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, 14, 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.

这篇论文研究了一种被称为喋喋不休的玩具的动力学。具体来说，就是考察小环绕大环旋转的机制，大环的力给小环提供扭矩的机制，小环的运动是否和草裙舞一样箍。之前的工作 [13, 14, 15] 研究了振铃的动力学；但是，尚未进行详细分析。因此，为了理解上述机制，获得了运动方程和约束条件，并对运动进行了分析。为了简化问题，在无滑移条件下分析了由直杆和垫圈组成的模型。洗衣机的运动有两种模式：单点接触（1PC）模式和两点接触（2PC）模式。振铃小环的运动类似于 2PC 模式下的垫圈，而呼啦圈的运动类似于 1PC 模式下的垫圈。分析表明，具有两个接触点的垫圈的运动等效于自由落体运动。然而，在实践中，速度通过能量耗散达到恒定值。垫圈围绕穿过两个接触点的轴旋转。杆在垂直于垫圈平面的接触点处施加的分力提供了作用在质心的旋转扭矩。平行于水平面的力的分量是向心力，它引起质心的圆周运动。振铃小环的运动类似于 2PC 模式下的垫圈，而呼啦圈的运动类似于 1PC 模式下的垫圈。分析表明，具有两个接触点的垫圈的运动等效于自由落体运动。然而，在实践中，速度通过能量耗散达到恒定值。垫圈围绕穿过两个接触点的轴旋转。杆在垂直于垫圈平面的接触点处施加的分力提供了作用在质心的旋转扭矩。平行于水平面的力的分量是向心力，它引起质心的圆周运动。振铃小环的运动类似于 2PC 模式下的垫圈，而呼啦圈的运动类似于 1PC 模式下的垫圈。分析表明，具有两个接触点的垫圈的运动等效于自由落体运动。然而，在实践中，速度通过能量耗散达到恒定值。垫圈围绕穿过两个接触点的轴旋转。杆在垂直于垫圈平面的接触点处施加的分力提供了作用在质心的旋转扭矩。平行于水平面的力的分量是向心力，它引起质心的圆周运动。