental 6 8 10 12 DC voltage (V) 14 16
5 Normalized Frequency (kHz) 0.996 0.994 0.992 0.99 0.988 Model Experimental 6 7 8 9 DC voltage (V) 10 11 12 (c) (d) Figure 5.6: Comparison between the electrostatic stiﬀness softening predicted by the model when increasing the DC bias and the experimental results. (a) Spring softening for 3 V AC; (b) Spring softening for 5 V AC; (c) Spring softening for 7 V AC; (d) Spring softening for 10 V AC. 76 correction accounts for the variation of frequency due to the fabrication variability and the value of the Quality Factor. Once revised, the macromodels of the doubleended tuning forks predict the response of the devices. The ﬁrst test on the models is intended to compare the results on electrostatic spring softening. As can be observed while comparing Figure 5.6a to Figure 5.6d, the model predicts the evolution of the shift of the natural frequency when driving the DETF to resonance using parallel plates. The error increases with the increasing bias due to model precision. However, the maximum discrepancies in the frequency are of 3 · 10−3 , lower than the precision of the experimental results (9 · 10−3 ). The second test consists on predicting amplitude of oscillation. Figure 5.8 shows pictures of the response for diﬀerent DC bias voltages when a beam is driven with a 7 V AC voltage. The pictures are able to capture the diﬀerent amplitudes of oscillation until snapping occurs (Figure 5.8d). Comparison of amplitude between experimental results and model simulation is presented in Figure 5.7a to Figure 5.7d. Again, the simulated evolution captures the behavior of the experimental results. Discrepancies between model and experimental results are lower than 5%. However, it can be observed that the amplitudes of oscillation of the prototype can be higher than those predicted by numerical simulation of the model. Analysis of the results seems to point out that the model is accurate enough to predict the behavior, but better identiﬁcation of the parameters of the devices should be carried out. Speciﬁcally, better estimation of the Quality Factor is needed. Its magnitude is directly related to the ﬁnal amplitude of oscillation. 77 Experimental amplitude of oscillation AC = 3 V Experimental amplitude of oscillation AC = 5 V 0.8 0.45 0.7
0.4 Normalized amplitude of oscillation (y/g0) Normalized amplitude of oscillation (y/g0) 0.35 0.6 0.3 0.5 0.25 0.4 0.2 0.15 0.3 0.1 Model Experimental 0.05 10 12 14 16 18 20 22 DC voltage (V) 24 26 28 30 32 0.2 Model Experimental 12 14 16 18 20 DC voltage (V) 22 24 26 28 10 (a)
Experimental amplitude of oscillation AC = 7 V 0.8 (b)
Experimental amplitude of oscillation AC = 10 V 0.7 0.8 Normalized amplitude of oscillation (y/g0) 0.6 Normalized amplitude of oscillation (y/g0) Model Experimental 6 8 10 12 DC voltage (V) 14 16 18 0.7 0.5 0.6 0.4 0.5 0.3 0.4 0.2 0.3 0.1 0.2 5 6 7 8 9 10 DC voltage (V) 11 12 Model Experimental 13 14 (c) (d) Figure 5.7: Comparison between the amplitude of oscillation predicted by the model when increasing the DC bias and the experimental results. (a) Amplitude for 3 V AC; (b) Amplitude for 5 V AC; (c) Amplitude for 7 V AC; (d) Amplitude for 10 V AC. 78 (a) (b) (c) (d) Figure 5.8: Evolution of the amplitude of oscillation when increasing the DC bias; (a) Beam at rest; (b) Beam oscillating with 15 V DC and 7 V AC; (c) Beam oscillating with 18 V DC and 7 V AC; (d) Beam after snapping with 20 V DC and 7 V AC. To improve characterization, electrostatic sensing must be introduced, as well as, the driving control loop. The same studies were performed on the rest of the prototypes. Similar patterns on the amplitude and frequency were obtained in all the designs. 79 5.4 AC Dynamic Pullin Analysis The design of the prototypes was aimed to validate the conditions of stable actuation of doubleended tuning forks. Based on this goal, pullin experimental tests have been carried out to extract the AC Dynamic Pullin Voltage of the prototypes.
