Hiroshi Takano

This paper investigates the dynamics of a toy known as the chatter ring. Specifically, it examines the mechanism by which the small ring rotates around the large ring, the mechanism by which the force from the large ring provides torque to the small ring, and whether the motion of the small ring is the same as that of a hula hoop. The dynamics of a chatter ring has been investigated in previous work [13–15]; however, a detailed analysis has not yet been performed. Thus, to understand the mechanisms described above, the equations of motion and constraint conditions are obtained, and an analysis of the motion is performed. To simplify the problem, a model consisting of a straight rod and a washer ring is analyzed under the no-slip condition. The motion of a washer has two modes: the one point of contact (1PC) mode and two points of contact (2PC) mode. The motion of the small ring of the chatter ring is similar to that of a washer in the 2PC mode, whereas the motion of a hula hoop is similar to that of a washer in the 1PC mode. The analysis indicates that the motion of a washer with two points of contact is equivalent to free fall motion. However, in practice, the velocity reaches a constant value through energy dissipation. The washer rotates around an axis that passes through the two points of contact. The components of the forces exerted by the rod at the points of contact that are normal to the plane of the washer provide rotational torque acting at the center of mass. The components of the forces parallel to the horizontal plane are centripetal forces, which induce the circular motion of the center of mass.

An effective equation of motion of a rattleback is obtained from the basic equation of motion with viscous friction depending on slip velocity. This effective equation of motion is used to estimate the number of spin reversals and the rattleback’s shape that causes the maximum number of spin reversals. These estimates are compared with numerical simulations based on the basic equation of motion.