In
kinematics, Chebyshev's linkage is a
four-bar linkage
In the study of mechanisms, a four-bar linkage, also called a four-bar, is the simplest closed-chain movable linkage. It consists of four bodies, called ''bars'' or ''links'', connected in a loop by four joints. Generally, the joints are config ...
that converts
rotational motion
Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional ...
to approximate
linear motion
Linear motion, also called rectilinear motion, is one-dimensional motion along a straight line, and can therefore be described mathematically using only one spatial dimension. The linear motion can be of two types: uniform linear motion, with co ...
.
It was invented by the 19th-century mathematician
Pafnuty Chebyshev
Pafnuty Lvovich Chebyshev ( rus, Пафну́тий Льво́вич Чебышёв, p=pɐfˈnutʲɪj ˈlʲvovʲɪtɕ tɕɪbɨˈʂof) ( – ) was a Russian mathematician and considered to be the founding father of Russian mathematics.
Chebyshe ...
, who studied theoretical problems in kinematic
mechanisms. One of the problems was the construction of a linkage that converts a rotary motion into an approximate straight-line motion (a
straight line mechanism). This was also studied by
James Watt in his improvements to the
steam engine, which resulted in
Watt's linkage.
Cornell university
– Cross link straight-line mechanism
Equations of motion
The motion of the linkage can be constrained to an input angle that may be changed through velocities, forces, etc. The input angles can be either link ''L''2 with the horizontal or link ''L''4 with the horizontal. Regardless of the input angle, it is possible to compute the motion of two end-points for link ''L''3 that we will name A and B, and the middle point.
:
:
while the motion of point B will be computed with the other angle,
:
:
And ultimately, we will write the output angle in terms of the input angle,
:
Consequently, we can write the motion of point P, using the two points defined above and the definition of the middle point.
:
:
Input angles
The limits to the input angles, in both cases, are:
:
:
Usage
Chebyshev linkages did not receive widespread usage in steam engines, but are commonly used as the 'Horse head' design of level luffing crane. In this application the approximate straight movement is translated away from the line's midpoint, but it is still essentially the same mechanism.
See also
* Chebyshev lambda linkage, the cognate of the Chebyshev linkage.
*Four-bar linkage
In the study of mechanisms, a four-bar linkage, also called a four-bar, is the simplest closed-chain movable linkage. It consists of four bodies, called ''bars'' or ''links'', connected in a loop by four joints. Generally, the joints are config ...
* Straight line mechanism
References
External links
{{Commons category, Chebyshev linkage
Cornell university, ''"How to draw a straight line, by A.B. Kempe, B.A."''
using the Molecular Workbench software
A Geogebra
simulation of the linkage
A 3D video
of the linkage
Linkages (mechanical)
Linear motion
Straight line mechanisms