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The term linkage is referred to any mechanism that is a combination of links or bars which are connected by pins, sliders, etc. The basic system of a crank mechanism is the four-bar linkage (or quadric-crank mechanism, Fig.1), consisting four links connected by pin joints which form pivots. The dimensions the individual links are given, and which of the four links is made the stationery frame, determine whether particular links will perform complete revolutions or merely oscillatory (to-and-fro rocking) movements.

Mechanical linkages are a fundamental part of machine design, and yet many simple linkages were neither well understood nor invented until the 19th century. Consider a stick: it has six degrees of freedom, three of which are the coordinates of its centre in space, the other three describing its rotation. Once nudged between a boulder and fulcrum it is constrained to a particular motion, to act as a lever to move the boulder. When more links are added and joined in various ways their collective motion can be further defined. Very complicated and precise motions can be designed into a linkage with only a few parts.

The Industrial Revolution was the golden age of mechanical linkages. Mathematical, engineering and manufacturing advances provided both the need and the ability to create new mechanisms. Many simple mechanisms that seem obvious today required some of the greatest minds of the era to create. Leonhard Euler was one of the first mathematicians to study linkage synthesis, and James Watt worked very hard to invent the Watt linkage to support his steam engine's piston. Chebyshev worked on mechanical linkage design for over thirty years, which led to his work on polynomials. New linkage inventions, designed by need, were instrumental in cloth making, power conversion and speed regulation. The ability of a mechanism to produce accurate linear motion, without a reference guide way, took years to solve.

For example, if the bottom link in Fig.1 is stationery and thus constitutes the frame of the mechanism, the shorter of the two links attached to it can rotate through 360 degrees (and is accordingly termed the crank), whereas the other link attached to the frame (and connected to the crank by the fourth link, termed the coupler) can only oscillate about its pivot and is accordingly referred to as the lever (or rocker arm). The amplitude of the lever will be accordingly smaller as the crank is shorter.

On the basis of this principle, it is possible to construct a mechanism in which the length of the crank can be measured while it is in motion. As shown in Fig.2, the lever may be connected to a ratchet wheel and pawl, so that the driven shaft (on which the ratchet wheel is mounted) rotates intermittently in one direction only. By varying of the length of the crank (by means of the slot), the amplitude of the lever can be varied from almost zero to a maximum, when the point A of the coupler is at the lower or at the upper end of the slot respectively.

If the shortest link of a four-bar linkage is held stationary (Fig.3), the resulting mechanism is called a drag-link mechanism. Here both of the links (crank and lever) attached to the stationary frame can perform complete revolutions. When the left-hand crank rotates at constant speed, the right-hand crank (originally the lever) rotates at varying speed.

A special case is the parallel linkage, in which the frame and coupler are of equal length and the two cranks are likewise of equal length (Fig.4). If the two cranks rotate in opposite directions, the mechanism is known as an antiparallel linkage (Fig.5). The drafting machine (Fig.6) comprises two parallel linkages which provide parallel motion of the straight edges. The same principle is applied to the toolbox illustrated in Fig.7.

The pantograph (Fig.8a) utilizes a parallel linkage for the proportional enlargement or reduction in scale of a given drawing. The points B and C of this mechanism trace figures that are similar in shape but differ in scale. The linear dimensions of the two figures are proportional to the respective distances of the points B and C to the pivot A.

For Example, if the distance CA is four times the distance BA, and if B is made to trace the outlines of a drawing, then a pencil at C will reproduce this drawing at four times the original size, and vice versa. A multiple pantograph is shown in Fig.8b. Here all the point located on the horizontal line- i.e., B, C, C1 and C2 will trace similar figures of varying size, depending on the respective distances from those points to the pivot at A. The familiar device known as lazy tongs, used for such purposes as lamp or telephone supports, consists of an assembly of parallel linkages. Fig.9 shows a typical application of a four-bar linkage for producing to-and-fro swinging motion of a fan.

An interesting feature of the beam-and-crank mechanism (Fig.10) is that whereas all points of the crank and lever trace only circular paths, points on or associated with the coupler trace paths that may have a wide variety of shapes, depending on where these points are located. This principle may be utilized for obtaining motions conforming to paths of particular shape.

If the trace of a point C on the coupler AB in the mechanism illustrated in Fig.11 locally conforms to a circular curve with radius r, it is possible to connect at C a link CM, of length r, which will produce a temporary standstill of the oscillating lever attached to it. So long as the point C travels along the circular curved portion of the trace, the link CM will rotate only about the point M, without causing displacement of this point.

Electronic technology has replaced many linkage applications taken for granted today, such as mechanical computation, typewriting and machining. However, modern linkage design continues to advance, and designs that used to occupy an engineer for days are now optimized with a computer in seconds.