Simplified formulae to investigate flexural vibration characteristics of piezoelectric tubes in ultrasonic micro-actuators

Based on the Rayleigh energy theory combining with Timoshenko beam model, the flexural vibration characteristics of piezoelectric tubes in ultrasonic micro-actuators are investigated. Additionally, the simplified formulae are derived to study the fundamental flexural resonance frequencies of the piezoelectric tubes with free-free ends and cantilevers. By changing the sizes of the tubes and the mass loads at the free ends, the variations of the flexural resonance frequencies of the piezoelectric tubes and cantilevers are calculated theoretically. To verify accuracy of the simplified formulae, by changing the lengths of the tubes and the mass loads the flexural resonance frequencies of the piezoelectric tube with free-free ends are measured experimentally. The theoretical results agree well with the experimental measurements, which demonstrate that the simplified formulae are accurate and effective for analyzing the flexural vibration characteristics of the piezoelectric tubes in the ultrasonic micro-actuators.