Chebyshev linkage

Chebyshev linkage

The Chebyshev linkage is a mechanical linkage that converts rotational motion to approximate straight-line motion.

It was invented by the 19th century mathematician Pafnuty Chebyshev 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. This was also studied by James Watt in his improvements to the steam engine.

The straight line linkage confines the point P — the midpoint on the link L₃ — on a straight line at the two extremes and at the center of travel. (L₁, L₂, L₃, and L₄ are as shown in the illustration.) Between those points, point P deviates slightly from a perfect straight line. The proportions between the links are

$L_{1}:L_{2}:L_{3}=2:2.5:1=4:5:2.\,$

Point P is in the middle of L₃. This relationship assures that the link L₃ lies vertically when it is at the extremes of its travel.

The lengths are related mathematically as follows:

$L_{4}=L_{3}+{\sqrt {L_{2}^{2}-L_{1}^{2}}}.\,$

See also

• Watt's linkage, a similar straight-line mechanism with the direction of one of the arms reversed.
• Straight line mechanism
• Peaucellier–Lipkin linkage
• Four-bar linkage

External links

• Cornell university, "How to draw a straight line, by A.B. Kempe, B.A."
• A simulation using the Molecular Workbench software