Fig. 79.
When the teeth of the wheel are curved to partly envelop the worm circumference it has been found, from experiments made by Robert Briggs, that the worm and the wheel will be more durable, and will work with greatly diminished friction, if the pitch line of the worm be located to increase the length of face and diminish that of the flank, which will decrease the length of face and increase the length of flank on the wheel, as is shown in [Fig. 79]; the location for the pitch line of the worm being determined as follows:—
Fig. 80.
The full radius of the worm is made equal to twice the pitch of its teeth, and the total depth of its teeth is made equal to .65 of its pitch. The pitch line is then drawn at a radius of 1.606 of the pitch from the worm axis. The pitch line is thus determined in [Fig. 76], with the result that the area of tooth face and of worm surface is equalized on the two sides of the pitch line in the figure. In addition to this, however, it may be observed that by thus locating the pitch line the arcs both of approach and of recess are altered. Thus in [Fig. 80] is represented the same worm and wheel as in [Fig. 79], but the pitch lines are here laid down as in ordinary gearing. In the two figures the arcs of approach are marked by the thickened part of the generating circle, while the arcs of recess are denoted by the dotted arc on the generating circle, and it is shown that increasing the worm face, as in [Fig. 79], increases the arc of recess, while diminishing the worm flank diminishes the arc of approach, and the action of the worm is smoother because the worm exerts more pulling than pushing action, it being noted that the action of the worm on the wheel is a pushing one before reaching, and a pulling one after passing, the line of centres.
Fig. 81.
It may here be shown that a worm-wheel may be made to work correctly with a square thread. Suppose, for example, that the diameter of the generating circle be supposed to be infinite, and the sides of the thread may be accepted as rolled by the circle. On the wheel we roll a straight line, which gives a cycloidal curve suitable to work with the square thread. But the action will be confined to the points of the teeth, as is shown in [Fig. 81], and also to the arc of approach. This is the same thing as taking the faces off the worm and filling in the flanks of the wheel. Obviously, then, we may reverse the process and give the worm faces only, and the wheel, flanks only, using such size of generating circle as will make the spaces of the wheel parallel in their depths and rolling the same generating circle upon the pitch line of the worm to obtain its face curve. This would enable the teeth on the wheel to be cut by a square-threaded tap, and would confine the contact of tooth upon tooth to the recess.