AC Pullin Voltage Comparison 60 Q=30 Q=70 Q=100 Q=500 Q=1000 experimental data 50 30 40 DC voltage (V) 70 Experimental AC Dynamic Pullin
30 100 20 10 500 1000
0 0 2 4 6 8 10 12 AC voltage (V) 14 16 18 20 Figure 5.9: Comparison of the values for which the experimental DETF was pulledin. The trend of the experimental results agree with that of the simulations. Figure 5.9 plots the needed combination of AC and DC voltages to produce dynamic pullin in the experimental tests. As can be observed, the values of the experimental AC Dynamic Pullin follow the predicted ones. The discrepancies between DC voltage values are about 20 %. Analysis of the results indicates that better estimation of the prototype parameters is needed for a more exact prediction. It must be pointed out that obtaining snapping values is diﬃcult because it risks damaging the device. For this reason, only three values were obtained. However, study of the evolution of the amplitude allows to predict the estimated pullin value. If the evolution of the amplitude of oscillation of the experimental prototype is examined against the conditions of maximum amplitude of oscillation 80 (Figure 3.19a), it can be seen that even with the variability and reduced number of examples, in each case, the pullin condition is validated. Figure 5.10 shows in the same plot the maximum amplitude that ensures stable actuation for each set of DC and AC voltages and the actual amplitude of oscillation of the DETF prototype. As can be observed, these values allow to predict when the snapping will occur. In the ﬁrst case, Figure 5.10a, the amplitude during the experimentation remains far away from the unstable region and pullin is not reached. However, in the rest of the cases, Figure 5.10b to Figure 5.10d, the amplitude increases until it reaches the unstable region. These experimental values correspond to the AC pullin tests, when snapping was forced. For 5 V AC, the amplitude before snapping was 80% of the initial gap, well on the uncertain region (y > 0.75 g0 ). The same conclusion is reached when looking at the 7 V AC point, where the amplitude was .8g0 , close to the unstable region. Or at the 10 V AC point, where the amplitude was .9g0 > 0.85 g0 . The experimental results validate the conditions of stable actuation derived in Chapter 3. Overshooting the unstable equilibrium of the energy for VD + VAC has been shown to cause the snapping of the electrodes. Moreover, numerical simulation has demonstrated that can be used to predict the results with simpliﬁed models. Although better parameter estimation may be needed, numerical simulation has shown its usefulness when designing doubleended tuning forks with electrostatic actuation. 81 Experimental amplitude of oscillation AC = 3 V Experimental amplitude of oscillation AC = 5 V 0.9 0.9 0.8 Normalized amplitude of oscillation (y/g0) Normalized amplitude of oscillation (y/g0) Max Stable Amplitude Oscillation Amplitude 5 10 15 20 DC voltage (V) 25 30 35 0.8 0.7 0.7 0.6 0.6 0.5 0.5 0.4 0.4 0.3 0.2 0.3 0.1 0.2 5 10 15 DC voltage (V) 20 Max Stable Amplitude Oscillation Amplitude 25 30 (a)
Experimental amplitude of oscillation AC = 7 V (b)
Experimental amplitude of oscillation AC = 10 V 0.9 0.9 0.8 Normalized amplitude of oscillation (y/g0) 0.8 Normalized amplitude of oscillation (y/g0) 0.7 0.7 0.6 0.6 0.5 0.5 0.4 0.4 0.3 0.3
0.2 0.1 2 4 6 8 10 12 DC voltage (V) 14 Max Stable Amplitude Oscillation Amplitude 16 18 20 0.2 2 4 6 8 DC voltage (V) 10 Max Stable Amplitude Oscillation Amplitude 12 14 (c) (d) Figure 5.10: Comparison between the experimental amplitude of oscillation and the maximum stable amplitude for the diﬀerent combination of voltages. (a) Evolution for 3 V AC. The amplitude does not reach the unstable region; (b) Evolution for 5 V AC. Snapping occurs when the amplitude reaches the unstable region at 29 V DC; (c) Evolution for 7 V AC. Snapping occurs for 19 V DC; (d) Evolution for 10 V AC. Snapping occurs for 15 V DC; 82 Chapter 6 Conclusions and Future Work
