Page:The Kinematics of Machinery.djvu/283

 UNIVERSAL JOINT, 261

The chain which constitutes this joint has four links, which are marked in the figure with the letters a, b, o, and d. The link a is paired with b by the turning-pair 2. Normal to this turning-pair is another, 3, which has its open cylinder in the fork of b, and its full cylinder in the sloping arm of the piece c ; the link b must therefore be written C + ... _L ... 0~. It must be noted that the lower and upper arms of the fork form together one piece only, and must be reckoned as such ; the same is true of the two ends of the arm of c, which kinematically form a single element only. The piece c consists of two solid cylinders, 3 and 4, having their axes crossing at right angles, and it must therefore be written C + ... J. ... C +. The third link, the fork and spindle d, is similar to &, and will be written in the same way. The fourth link a, lastly, consists of two open cylinders, 1 and 2, oblique to each other, and so must be written C~ ... Z. ... C~ ; it is a fixed link, as its form in the figure shows. The complete formula, therefore (to which we have added the letters and numbers used above to dis- tinguish the links and pairs), runs thus :

c+ ..,... c~ c+ ..._L... etc-. .,j_... cc- .. ./.... c-

23 4 f~ ""2

b c d a

There is one geometrical property of the chain which is not shown by our formula, namely that the axes of the pairs 1, 2, 3, and 4 have a common point of intersection. But unless the chain possessed this property it would not be possible, on our supposition that all its pairs are closed. No special indication of this property is therefore commonly necessary. Our formula shows, however, that the three links b, c, and d are again identical. This circum- stance is very notable, and we shall later on have to deal with it in another form ; the common construction of the joint so entirely conceals it as to make it almost unrecognizable.

The belt train, the kinematic nature of which we have already examined, will be written as follows :

T ii L T^ R+ I. . C + C~ II C~C + .... I ...-R+,

p P' -^ U T

The tension organ used here is a flat band, and is therefore marked with the suffix p (prismatic) ; it rolls both on to and off each