§ 78.* An example of an approximate straight-line motion composed of three bars fixed to a frame is shown in fig. 114. It is due to P. L. Tchebichev of St Petersburg. The links AB and CD are equal in length and are centred respectively at A and C. The ends D and B are joined by a link DB. If the respective lengths are made in the proportions AC : CD : DB = 1 : 1.3 : 0.4 the middle point P of DB will describe an approximately straight line parallel to AC within limits of length about equal to AC. C. N. Peaucellier, a French engineer officer, was the first, in 1864, to invent a linkwork with which an exact straight line could be drawn. The linkwork is shown in fig. 115, from which it will be seen that it consists of a rhombus of four equal bars ABCD, jointed at opposite corners with two equal bars BE and DE. The seventh link AF is equal in length to halt the distance EA when the mechanism is in its central position. The points E and F are fixed. It can be proved that the point C always moves in a straight line at right angles to the line EF. The more general property of the mechanism corresponding to proportions between the lengths FA and EF other than that of equality is that the curve described by the point C is the inverse of the curve described by A. There are other arrangements of bars giving straight-line motions, and these arrangements together with the general properties of mechanisms of this kind are discussed in How to Draw a Straight Line by A. B. Kempe (London, 1877).

Fig. 116.
Fig. 117.

§ 79.* The Pantograph.—If a parallelogram of links (fig. 116), be fixed at any one point a in any one of the links produced in either direction, and if any straight line be drawn from this point to cut the links in the points b and c, then the points a, b, c will be in a straight line for all positions of the mechanism, and if the point b be guided in any curve whatever, the point c will trace a similar curve to a scale enlarged in the ratio ab : ac. This property of the parallelogram is utilized in the construction of the pantograph, an instrument used for obtaining a copy of a map or drawing on a different scale. Professor J. J. Sylvester discovered that this property of the parallelogram is not confined to points lying in one line with the fixed point. Thus if b (fig. 117) be any point on the link CD, and if a point c be taken on the link DE such that the triangles CbD and DcE are similar and similarly situated with regard to their respective links, then the ratio of the distances ab and ac is constant, and the angle bac is constant for all positions of the mechanism; so that, if b is guided in any curve, the point c will describe a similar curve turned through an angle bac, the scales of the curves being in the ratio ab to ac. Sylvester called an instrument based on this property a plagiograph or a skew pantograph.

The combination of the parallelogram with a straight-line motion, for guiding one of the points in a straight line, is illustrated in Watt’s parallel motion for steam-engines. (See [Steam-engine].)

§ 80.* The Reuleaux System of Analysis.—If two pieces, A and B, (fig. 118) are jointed together by a pin, the pin being fixed, say, to A, the only relative motion possible between the pieces is one of turning about the axis of the pin. Whatever motion the pair of pieces may have as a whole each separate piece shares in common, and this common motion in no way affects the relative motion of A and B. The motion of one piece is said to be completely constrained relatively to the other piece. Again, the pieces A and B (fig. 119) are paired together as a slide, and the only relative motion possible between them now is that of sliding, and therefore the motion of one relatively to the other is completely constrained. The pieces may be paired together as a screw and nut, in which case the relative motion is compounded of turning with sliding.

Fig. 118.Fig. 119.

These combinations of pieces are known individually as kinematic pairs of elements, or briefly kinematic pairs. The three pairs mentioned above have each the peculiarity that contact between the two pieces forming the pair is distributed over a surface. Kinematic pairs which have surface contact are classified as lower pairs. Kinematic pairs in which contact takes place along a line only are classified as higher pairs. A pair of spur wheels in gear is an example of a higher pair, because the wheels have contact between their teeth along lines only.

A kinematic link of the simplest form is made by joining up the halves of two kinematic pairs by means of a rigid link. Thus if A1B1 represent a turning pair, and A2B2 a second turning pair, the rigid link formed by joining B1 to B2 is a kinematic link. Four links of this kind are shown in fig. 120 joined up to form a closed kinematic chain.

Fig. 120.

In order that a kinematic chain may be made the basis of a mechanism, every point in any link of it must be completely constrained with regard to every other link. Thus in fig. 120 the motion of a point a in the link A1A2 is completely constrained with regard to the link B1B4 by the turning pair A1B1, and it can be proved that the motion of a relatively to the non-adjacent link A3A4 is completely constrained, and therefore the four-bar chain, as it is called, can be and is used as the basis of many mechanisms. Another way of considering the question of constraint is to imagine any one link of the chain fixed; then, however the chain be moved, the path of a point, as a, will always remain the same. In a five-bar chain, if a is a point in a link non-adjacent to a fixed link, its path is indeterminate. Still another way of stating the matter is to say that, if any one link in the chain be fixed, any point in the chain must have only one degree of freedom. In a five-bar chain a point, as a, in a link non-adjacent to the fixed link has two degrees of freedom and the chain cannot therefore be used for a mechanism. These principles may be applied to examine any possible combination of links forming a kinematic chain in order to test its suitability for use as a mechanism. Compound chains are formed by the superposition of two or more simple chains, and in these more complex chains links will be found carrying three, or even more, halves of kinematic pairs. The Joy valve gear mechanism is a good example of a compound kinematic chain.