Page:The Kinematics of Machinery.djvu/68

We may now proceed to the combination of three or four pairs of elements. Suppose the pairs

$ab$ $cd$ $ef$ $gh$

to be given. Let each element of each of these be joined to one element of another pair,—then every pair keeps its own peculiarity and at the same time has another added to it. The combination may take place in a number of different ways, for example in the same order as above.

$b-cd-ef-gh-a$ ;

or in the order

$b-dc-ef-hg-a$,

and so on. The whole now forms a linkage returning upon itself, like an endless chain, consisting simply of single links connected together. A combination of pairs of elements in this way we shall call a chain, or more fully a kinematic chain. The body which is formed by the junction of the elements of two different pairs is then a link of the kinematic chain. Every link of the above-mentioned chain consists of two elements, so that the chain here has as many links as it contains pairs.

In the chain every two adjacent links have a definite relative motion, that namely which belongs to the pair of elements connecting them. But two links which are connected by a third do not possess definite reciprocal motions except under certain conditions. Such motions can occur only if the chain be so arranged that every alteration in the position of a link relatively to the one next to it be accompanied by an alteration in the position of every other link relatively to the first. In a kinematic chain which possesses this peculiarity, each link has only one relative motion to each other link; if, that is to say, any relative motion occur in the chain, all the links are constrained to execute determinate relative motions. Such a kinematic chain I call a constrained closed—or simply a closed—chain.

We may take as an illustration the simple chain shown in figure 10. It consists of four similar pairs $ab,cd,ef,gh$, each being a cylindrical pin fitting a corresponding eye, the axes of all being parallel. Here each link has only motion in a circle relatively to the one next to it. Every turning of $ha$ relatively to $gf$ must necessarily be accompanied by alterations in the positions of $bc$ and of $de$—the chain is therefore closed.