6.1 Conclusions New MEMS devices are continuously ﬁnding their place in the sensors market. In this framework, doubleended tuning forks oscillators have the needed characteristics to improve many applications. Frequency output, high sensitivity, large bandwidth and robustness to noise are some of the advantageous properties of micromachined doubleended tuning forks which are expected to contribute in the development of new devices ranging from onchip frequency references to resonant sensors such as accelerometers. The development of the present thesis has been aimed to understand and clarify the conditions of stable electrostatic actuation of doubleended tuning forks, to this goal the following results have been achieved: • Numerical simulation has been shown to be the only available method to predict the evolution of doubleended tuning forks driven with electrostatic parallel actuators. Models of the structure has been deduced and simulated. The experimental data set has conﬁrmed the usefulness of this method to predict 83 the behavior of the devices. • The conditions of stable electrostatic actuation of DETF has been analyzed using energy methods. This analysis allows to determine the maximum stable amplitude of oscillation of the device for a given actuation voltage. • An algorithm has been developed that allows to determine the stability of the doubleended tuning fork when driven to resonance. The AC Dynamic Pullin Voltage can be determined using the algorithm. This value indicates the maximum DC voltage that can be applied once chosen the AC driving voltage. Numerical simulation has been used to validate the algorithm. The work has been complemented with design and fabrication of doubleended tuning forks prototypes: • The design issues involving the fabrication of doubleended tuning forks in surfacemicromachining technology have been analyzed. This study has led to the fabrication of a family of prototypes to test stable actuation. • The fabricated prototypes have been tested. Analysis of the properties reveals important diﬀerences between the analytical data and the experiments. More insight is needed on the fabrication process to understand the sources of the variability and the unavoidable limits that must be taken into account in the design process. • The prototypes have been used to validate the simulation approach, as well as the analytically predicted values of the AC Dynamic Pullin Voltage. The 84 results show that voltages clearly lower than the Static Pullin Voltage can lead the structure to snapping if the beam is oscillated at its natural frequency. 6.2 Future Work In the study of DETF Oscillators a lot of work is left to be done. This section summarizes the next steps that should be made. Some of them are developments that continue the study presented here, while other present new analysis that should be carried out on the fabricated prototypes. Moreover, some applications where DETF can be used are presented. Those prototypes were fabricated in the same run but have not been tested yet. 6.2.1 AC Dynamic Pullin Characterizations Work remain to be done in improving the analytical models and to be able to characterize the AC Dynamic Pullin Voltage more accurately. To do so, electric signal extraction from the devices is mandatory. In the same direction, the devices should be tested under vacuum conditions. Figure 6.1 shows the MMR Vacuum testing chamber available at the Microsystems Lab. This probestation allows to probe microdevices under the pressure of few miliTorr. Moreover, the pullin characterization should be extended to lateral comb ﬁngers. Under air this is not possible because high voltages are required to drive the structure what would lead to melting of the structure when snapping occurs. Under vacuum lower voltages are needed, allowing this extension. 85 Figure 6.1: Vacuum testing chamber. 6.2.2 Oscillator Electronics Next step on the design of a DETF oscillator is to build the driving and sensing electronics needed for the oscillator. The simplest way to drive a beam to its resonant frequency is to build a positive feedback loop. At resonance, the force and the position of the structure present a 90degree phase shift. If the system is driven to instability while using a force phaseshifted 90 degrees from the position, the frequency leads to resonance. To obtain the force, the output current of the beam can be used, because it is proportional to the velocity, and 90phase shifted from the position by deﬁnition [Roessig, 1998]. Figure 6.2 shows the required circuitry. The output current is converted to voltage using a Transimpedance Ampliﬁer of gain Ramp . The output of the ampliﬁer is then fed to PhaseLocked Loop (PLL) [Gardner, 1979]. A PLL produces a square wave signal of ﬁxed amplitude at the frequency of the input voltage. This output can be used to close the loop and drive the beam. The ﬁxed output amplitude allows to choose the amplitude of oscillation of the beam. Discrete components can be used to build the circuit. In the proposed scheme, 86 Figure 6.2: DETF Oscillator electronics National Semiconductor Low Noise Ampliﬁer LM837N and PhaseLocked Loop LM565C have been used. The testing of the circuit shows that it is possible to extract currents of the order of nanoamperes and drive the structure with signals up to 12 V pick to pick. Using this circuitry, frequency output can be directly extarcted if the DETF wants to be used as a frequency source (Pin 5, Figure 6.2). In the same way, demodulated frequency can be extracted (Pin 7, Figure 6.2), that is useful if the DETF is used as a resonant sensor. 