Inkinematics,Chebyshev's linkage is afour-bar linkage that convertsrotational motion to approximatelinear motion.

It was invented by the 19th-century mathematicianPafnuty Chebyshev, who studied theoretical problems in kinematicmechanisms. One of the problems was the construction of a linkage that converts a rotary motion into an approximate straight-line motion (astraight line mechanism). This was also studied byJames Watt in his improvements to thesteam engine, which resulted inWatt's linkage.^[1]

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 L₂ with the horizontal or link L₄ with the horizontal. Regardless of the input angle, it is possible to compute the motion of two end-points for link L₃ that we will name A and B, and the middle point.

${\displaystyle x_A=L_2\cos ({\phi }_1)}$

${\displaystyle y_A=L_2\sin ({\phi }_1)}$

while the motion of point B will be computed with the other angle,

${\displaystyle x_B=L_1-L_4\cos ({\phi }_2)}$

${\displaystyle y_B=L_4\sin ({\phi }_2)}$

And ultimately, we will write the output angle in terms of the input angle,

${\displaystyle {\phi }_2=\arcsin \left[\frac{L_2\sin ({\phi }_1)}{\overline{A{O}_2}}\right]-\arccos \left(\frac{L_4^2+{\overline{A{O}_2}}^2-L_3^2}{2L_4\overline{A{O}_2}}\right)}$

Consequently, we can write the motion of point P, using the two points defined above and the definition of the middle point.

${\displaystyle x_P=\frac{x_A+x_B}{2}}$

${\displaystyle y_P=\frac{y_A+y_B}{2}}$

The limits to the input angles, in both cases, are:

${\displaystyle {\phi }_{\text{min}}=\arccos \left(\frac{4}{5}\right)\approx {36.8699}^{\circ }.}$

${\displaystyle {\phi }_{\text{max}}=\arccos \left(\frac{-1}{5}\right)\approx {101.537}^{\circ }.}$

Chebyshev linkages did not receive widespread usage in steam engines,^{[citation needed]} 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.

• Chebyshev lambda linkage, thecognate of the Chebyshev linkage.
• Four-bar linkage
• Straight line mechanism

Wikimedia Commons has media related toChebyshev linkage.

• Cornell university,"How to draw a straight line, by A.B. Kempe, B.A."
• A simulationArchived 2011-07-25 at theWayback Machine using the Molecular Workbench software
• A Geogebra simulation of the linkage
• A 3D video of the linkage

• Linkages (mechanical)
• Linear motion
• Straight line mechanisms

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