Page:The Kinematics of Machinery.djvu/507

 CLOSURE OF CHAINS. 485

Our investigations have, lastly, furnished us with a most im- portant theoretic result connected with the general nature of the closure of a kinematic chain or pair of elements. They have shown us that in every description of kinematic chain, from the most complex to the simplest, we have to distinguish three kinds of closure, namely:

1. Normal constrained closure,

2. Unconstrained closure,

3. Fixed closure.

In all three cases the conditions are fulfilled that the chain returns upon itself, and that proper pairing occurs between each link and its neighbour.

Under constrained closure all the relative motions of the links are perfectly determinate.

Under unconstrained closure these relative motions are made indeterminate by the addition of links to the chain.

Under fixed closure the motions of the links are entirely prevented.

All these kinds of closure are used in practice. The first and most important occurs in every machine, and forms a characteristic feature of it. The second we find in disengaging apparatus, where the action of a portion of the machine is stopped or reversed. The last kind of closure is used both for this purpose and with the object of preventing motions taking place within any single link, or, in other words, for making separate pieces into one link. The common constrained closure lies between the two other cases, and this is sometimes an assistance to us in finding out among the possible closed arrangements of links in any chain the important special case of constrained connection.

The apparent work of the machine designer consists in utilising these three methods of closure in different ways and for different objects. In reality, however, they are to him only means which he employs to solve the problem placed before him in the com- plete machine. We must now proceed to examine the general propositions which present themselves in connection with this subject.