6.2.3 Frequency Shift Detection As has been already introduced, one of the qualities of a DETF is its sensitivity to axial forces. When a tension is applied along the axis of the beam, the structural properties change, producing a shift in the natural frequency (Figure 6.3). The prototypes have been designed including a parallel plate capacitor on one end in order to characterize the sensitivity of the designs to axial forces. 87 (a) (b) Figure 6.3: Vibrating beam frequency shift under the action of an axial force Characterization of force sensitivity and its relationship to DETF designs and natural frequency is needed to delimit the applications where DETFs can be used. In the development of this analysis, two applications were under study: Tunable DETF Oscillators and Angular Accelerometers. 6.2.3.1 Tunable DETF Oscillators The ability of building tunable oscillators as onchip frequency references is under study. This kind of oscillators would be highly valuable for Onchip Adaptive Control strategies, where the properties of a device are extracted driving it into resonance. In this case, a frequency source is locked to the input of the device driving the structure to resonance. The parameters of the structure can be extracted from the information in the resonant state. 88 The sensitivity of DETFs to axial forces can be used for tuning the frequency output in a small range. This fact can be useful when the frequency signal is locked to another device, because frequency matching is needed. It has been shown that fabrication variability can produce changes in the frequency of up to 20%. Even considering that the DETF and the device that has to be locked suﬀer the same variation, a mismatch of 5% is intrinsic in the process fabrication. If the central frequency is 50 kHz, that could mean 2.5 kHz of diﬀerence. Tuning can compensate for these variations and allow the matching of frequencies of both structures. Figure 6.4: DETF with force tuning in one of the ends. Prototypes have been fabricated to investigate the range of frequencies and tunability range that can be obtained using surface micromachining technologies. The frequency would be tuned using the parallel plate placed in one of the end of the DETFs (Figure 6.4). 6.2.3.2 Angular Accelerometers Inertial navigation, automotive industry and electronics industry are diﬀerent ﬁelds that could greatly beneﬁt if a highprecision angular accelerometer is implemented. 89 3Dinertial devices must include three linear displacement sensors, usually linear accelerometers, and three angular rotation sensors. Car applications such as rollover detection or active suspensions need to detect the rotation of the car. In other ﬁelds, like consumer electronics, it is common to use rotatory devices such as hard disk drives and CD players. In order to have complete control of the rotation, an angular sensor must be embedded in the device. (a) (b) Figure 6.5: (a) Schematic of a linear accelerometer using a DETF; (b) Schematic of an angular accelerometer using the same principle. In some applications, angular movement can be sensed out using gyroscopes. A gyroscope allows to extract directly angular velocity or position using the inertial Coriolis force that appears in a moving system when it is rotated [Clark, 1997], [M.W.Putty, 1995]. accelerations. For this reason, in the macroworld the literature provides several ways to build an angular accelerometer. Diﬀerent physics principles can be used: mechanics, ﬂuids, electromagnetism, piezoelectrics [Meydan, 1997]. Nevertheless, in a lot of the applications, e.g. vibration control of hard disk drives, the angular accelerometer needs to be embedded in the device. This introduces 90 However, gyroscopes cannot be used under high angular constrains on size and cost that are not satisﬁed