Sung C., Chao P., Wang C.

The motion of a hula hoop is commonly regarded as the circular oscillation which a ring undergoes around a moving human body. Based on simple dynamics, the hula-hoop motion is made possible by the interactive force between the moving ring and human body. Inspired by the generic concept of the hula-hoop motion, this study proposes a novel design of a motion transformer that is capable of transforming linear reciprocating motions to rotary ones based on the concepts similar to the hula-hoop motion. To the end of the transformer design, the dynamics of a generic hula hoop is first explored. With basic understandings on the hula-hoop motion, the transformer is proposed with essentially a main mass sprung in one direction and a free-moving mass attached at one end of a rod while the other end hinged onto the main mass. As the main mass oscillates harmonically under a sinusoidal external excitation and the initial conditions of the free-moving mass are carefully selected, the free-moving mass rotates around the hinge at same frequency to perform hula-hoop motion. To simplify the analysis, we assume that the main mass moves in one degree of freedom and every kinematic joint in the system has no friction between contact surfaces. Then, two nonlinear differential equations that govern the motion of the main mass and the free-moving mass, respectively, are developed for the system. Using the method of harmonic balance solves the approximate analytical solutions of the two equations. The stability of each solution and the corresponding stable region are then evaluated by employing Floquet theory. Finally, we construct an experimental setup, which composes of a linear motor for driving the main mass and a servomechanism for establishing the initial conditions of the free-moving mass, for the purpose of validating the proposed design and analysis.