Rotating bouncing disks, tossing pizza dough, and the behavior of ultrasonic motors

Pizza tossing and certain forms of standing-wave ultrasonic motors (SWUMs) share a similar process for converting reciprocating input into continuous rotary motion. We show that the key features of this motion conversion process such as collision, separation and friction coupling are captured by the dynamics of a disk bouncing on a vibrating platform. The model shows that the linear or helical hand motions commonly used by pizza chefs and dough-toss performers for single tosses maximize energy efficiency and the dough's airborne rotational speed; on the other hand, the semielliptical hand motions used for multiple tosses make it easier to maintain dough rotation at the maximum speed. The system's bifurcation diagram and basins of attraction also provide a physical basis for understanding the peculiar behavior of SWUMs and provide a means to design them. The model is able to explain the apparently chaotic oscillations that occur in SWUMs and predict the observed trends in steady-state speed and stall torque as preload is increased.