by the available products on the market. The idea is to use the doubleended tuning fork force detection capability to sense the inertial force generated by a proof mass under the action of the inertial acceleration that wants to be measured. (a) (b) Figure 6.6: (a) SEM of the Asymmetric Vibratory Angular Accelerometer; (b) Closeup of the proof mass of the device. DETFs have already been used to sense linear acceleration. First applications made used of quartz doubleended tuning forks [Reedy and Kass, 1990]. And later, the use of bulkmicromachining [Leonardson and Foote, 1998] and surfacemicromachined [Roessig et al., 1999] DETF to sense linear acceleration was presented. In order to sense acceleration, the DETF is attached to a proof mass that converts the linear acceleration to be sensed into the inertial force that is applied to the DETF (Figure 6.5a). Under this axial load, the frequency of oscillation of the DETF changes and is measured through the comb drives, obtaining a frequency shift as output that is proportional to the linear acceleration to be sensed. 91 (a) (b) Figure 6.7: (a) SEM of the Symmetric Vibratory Angular Accelerometer; (b) Closeup of a DETF attached to the proof mass. The goal is to use the same principle to detect angular acceleration. In this case, a proof mass is build to be sensitive to angular accelerations perpendicular to the substrate of the chip. Under this acceleration, the proof mass, due to its inertia, tends to rotate generating a moment that is converted to linear force applied to the DETF (Figure 6.5b). In the same way as in the linear case, this force produces a frequency shift in the oscillation of the DETF and the variation of the output signal is proportional to the angular acceleration that is measured. Two prototypes of the device have been developed using surface micromachining technology and they have been fabricated in the Cronos MUMPs run 42. Figure 6.6a shows an asymmetric Vibratory Angular Accelerometer and Figure 6.6b shows a symmetric Vibratory Angular Accelerometer. Analysis and testing of the devices is a part of the future plans. 92 Bibliography
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This appendix lists the Mﬁles used to numerically simulate the dynamic behavior of the doubleended tuning forks. The ﬁrst ﬁle, snap detf.m, implements the numerical integration and obtains the results, while the second ﬁle, eq snap detf.m, includes the dynamic equations of the system. Simulations have been carried out using Matlab 6.0 of Mathworks. A.1 snap detf.m %t r a n s i e n t study f o r DETF clear a l l ; close a l l ; ’ ∗∗∗∗∗∗ s n a p d e t f ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ’ global k m b Q z0 x0s x0d E rho g0 e0 global Vc Va Q ws f w nd Vs ns g 0 l Ramp %aux Ramp=10ˆ9; %Design p a r a m e t e r s %c o n s t a n t s 97 E=170 e9 ; rho =2300; g0=2e − 6; e0 =8.85 e − 12; z0=2e − 6; nd=10; ns =12; x0d=2e −6∗ nd ; x0s=2e −6∗ ns ; g 0 l =2e − 6; %lumped model %LM1B mumps2 k =3.27; m=.376 e − 11; x0d=114 e − 6; ns =12; %a c t u a t i o n and s e n s i n g Vc=56; Va=0; Vs=0; %Q u a l i t y f a c t o r Q=500; b=sqrt ( k ∗m/Qˆ 2 ) ; %f r e q u e n c y w=sqrt ( k/m) ; ws=w; f=w/2/ pi f s=ws /2/ pi sampling =.0000001; Y0 = [ 0 0 ] ; OPTIONS = o d e s e t ( ’ RelTol ’ , 1 e − 4, ’ AbsTol ’ , 1 e − 9); [ t ,Y]= ode45 ( ’ e q s n a p d e t f ’ , 0 : s a mp l i n g : . 0 0 5 , Y0 , OPTIONS ) ; % % % % %P o s i t i o n% % % % % % % %%%% %%%%%%% figure (1) plot ( t ,Y ( : , 1 ) / g0 ) xlabel ( ’ time ( s e c ) ’ ) ylabel ( ’ n o r m a l i z e d a m p l i t u d e ( y/ g0 ) ’ ) 98 a xis t i g h t %%%% % % % INFORMATION % % % % % % % %%%%%%% amp=(max(Y( s i z e (Y, 1 ) / 2 : s i z e (Y, 1 ) , 1 ) ) − min (Y( s i z e (Y, 1 ) / 2 : s i z e (Y, 1 ) , 1 ) ) ) / 2 ampr=amp/ g0 h i g h s t e a d y=max(Y( s i z e (Y, 1 ) / 2 : s i z e (Y , 1 ) , 1 ) ) / g0 h i g h=max(Y ( : , 1 ) ) / g0 %e n e r g y e q u i l i b r i u m p o i n t s g0u=g0 ∗ 1 e6 ; e0u=e0 ∗ 1 e6 ; x0du=x0d ∗ 1 e6 ; z0u=z0 ∗ 1 e6 ; Vmax=Vc+Va ; maxy=roots ( [ k − 2 ∗ g0u ∗ k k ∗ g0u ˆ2 − e0u ∗ x0du ∗ z0u ∗Vmaxˆ 2 / 2 ] ) / g0u Vmin=Vc−Va ; miny=roots ( [ k − 2 ∗ g0u ∗ k k ∗ g0u ˆ2 − e0u ∗ x0du ∗ z0u ∗ Vmin ˆ 2 / 2 ] ) / g0u %s n a p p i n g v o l t a g e Vpi=sqrt ( 8 ∗ k ∗ g0 ˆ3/27/ e0 / z0 / x0d ) %s n a p p i n g d i s t a n c e − i n f l e x i o n p o i n t gap=(g0 −(e0 ∗ z0 ∗ x0d ∗Vmaxˆ2/ k ) ˆ ( 1 / 3 ) ) / g0 %Dynamic p u l l − i n v o l t a g e Vdpin=sqrt ( k ∗ g0 ˆ3/4/ e0 / z0 / x0d ) ’ ∗∗∗∗∗∗ end s n a p d e t f ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ’ ’ ∗∗∗∗∗∗ c h e c k i n g f r e q u e n c y ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ’ f t =1/ s a m p l i n g / 1 0 ˆ 3 ; %kHz i n t e r v a l =1/ f t ; N=128 ∗ 256 data =4 ∗ 256 Z = f f t (Y ( : , 1 ) , N ) ; Pyy = Z . ∗ conj (Z ) / N; f f = f t ∗ ( 0 : data ) /N; 99 [ v1 , v i ]=max( Pyy ( 2 : data + 1 ) ) ; f r e q u e n c y= f f ( v i +1) ’ ∗∗ end c h e c k i n g f r e q u e n c y ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ’ A.2 eq snap detf.m function YD=e q s n a p d e t f ( t ,Y) global k m b Q z0 x0s x0d E rho g0 e0 Vc global Va ws nd Vs ns x 0 d l g 0 l Ramp %p o s i t i o n and v e l o c i t y y=Y( 1 ) ; yd=Y( 2 ) ; %v a r i a t i o n o f c a p a c i t a n c e f o r t h e d i f f e r e n t p r o t o t y p e s Cld =(( e0 ∗ z0 ) / ( g0 ) ) ∗ 2 ∗ nd ; C l s =(( e0 ∗ z0 ) / ( g0 ) ) ∗ 2 ∗ ns ; Cld2 =(( e0 ∗ z0 ) / ( g0 ) ) ∗ 2 ∗ nd + ( ( e0 ∗ z0 ∗ x0d ) / ( g 0 l −y ) ˆ 2 ) ; Cl s 2 =(( e0 ∗ z0 ) / ( g0 ) ) ∗ 2 ∗ ns + ( ( e0 ∗ z0 ∗ x0s ) / ( g 0 l+y ) ˆ 2 ) ; Cpd=(( e0 ∗ z0 ∗ x0d ) / ( g0−y ) ˆ 2 ) ; Cps=(( e0 ∗ z0 ∗ x0s ) / ( g0+y ) ˆ 2 ) ; %e l e c t i o n o f c a p a c i t a n c e f o r t h e s i m u l a t i o n Cd=Cpd ; Cs=C l s ; %S e n s i n g b i a s Vss=Vc ; %A m p l i f i e r g a i n Ramp=b/ Vss ˆ2/ Cs/Cd ; i f t <.000001 %I n i t i a l e x c i t a t i o n V=Vc+Va ∗ sin ( ws ∗ t ) ; else %Sensed c u r r e n t i s=Vss ∗ Cs ∗ yd ; Vas=Ramp∗ i s ; Vas=sign ( Vas ) ∗ Va ; 100 %D r i v i n g v o l t a g e V=Vc+Vas ; end %E l e c t r o s t a t i c f o r c e a t d r i v i n g and s e n s i n g Fd=1/2 ∗Cd∗Vˆ 2 ; Fs=1/2 ∗ Cs ∗ Vs ˆ 2 ; i f y > g0 %sn a p p i n g c o n d i t i o n YD( 1 , 1 ) = 0 ; YD( 2 , 1 ) = 0 ; else %E q u a t i o n s o f motion YD( 1 , 1 ) = yd ; YD( 2 , 1 ) = ( Fd−Fs−k ∗ y−b ∗ yd ) /m; end 101 Appendix B Maple listings
This appendix lists an example of the ﬁles used to calculate the analytical parameters of a doubleended tuning fork. In the example, tk lat bal 1b.mws, the included data corresponds to a lateral comb actuated DETF. Moreover, AC dynamic pullin.mws includes an implementation of the AC Dynamic Pullin Algorithm. The ﬁles have been generated in Maple 6.0 of Waterloo Maple Inc. B.1 tk lat bal 1b.mws Tuning Fork Design L a t e r a l combs : b a l a n c e d , 1 − s i d e d > restart ; Parameters − Co n s t a n t s > E:=170 e9 : rho : = 2 3 3 0 : > L:=200 e − 6:w:=2 e − 6: t :=2 e − 6: > In :=wˆ3 ∗ t / 1 2 :A:= t ∗w : > F a x i a l :=0 e − 6: F r e s i d u a l : = 0 : > A comb : = 1 1 3 8 e − 12;M comb:= rho ∗ A comb ∗ t ; > M beam:=A∗ L ∗ rho ; > M comb/M beam ; 102 1 s t Mode o s c i l l a t i o n − Rao ; > > > > > alpha :=−1.018: beta : = 4 . 7 3 : h:=−.618: Def :=h ∗ ( s i n h ( b e t a ∗ e)− s i n ( b e t a ∗ e)+ a l p h a ∗ ( c o s h ( b e t a ∗ e)− c o s ( b e t a ∗ e ) ) ) ; D e f e := d i f f ( Def , e ) ; D e f e e := d i f f ( D e f e , e ) ; Def comb := e v a l f ( s u b s ( e = . 5 , e v a l f ( Def ) ) ) ; Frequency v a l u e s ( Hz ) > K e f f :=E∗ In /Lˆ3 ∗ i n t ( D e f e e ˆ 2 , e =0..1)+ ( F a x i a l+F r e s i d u a l ) /L ∗ i n t ( D e f e ˆ 2 , e = 0 . . 1 ) ; > M e f f := rho ∗A∗ L ∗ i n t ( Def ˆ 2 , e =0..1)+M comb ∗ Def comb ˆ 2 ; > omega n := s q r t ( K e f f / M e f f ) ; > f KHz := e v a l f ( omega n /1 e3 /2/ Pi ) ; > s e n s i t i v i t y w :=1/(2 ∗ omega n ∗ M e f f ∗ L) ∗ i n t ( D e f e ˆ 2 , e = 0 . . 1 ) ; > s e n s i t i v i t y f H z n N := e v a l f ( s e n s i t i v i t y w /2/ Pi /1 e9 ) ; Q factor > > > > > > > > > > Nd: = 1 2 : Ns : = 1 5 : o v e r l a p :=6 e − 6: P:=1000 e − 3: g c o u :=2 e − 6: A ov :=2 ∗ (Nd+Ns ) ∗ o v e r l a p ∗ t ; A beam:=L ∗w; A cou :=A beam+A comb+A ov ; z g A A A s q := t : g s q 1 :=2 e − 6: s q 2 :=4 e − 6: bsq : = t ∗w: sq1 :=2 ∗ Ns ∗w∗ t ; sq2 :=2 ∗ Ns ∗w∗ t+A bsq ; > b cou : = 3 . 7 e −4∗P∗ A cou ; > b s q : = 7 ∗ 3 . 7 e −4∗P∗ A sq1 ∗ z s q ˆ2/ g s q 1 ˆ2 + 7 ∗ 3 . 7 e −4∗P∗ A sq2 ∗ z s q ˆ2/ g s q 2 ˆ 2 ; > b p := b cou+b s q ; > Q p:= omega n ∗ ( M e f f ) / b p ; > > > > mu: = 1 7 . 9 e − 6: b c o u a i r :=mu∗ A cou / g c o u ; b s q a i r :=7 ∗mu∗ A sq1 ∗ z s q ˆ2/ g s q 1 ˆ3+7 ∗mu∗ A sq2 ∗ z s q ˆ2/ g s q 2 ˆ 3 ; b a i r := b c o u a i r+b s q a i r ; 103 > Q a i r := omega n ∗ ( M e f f ) / b a i r ; E l e c t r o s t a t i c Driving & Sensing Parameters > Nd: = 1 2 : Ns : = 1 5 : e p s i l o n : = 8 . 8 5 e − 12: g :=2 e − 6: t :=2 e − 6: a l p h a : = 1 . 