Coupled Flexural/Torsional Vibrations of a Piezoelectrically-Actuated Vibrating Beam Gyroscope

In this paper, the coupled flexural-torsional vibrations of a cantilever beam with rigid mass attached to its end are studied. The beam base is subjected to a rotational motion, forming a vibrating beam gyroscope undergoing bending-twisting vibrations. The primary (flexural) vibrations are produced in the beam using a piezoelectric patch actuator. Due to the gyroscopic effect, secondary (torsional) vibrations are induced in the beam. First, a detailed mathematical modeling of the system is developed using extended Hamilton's Principle. Partial differential equations of the motion for both bending and torsion vibrations are derived, which are coupled through the gyroscopic terms. Expression for the frequency equation is presented, and the effect of the base rotation on the system natural frequencies is studied. Finally, the system governing equations are solved and simulated using assumed mode model to analyze the relationship between the base rotation and gyroscopic coupling. It is shown that the system natural frequencies are nearly independent of the magnitude of the base rotation for small rotation velocities. Furthermore, the natural frequencies depend on the magnitude as well as dimensions of the end mass. Also, it is shown that the gyroscopic effect increases with increase in base rotation rate and primary excitation amplitude.© 2006 ASME

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