Rolling cone motion is the rolling motion generated by a

cone A cone is a three-dimensional geometric shape that tapers smoothly from a flat base (frequently, though not necessarily, circular) to a point called the apex or vertex. A cone is formed by a set of line segments, half-lines, or lines co ...

rolling over another cone. In rolling cone motion, at least one of the cones is

convex Convex or convexity may refer to: Science and technology * Convex lens, in optics Mathematics * Convex set, containing the whole line segment that joins points ** Convex polygon, a polygon which encloses a convex set of points ** Convex polytop ...

, 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

\pi

). The distinguishing characteristic of a rolling cone, in relation to other axially symmetrical rollers (

cylinder A cylinder (from ) has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes. In elementary geometry, it is considered a prism with a circle as its base. A cylinder may also be defined as an infi ...

sphere A sphere () is a Geometry, geometrical object that is a solid geometry, three-dimensional analogue to a two-dimensional circle. A sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...

, round disk), is that while rolling on a flat surface, the cone's

center of gravity In physics, the center of mass of a distribution of mass in space (sometimes referred to as the balance point) is the unique point where the weighted relative position of the distributed mass sums to zero. This is the point to which a force may ...

performs a circular motion rather than a

linear Linearity is the property of a mathematical relationship ('' function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear ...

one. Another unique characteristic is that one of its points (its apex) is at rest throughout the entire motion.

Kinematics

The motion of a rolling cone can be described as a superposition of a

rotational motion Rotation, or spin, is the circular movement of an object around a ''axis of rotation, central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A t ...

of the cone around its axis of symmetry, and a

rotary motion Rotation around a fixed axis is a special case of rotational motion. The fixed- axis hypothesis excludes the possibility of an axis changing its orientation and cannot describe such phenomena as wobbling or precession. According to Euler's rot ...

of its axis around the axis of symmetry of the stationary cone. The ratio between the

angular velocities In physics, angular velocity or rotational velocity ( or ), also known as angular frequency vector,(UP1) is a pseudovector representation of how fast the angular position or orientation of an object changes with time (i.e. how quickly an object ...

of these two motions is given by: :

\frac =

where

\alpha

and

\beta

are the half apex angles of the stationary cone and the rolling cone, respectively,

\omega_1

is the angular velocity of the rolling cone's axis of symmetry around the axis of symmetry of the stationary cone, and

\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 (i.e.

\alpha=\frac

), this ratio becomes

\frac{\sin\beta}

. For example, a cone having an apex angle of 60 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.

Use

One of the most practical applications of rolling cones is the use of

tapered roller bearing Tapered roller bearings are rolling element bearings that can support axial forces (i.e., they are good thrust bearings) as well as radial forces. Description The inner and outer ring raceways are segments of cones and the rollers are taper ...

s in rotating devices. Tapered bearings can bear higher loads than

ball bearing A ball bearing is a type of rolling-element bearing that uses balls to maintain the separation between the bearing races. The purpose of a ball bearing is to reduce rotational friction and support radial and axial loads. It achieves this ...

s in both radial and axial directions, and therefore are more frequently used as wheel bearings in most wheeled land vehicles. In Conveyor systems, conical rollers are sometimes used when there's a need to create a curved path. A common example is belt conveyors in airport terminals where there's a need to move the luggage in loops. In the 18th and 19th century rolling cone motion was used in the process of olive oil extraction. The olives were put in a large circular basin and heavy metal cones were rolled upon them. The fact that a cone can roll in circles without sliding made it more efficient to use conical roller millstones.

References

*Sir William Thomson and Peter Guthrie Tait (2003).
Principles of mechanics and dynamics
. pp. 79–81.

External links

Simulation appelet by Eugene Butikov. Covers also the case of a concave cone.

A review of traditional olive oil extracting techniques including conical rollers millstones. Motion (physics)