6 1 : > omega:= omega n ; > dC dx d := e p s i l o n ∗ 2 ∗ Nd∗ t /g ; dC dx s := e p s i l o n ∗ 2 ∗ Ns ∗ t /g ; Voltages > VD: = 3 0 ; vd : = 3 ; b:= b a i r : s t a t i c displacement > x s :=1/2 ∗ dC dx d / K e f f ∗ (VDˆ2+1/2 ∗ vd ˆ2) − 1/2 ∗ dC dx s / K e f f ∗VSˆ 2 ; > x s s := su b s ( { VS=VD } , e v a l f ( x s ) ) ; o s c i l l a t i o n amplitude > x 1 := dC dx d ∗VD∗ vd/ s q r t ( ( K e f f −M e f f ∗ omegaˆ2)ˆ2+bˆ2 ∗ omega ˆ 2 ) ; > p s i 1 := a r c t a n ( b ∗ omega / ( K e f f −M e f f ∗ omega ˆ 2 ) ) ; > x 2:=− dC dx d ∗ vd ˆ2/4/ s q r t ( ( K e f f −4∗ M e f f ∗ omega ˆ2)ˆ2+4 ∗ bˆ2 ∗ omega ˆ 2 ) ; > p s i 2 := a r c t a n ( 2 ∗ b ∗ omega / ( K e f f −4∗ M e f f ∗ omega ˆ 2 ) ) ; s e n s i n g a m p l i t u d e (A) > i 1 :=Vs ∗ dC dx s ∗ dC dx d ∗VD∗ vd/ s q r t ( ( K e f f −M e f f ∗ omegaˆ2)ˆ2+ bˆ2 ∗ omega ˆ 2 ) ∗ omega ; C a p a c i t a n c e , C v a r i a t i o n , f o r c e s and c u r r e n t s > > o v l :=6 e − 6:VV: = 5 :amp:=1 e − 6: > C d:= e p s i l o n ∗ 2 ∗ Nd∗ t ∗ o v l /g ; C s := e p s i l o n ∗ 2 ∗ Ns ∗ t ∗ o v l /g ; > dC dx d := e p s i l o n ∗ 2 ∗ Nd∗ t /g ; dC dx s := e p s i l o n ∗ 2 ∗ Ns ∗ t /g ; > F d :=1/2 ∗ dC dx d ∗VVˆ 2 ; F s :=1/2 ∗ dC dx s ∗VVˆ 2 ; > i d :=VV∗ dC dx d ∗amp∗ omega n ; i s :=VV∗ dC dx s ∗amp∗ omega n ; Amplifier design > R:= b a i r /VD/Vs/ dC dx d / dC dx s ; > R p:= b p /VD/Vs/ dC dx d / dC dx s ; B.2 AC dynamic pullin.mws S t a b i l i t y Analysis AC Dynamic P u l l − i n Algorithm 104 P arameters > Vd: = 1 5 ; Vac : = 3 ;Va:=Vd−Vac ; Vb:=Vd+Vac ; g0 :=2 e − 6: ds :=114 e − 6: t :=2 e − 6:k : = 3 . 2 7 :m: = . 3 7 6 e − 11: e0 : = 8 . 8 5 e − 12; P o t e n t i a l energy p r o f i l e s > eqa :=k /2 ∗ yˆ2− t ∗ ds ∗ e0 ∗ Vaˆ 2 / 2 / ( g0−y ) : > eqb :=k /2 ∗ yˆ2− t ∗ ds ∗ e0 ∗Vbˆ 2 / 2 / ( g0−y ) : > p l o t ( [ eqa , eqb ] , y=−1.7 e − 6 . . 1 . 7 e − 6); Center o f o s c i l l a t i o n > eqdab :=k ∗ y−t ∗ ds ∗ e0 ∗ ( ( Va+Vb) / 2 ) ˆ 2 / 2 / ( g0−y ) ˆ 2 : > sdab := s o l v e ( eqdab , y ) : sdab [ 1 ] ; Calculation unstable equilibrium > eqdb :=k ∗ y−t ∗ ds ∗ e0 ∗Vbˆ 2 / 2 / ( g0−y ) ˆ 2 : > sdb := s o l v e ( eqdb , y ) ; > s d b i n t : = . 9 ∗ sdb [ 2 ] ; Energy a t P oi nt 1 and 2 > v1 := s ub s ( { y=s d b i n t } , e v a l f ( eqb ) ) ; > v2 := s ub s ( { y=s d b i n t } , e v a l f ( eqa ) ) ; Slope > Q: = 7 0 ;B:= s q r t ( k ∗m) /Q; > s l p a :=B ∗ ( s d b i n t −sdab [ 1 ] ) ∗ s q r t ( k/m) ; C a l c u l a t i o n Po in t 3 > i n t a := s o l v e ( eqa=v2+s l p a ∗ ( y− s d b i n t ) , y ) ; i n t a [ 1 ] ; Energy a t P oi nt 4 > v 4 i n t := s ub s ( { y=i n t a [ 1 ] } , e v a l f ( eqb ) ) ; C a l c u l a t i o n Po in t 5 > i n t b := s o l v e ( eqb=v 4 i n t − s l p a ∗ ( y− i n t a [ 1 ] ) , y ) ; i n t b [ 2 ] ; > v 5 i n t := s ub s ( { y=i n t b [ 2 ] } , e v a l f ( eqb ) ) ; S t a b i l i t y t e s t ( s t a b l e i f s t < 0) > s t := i n t b [ 2 ] − s d b i n t ; Plot of the o s c i l l a t i o n loop 105 > p l o t ( [ v1+s l p a ∗ ( y− s d b i n t ) , eqa , v 3 i n t − s l p a ∗ ( y− i n t a [ 1 ] ) , eqb ] , y=−1.9 e − 6 . . 1 . 9 e − 6); 106 Appendix C Ansys listings
This appendix lists examples of the input ﬁles used to extract the oscillation modes and frequency sensitivity of the doubleended tuning forks. The examples include the data of a lateralcombactuated DETF. The ﬁrst ﬁle, lb1b beam4.inp,implements the extraction of modes and the second ﬁle, lb1b beam4 sens.inp, includes the routine to extract the force sensitivity. examples use BEAM4 as structural element. The ﬁles have been executed using ANSYS/ED 5.6 of Ansys Inc. Both C.1 lb1b beam4.inp !MODAL ANALYSIS ! beam l b 1 b FINISH /CLEAR /GRA,POWER /GST,ON /PREP7 ! parameters b l =200 ! beam l e n g t h 107 s 1 l =37 s 2 l =114 f s e p =8 f l =12 ! comb arm ! comb l o n g ! f i n g e r s sep ! f i n g e r s long ! beam K, 1 , 0 , bl / 2 , 0 , K, 2 , 0 , − b l / 2 , 0 , ! comb arm K, 3 , 0 , 0 , 0 K, 4 , − s 1 l , 0 , 0 LSTR , 1, LSTR , 3, LSTR , 3, 3 2 4 ! aux KGEN , 8 , 4 , , , , f s e p , , , 0 KGEN, 8 , 4 , , , , − f s e p , , , 0 K, 1 9 , − s 1 l , − s 2 l / 2 , 0 K, 2 0 , − s 1 l , s 2 l / 2 , 0 !SUPORT LSTR , LSTR , LSTR , LSTR , LSTR , LSTR , LSTR , LSTR , LSTR , LSTR , LSTR , LSTR , LSTR , LSTR , LSTR , LSTR , 20, 11, 10, 9, 8, 7, 6, 5, 4, 12, 13, 14, 15, 16, 17, 18, 11 10 9 8 7 6 5 4 12 13 14 15 16 17 18 19 K, 2 1 , − ( s 1 l+ f l ) , − ( s 2 l /2 − 1),0 KGEN , 1 5 , 2 1 , , , , 8 , , , 0 108 ! EXT FINGERS LSTR , 11, LSTR , 10, LSTR , 9, LSTR , 8, LSTR , 7, LSTR , 6, LSTR , 5, LSTR , 4, LSTR , 12, LSTR , 13, LSTR , 14, LSTR , 15, LSTR , 16, LSTR , 17, LSTR , 18, 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 K, 3 6 , − ( s 1 l − f l ) , ( s 2 l /2 − 1),0 KGEN, 6 , 3 6 , , , , − 8 , , , 0 K, 4 2 , − ( s 1 l − f l ) , − ( s 2 l /2 − 1),0 KGEN , 6 , 4 2 , , , , 8 , , , 0 ! INT FINGERS LSTR , 11, LSTR , 10, LSTR , 9, LSTR , 8, LSTR , 7, LSTR , 6, LSTR , LSTR , LSTR , LSTR , LSTR , LSTR , 13, 14, 15, 16, 17, 18, 36 37 38 39 40 41 47 46 45 44 43 42 !ELEMENT ELECTION !