Rolling cone motion

Rolling cone motion is the rolling motion generated by a cone rolling over another cone. In rolling cone motion, at least one of the cones is convex, while the other cone may be either convex, or concave, or a flat surface (a flat surface can be regarded as a special case of a cone whose apex angle equals ${\displaystyle \pi }$). The distinguishing characteristic of a rolling cone, in relation to other axially symmetrical rollers (cylinder, sphere, round disk), is that while rolling on a flat surface, the cone's center of gravity performs a circular motion rather than a linear one. Another unique characteristic is that one of its points (its apex) is at rest throughout the entire motion.

The motion of a rolling cone can be described as a combination of a rotational motion of the cone around its axis of symmetry, and a rotary motion of its axis around the axis of symmetry of the stationary cone. The ratio between the angular velocities of these two motions is given by:

where ${\displaystyle \alpha }$ and ${\displaystyle \beta }$ are the half apex angles of the stationary cone and the rolling cone, respectively, ${\displaystyle \omega _{1}}$ is the angular velocity of the rolling cone's axis of symmetry around the axis of symmetry of the stationary cone, and ${\displaystyle \omega _{2}}$ is the angular velocity of the rolling cone around its own axis of symmetry. In the special case of a cone rolling on a flat surface, this ratio becomes ${\displaystyle \sin \alpha }$, where ${\displaystyle \alpha }$ is the cone's half apex angle. For example, a cone having an apex angle of 120 degrees, while being rolled on a flat surface, will perform exactly two full rotations around its axis of symmetry before returning to its original position.

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