Chebyshev Mechanism

Animated scheme of the Chebyshev mechanism.

The Chebyshev mechanism , also known as the Chebyshev parallelogram (the spellings Chebyshev , Chebyshev or Chebyshev are also found in transcriptions ), is a three-part coupling gear that converts the rotary motion into an approximately straight one ( straight line ). It was discovered in the 19th century by the Russian mathematician Pafnuti Lwowitsch Chebyshev , who was researching the theoretical problems of kinematic mechanisms. The conversion of a rotary motion into a linear motion is part of the more general problem of converting a rotary motion into any motion.

The range of linear movement is defined by the point - the center point of the rocker - that moves between the two end points of this gear. The deviation from the ideal linear movement is small on this route. ${\displaystyle P}$ ${\displaystyle L_{3}}$

The relationship between the individual lengths corresponds to the following expression:

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

From the relationship shown, it follows that the rocker is vertical when it has reached one of the end points of its movement. ${\displaystyle L_{3}}$

Mathematically, the lengths are related as follows:

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

Limits the input angle

Limiting the input angle if L2 is the input swing arm

The limits of the input angle are in both cases (L2 or L4 as input angle):

${\displaystyle \varphi _{\text{min}}=\arccos \left({\frac {4}{5}}\right)\approx 36.87^{\circ }.\,}$

${\displaystyle \varphi _{\text{max}}=\arccos \left({\frac {-1}{5}}\right)\approx 101.5^{\circ }.\,}$

other names

• Chebyshev parallelogram
• Cross link from Chebyshev

Similar mechanisms

The Chebyshev lambda mechanism (shown here twice, blue and green) shows the identical movement path

• Chebyshev Lambda Mechanism
• Hoeck Mechanism

Individual evidence

1. ^ Francis Moon: Cross Link Straight-line Mechanism. Cornell University, January 1, 2001, accessed February 13, 2014 . 
2. ↑ Straight Line Generators - Tchebicheff's Linkage. Brock institute for advanced studies, September 3, 2007, accessed February 13, 2014 . 
3. ↑ Digital mechanism and gear library: Applied gear theory Schroedel, Hannover a. a., 1952. page 178

Web links

Commons : Chebyshev linkage  - collection of images, videos and audio files

• Chebyshev Mechanisms website (Russian, English, French)