∗ ET , 1 ,BEAM4 !∗ KEYOPT, 1 , 2 , 0 KEYOPT, 1 , 6 , 0 109 KEYOPT, 1 , 7 , 0 KEYOPT, 1 , 9 , 0 KEYOPT, 1 , 1 0 , 0 !∗ !∗ !CONSTANT DEFINITION R, 1 , 4 , 1 6 / 1 2 , 1 6 / 1 2 , 2 , 2 , , R, 2 , 6 , 2 ∗ 2 7 / 1 2 , 3 ∗ 8 / 1 2 , 2 , 3 , , R, 3 , 8 , 2 ∗ 6 4 / 1 2 , 4 ∗ 8 / 1 2 , 2 , 4 , , !∗ !∗ ! PROPERTIES UIMP , 1 , EX , , , 1 7 0 e3 , UIMP , 1 ,NUXY , , , . 2 2 , UIMP , 1 , DENS , , , 2 3 3 0 e − 18, !∗ !MESHING TYPE , 1 MAT, REAL , ESYS , SECNUM, !BEAM ESIZE , 2 0 , 0 , LMESH, 1 , 2 1 1 0 TYPE , 1 MAT, REAL , ESYS , SECNUM, ! FINGERS ESIZE , , 2 , 1 1 0 110 LMESH, 2 0 , 4 6 TYPE , 1 MAT, REAL , ESYS , SECNUM, !SUPORT1 ESIZE , , 2 , LMESH, 3 , 1 3 0 TYPE , 1 MAT, REAL , ESYS , SECNUM, !SUPORT2 ESIZE , , 1 0 , LMESH, 4 , 1 9 FINISH /SOLU 1 2 0 DK , 1 , , 0 , , 0 , ALL , , , , , , DK , 2 , , 0 , , 0 , ALL , , , , , , !∗ ANTYPE, 2 !MODAL ANALYSIS !∗ MODOPT, SUBSP, 4 EQSLV,FRONT MXPAND, 0 , , , 0 LUMPM, 0 PSTRES, 0 !∗ MODOPT, SUBSP , 4 , 1 0 0 0 0 , 4 0 0 0 0 0 , ,OFF RIGID , ALL SUBOPT , 8 , 4 , 7 , 0 , 0 , ALL !∗ /STATUS,SOLU 111 SOLVE C.2 lb1b beam4 sens.inp !FREQUENCY SENSITIVITY ANALYSIS ! beam l b 1 b FINISH /CLEAR /GRA,POWER /GST,ON /PREP7 ! parameters b l =200 ! beam l e n g t h s 1 l =37 ! comb arm s 2 l =114 ! comb l o n g f s e p =8 f l =12 ! f i n g e r s sep ! f i n g e r s long ! beam K, 1 , 0 , bl / 2 , 0 , K, 2 , 0 , − b l / 2 , 0 , ! comb arm K, 3 , 0 , 0 , 0 K, 4 , − s 1 l , 0 , 0 LSTR , 1, LSTR , 3, LSTR , 3, 3 2 4 ! aux KGEN , 8 , 4 , , , , f s e p , , , 0 KGEN, 8 , 4 , , , , − f s e p , , , 0 K, 1 9 , − s 1 l , − s 2 l / 2 , 0 K, 2 0 , − s 1 l , s 2 l / 2 , 0 !SUPORT LSTR , LSTR , 20, 11, 11 10 112 LSTR , LSTR , LSTR , LSTR , LSTR , LSTR , LSTR , LSTR , LSTR , LSTR , LSTR , LSTR , LSTR , LSTR , 10, 9, 8, 7, 6, 5, 4, 12, 13, 14, 15, 16, 17, 18, 9 8 7 6 5 4 12 13 14 15 16 17 18 19 K, 2 1 , − ( s 1 l+ f l ) , − ( s 2 l /2 − 1),0 KGEN , 1 5 , 2 1 , , , , 8 , , , 0 !EXT FINGERS LSTR , 11, LSTR , 10, LSTR , 9, LSTR , 8, LSTR , 7, LSTR , 6, LSTR , 5, LSTR , 4, LSTR , 12, LSTR , 13, LSTR , 14, LSTR , 15, LSTR , 16, LSTR , 17, LSTR , 18, 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 K, 3 6 , − ( s 1 l − f l ) , ( s 2 l /2 − 1),0 KGEN, 6 , 3 6 , , , , − 8 , , , 0 K, 4 2 , − ( s 1 l − f l ) , − ( s 2 l /2 − 1),0 KGEN , 6 , 4 2 , , , , 8 , , , 0 ! INT FINGERS LSTR , 11, LSTR , 10, 36 37 113 LSTR , LSTR , LSTR , LSTR , LSTR , LSTR , LSTR , LSTR , LSTR , LSTR , 9, 8, 7, 6, 13, 14, 15, 16, 17, 18, 38 39 40 41 47 46 45 44 43 42 !ELEMENT ELECTION !∗ ET , 1 ,BEAM4 !∗ KEYOPT, 1 , 2 , 0 KEYOPT, 1 , 6 , 0 KEYOPT, 1 , 7 , 0 KEYOPT, 1 , 9 , 0 KEYOPT, 1 , 1 0 , 0 !∗ !∗ !CONSTANT DEFINITION R, 1 , 4 , 1 6 / 1 2 , 1 6 / 1 2 , 2 , 2 , , R, 2 , 6 , 2 ∗ 2 7 / 1 2 , 3 ∗ 8 / 1 2 , 2 , 3 , , R, 3 , 8 , 2 ∗ 6 4 / 1 2 , 4 ∗ 8 / 1 2 , 2 , 4 , , !∗ !∗ ! PROPERTIES UIMP , 1 , EX , , , 1 7 0 e3 , UIMP , 1 ,NUXY , , , . 2 2 , UIMP , 1 , DENS , , , 2 3 3 0 e − 18, !∗ !MESHING TYPE , 1 114 MAT, REAL , ESYS , SECNUM, !BEAM ESIZE , 2 0 , 0 , LMESH, 1 , 2 1 1 0 TYPE , 1 MAT, REAL , ESYS , SECNUM, ! FINGERS ESIZE , , 2 , LMESH, 2 0 , 4 6 1 1 0 TYPE , 1 MAT, REAL , ESYS , SECNUM, !SUPORT1 ESIZE , , 2 , LMESH, 3 , 1 3 0 TYPE , 1 MAT, REAL , ESYS , SECNUM, !SUPORT2 ESIZE , , 1 0 , LMESH, 4 , 1 9 FINISH 1 2 0 /SOLU FK , 1 , FY, 1 0 ! Axial f o r c e 10uN = +Fy 115 DK , 2 , , 0 , , 0 , ALL , , , , , , p s t r e s , on !ACTIVATE STRESS RECORDING /REP !∗ ANTYPE, 0 ! STRESS ANALYSIS !∗ /STATUS,SOLU SOLVE FINISH / solu FKDELE , 1 , ALL DK , 1 , , 0 , , 0 , ALL , , , , , , !∗ ANTYPE, 2 !MODAL ANALYSIS !∗ MODOPT, SUBSP, 4 EQSLV,FRONT MXPAND, 0 , , , 0 LUMPM, 0 PSTRES, 1 !∗ MODOPT, SUBSP , 4 , 1 0 0 0 0 , 4 0 0 0 0 0 , ,OFF RIGID , ALL SUBOPT , 8 , 4 , 7 , 0 , 0 , ALL !∗ /STATUS,SOLU SOLVE 116 ...
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This note was uploaded on 11/17/2010 for the course MECHANICAL Master The taught by Professor Andreishkel during the Spring '10 term at UC Irvine.
 Spring '10
 AndreiShkel

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