Design and analysis of a micromechanical tuning fork gyroscope

This thesis investigates the feasibility of a micromechanical tuning fork as an angular rate sensor. The gyroscopic effect of a tuning fork, which is caused by the oscillating mass moment of inertia due to the vibrating tines, is described and demonstrated by a simple model. The gyroscopic response to an input rate about the longitudinal axis between the tines is described by a second order, linear periodic (LP) differential equation which is derived by Hamilton's variational method. If the periodic terms in the equation of motion are neglected, the gyroscopic output is shown to be oscillatory in phase with the vibrating tines and linearly related to the input rate, the system quality factor, and the mechanical gain of the gyroscope. The mechanical gain is the ratio of the oscillating component of the mass moment of inertia about the input axis to the nominal component. The periodic terms in the equation of motion are not negligable for a lightly damped, high frequency micromechanical system. A sufficient condition for stability of the LP system is derived from the properties of the Mathieu Equation. The stability condition is specified in terms of the product of the system quality factor and the mechanical gain, and it is checked by applying Floquet Theory. The response of the gyroscopically forced LP equation is solved numerically and also estimated by a Fourier Series solution. Both of these solutions correspond with the estimated linear response within the stability region. Mathematical models are derived for the electrostatic driving force between the two fork tines, the variable capacitance sensing of the gyroscopic rotation, the stiffness properties of the structure, and the air damping torque. In addition, possible error sources are analyzed including cross-axis sensitivity, external forces and vibration, unbalance torques, motorsensor coupling, amplifier noise, and Brownian Noise. The system mathematical model is implemented into a computer program and a baseline design configuration is obtained which is compatible with micromachining processes. The predicted performance of the baseline design is shown to be competitive with the double gimbal micromechanical gyroscope currently being developed. Thesis Supervisor: Dr. Walter M. Hollister Title: Professor of Aeronautics and Astronautics

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