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MathKit 6

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Linkages

A linkage in its simplest form is a system of rigid bars, or links, connected by hinged joints that allow two connected bars to rotate with respect to each other. Such linkages are used in a pantograph (a tool for copying a drawing to a desired scale), in a windshield wiper, in the piston system of an engine and so on. In 1877 Sir Alfred Kempe published an amazing theorem which, in a popularized form, can be worded as follows: there exists a plane linkage that can forge your signature with any desired precision. In fact, Kempe’s proof contained a flaw, which was corrected in 2002.

A huge contribution to the theory of linkages was made by the great Russian mathematician P. L. Chebyshev. He was especially interested in the mechanisms that convert circular and rectilinear motions into each other. One of these devices and its simpler predecessors are presented in the models below.

Four-Bar Linkage


In a four-bar linkage both endpoints of the mechanism are fixed, forming the "ground link." One hinged joint uniformly rotates around a circle and the other one NON-uniformly moves back and forth along an arc.

Crank-and-Rocker Mechanism

A crank-and-rocker mechanism is obtained from the four-bar linkage by attaching a rigid triangle to the floating link (the one in the middle). The path described by the outer vertex of this triangle heavily depends on its shape, as you can see by varying the triangle's side lengths.


Draw the path by pressing the Trace button.
To change the lengths of sides and links, drag the red points.

Chebyshev's Lambda Mechanism

Pafnuty Lvovich Chebyshev, the great Russian mathematician (1821-1894), who made a substantial contribution to numerous areas of mathematics, invented many interesting mechanisms based on the four-bar linkage. The "Lambda mechanism," called so because it resembles the Greek letter lambda, was intended to convert the rotational motion into rectilinear motion. The mechanism was later studied by Karl Hoecken and is often referred to as Hoecken's Mechanism.


Explore different curves traced by this linkage
using sliders to change the lengths of the links.

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