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A trammel of Archimedes is a mechanism that generates the shape of an ellipse. () It consists of two shuttles which are confined ("trammeled") to perpendicular channels or rails and a rod which is attached to the shuttles by pivots at fixed positions along the rod.
As the shuttles move back and forth, each along its channel, all points on the rod move in elliptical paths. The motion of the rod is termed elliptical motion. The semi-axes ''a'' and ''b'' of the ellipses have lengths equal to the distances from the point on the rod to each of the two pivots.
The straight lines described by the pivots are special cases of an ellipse, where the length of one axis is twice the distance between the pivots and that of the other is zero. All points on a circle with a diameter defined by the two pivots reciprocate in such straight lines. This circle corresponds to the smaller circle in a Tusi couple.
The point midway between the pivots orbits in a circle around the point where the channels cross. This circle is also a special case of an ellipse. Here the axes are of equal length. The diameter of the circle is equal to the distance between the pivots. The direction of travel around the orbit is opposite to the sense of rotation of the trammel. Thus, if a crank centred on the crossing point of the channels is used to engage the trammel at the midway point to drive it, the rotation of the crankpin and the trammel are equal and opposite, which in practical applications results in extra friction and accelerated wear. This is compounded by high forces owing to the short throw of the crank of only 1/4 the travel of the pivots.
Wooden versions of the trammel of Archimedes have been produced as devices to draw or cut ellipses, known as ellipsographs. Versions are also made as
An ellipsograph is a trammel of Archimedes intended to draw, cut, or machine ellipses, e.g. in wood or other sheet materials. An ellipsograph has the appropriate instrument (pencil, knife, router, etc.) attached to the rod. Usually the distances ''a'' and ''b'' are adjustable, so that the size and shape of the ellipse can be varied.
The history of such ellipsographs is not certain, but they are believed to date back to Proclus and perhaps even to the time of
Video of various trammel designs in actionPhoto of a Kentucky Do-NothingInstructions
on how to build a Kentucky Do-Nothing
Video
of a Do-Nothing made from Lego bricks
"Wonky Trammel of Archimedes"
An exploration of a generalized trammel.
* ttps://www.youtube.com/watch?v=7Fn-26Jmi5E ''Secrets of the Nothing Grinder''YouTube video by Mathologer Traditional toys Mechanisms (engineering) Linkages (mechanical) Conic sections Novelty items Educational toys Archimedes
A trammel of Archimedes is a mechanism that generates the shape of an ellipse. () It consists of two shuttles which are confined ("trammeled") to perpendicular channels or rails and a rod which is attached to the shuttles by pivots at fixed positions along the rod.
As the shuttles move back and forth, each along its channel, all points on the rod move in elliptical paths. The motion of the rod is termed elliptical motion. The semi-axes ''a'' and ''b'' of the ellipses have lengths equal to the distances from the point on the rod to each of the two pivots.
The straight lines described by the pivots are special cases of an ellipse, where the length of one axis is twice the distance between the pivots and that of the other is zero. All points on a circle with a diameter defined by the two pivots reciprocate in such straight lines. This circle corresponds to the smaller circle in a Tusi couple.
The point midway between the pivots orbits in a circle around the point where the channels cross. This circle is also a special case of an ellipse. Here the axes are of equal length. The diameter of the circle is equal to the distance between the pivots. The direction of travel around the orbit is opposite to the sense of rotation of the trammel. Thus, if a crank centred on the crossing point of the channels is used to engage the trammel at the midway point to drive it, the rotation of the crankpin and the trammel are equal and opposite, which in practical applications results in extra friction and accelerated wear. This is compounded by high forces owing to the short throw of the crank of only 1/4 the travel of the pivots.
Wooden versions of the trammel of Archimedes have been produced as devices to draw or cut ellipses, known as ellipsographs. Versions are also made as toy
A toy or plaything is an object that is used primarily to provide entertainment. Simple examples include toy blocks, board games, and dolls. Toys are often designed for use by children, although many are designed specifically for adults and pe ...
s or novelty item
A novelty item or simply novelty is an object which is specifically designed to serve no practical purpose, and is sold for its uniqueness, humor, or simply as something new (hence "novelty", or newness). The term also applies to practical items wi ...
s (sold under the name of Kentucky do-nothings, nothing grinders, do nothing machines, smoke grinders, or bullshit grinders). In these toys the drafting instrument is replaced by a crank handle, and the position of the sliding shuttles is usually fixed.
Mathematics
Let be the outer end of the rod, and , be the pivots of the sliders. Let and be the distances from to and to , respectively. Let us assume that sliders and move along the andcoordinate
In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is sign ...
axes, respectively. When the rod makes an angle with the ''x''-axis, the coordinates of point are given by
:
:
These are in the form of the standard parametric equations for an ellipse in canonical position. The further equation
:
is immediate as well.
The trammel of Archimedes is an example of a four-bar linkage with two sliders and two pivots, and is special case of the more general oblique trammel. The axes constraining the pivots do not have to be perpendicular and the points , and can form a triangle. The resulting locus
Locus (plural loci) is Latin for "place". It may refer to:
Entertainment
* Locus (comics), a Marvel Comics mutant villainess, a member of the Mutant Liberation Front
* ''Locus'' (magazine), science fiction and fantasy magazine
** ''Locus Award' ...
of is still an ellipse.
Ellipsographs
An ellipsograph is a trammel of Archimedes intended to draw, cut, or machine ellipses, e.g. in wood or other sheet materials. An ellipsograph has the appropriate instrument (pencil, knife, router, etc.) attached to the rod. Usually the distances ''a'' and ''b'' are adjustable, so that the size and shape of the ellipse can be varied.
The history of such ellipsographs is not certain, but they are believed to date back to Proclus and perhaps even to the time of Archimedes
Archimedes of Syracuse (;; ) was a Greek mathematician, physicist, engineer, astronomer, and inventor from the ancient city of Syracuse in Sicily. Although few details of his life are known, he is regarded as one of the leading scientists i ...
.
See also
*Beam compass
A beam compass is a compass with a beam and sliding sockets or cursors for drawing and dividing circles larger than those made by a regular pair of compasses. The instrument can be as a whole, or made on the spot with individual sockets (called ...
* Bourke engine
* John Farey Jr.
* Hypocycloid
* Hypotrochoid
* Tusi couple
* Useless machine
A useless machine or useless box is a device which has a function but its direct purpose is deliberately unknown. The best-known useless machines are those inspired by Marvin Minsky's design, in which the device's sole function is to switch itsel ...
* Scott Russell linkage
Notes
References
* J. W. Downs: ''Practical Conic Sections: The Geometric Properties of Ellipses, Parabolas and Hyperbolas''. Courier Dover 2003, , pp. 4–5 () * I. I. Artobolevskii ''Mechanisms for the Generation of Plane Curves''. Pergamon Press 1964, .External links
{{commonscat, Trammel of ArchimedesVideo of various trammel designs in action
on how to build a Kentucky Do-Nothing
Video
of a Do-Nothing made from Lego bricks
"Wonky Trammel of Archimedes"
An exploration of a generalized trammel.
* ttps://www.youtube.com/watch?v=7Fn-26Jmi5E ''Secrets of the Nothing Grinder''YouTube video by Mathologer Traditional toys Mechanisms (engineering) Linkages (mechanical) Conic sections Novelty items Educational toys Archimedes