The dynamics of combined axial-torsional standing-wave ultrasonic motors

Abstract

Piezoelectric ultrasonic motors have the potential to enable important applications such as endovasular surgical micro-robots due to their high torque and power density at the 0.1-1 mm diameter range. A type of ultrasonic motor that is suitable for miniaturization is the combined axial-torsional standing-wave (CATS) ultrasonic motor that generates the CATS stator motion via pretwisted beam vibration converters. The operation of the motor involves (1) the generation of an ellipse-like stator tip trajectory when the pretwisted-beam stator is excited to vibrate in a CATS motion by a piezoelectric transducer, and (2) the transfer of frictional torque when the rotor is pressed against the stator tip. To gain a better understanding of the CATS ultrasonic motor, centimeter-scale prototypes were fabricated and tested to determine the characteristics of the motor design. Theoretical models of the pretwisted beam stator and the torque transfer mechanism were also investigated to help us predict the effects of various design parameters. The axial and torsional resonance frequencies of the pretwisted-beam stator needs to be matched for an effcient generation of the CATS stator motion. To help designers select the right analysis method for the design process, we investigated the validity of common pretwisted beam theories that assume the warping function of a pretwisted beam is locally identical to that of a prismatic beam. Through a scaling analysis of the equations governing the warping function of pretwisted beams -- derived using semi-inverse method and Hamilton's principle -- we obtained a set of criteria for checking the validity of the assumption. These criteria allow us to determine at what geometries the use of prismatic warping function will result in poor predictions of the axial resonance frequency and that alternative modelling methods are needed. Existing models of CATS motors ignore the vertical displacement of the rotor, predicting periodic behaviours that are contrary to the apparently random oscillations observed in the motor's steady-state operation. Our incorporation of the rotor's vertical motion results in a bouncing-disk model that explains various behaviours of the motor prototype, including the oscillations in the transient speed-time curve, and the effect of preload on stall torque and steady-state speed. The nonlinear dynamical system formed by the bouncing disk model shows that different stator trajectories and interface properties can give rise to complex phenomena such as period doubling bifurcation, chaos, and extremely long period chattering orbits. Knowledge of the location and basins of attraction for these orbits gives us detailed understanding of the motor's behaviour that will help designers improve the performance of CATS ultrasonic motor.