When the wheels are required to transmit motion rather than power (as in the case of clock wheels), to move as frictionless as possible, and to place a minimum of thrust on the journals of the shafts of the wheels, the generating circle may be made nearly as large as the diameter of the pitch circle, producing teeth of the form shown in [Fig. 52]. But the minimum of friction is attained when the two flanks for the tooth are drawn into one common hypocycloid, as in [Fig. 53]. The difference between the form of tooth shown in [Fig. 52] and that shown in [Fig. 53], is merely due to an increase in the diameter of the generating circle for the latter. It will be observed that in these forms the acting length of flank diminishes in proportion as the diameter of the generating circle is increased, the ultimate diameter of generating circle being as large as the pitch circles.
Fig. 54.
[1]This form is undesirable in that there is contact on one side only (on the arc of approach) of the line of centres, but the flanks of the teeth may be so modified as to give contact on the arc of recess also, by forming the flanks as shown in [Fig. 54], the flanks, or rather the parts within the pitch circles, being nearly half circles, and the parts without with peculiarly formed faces, as shown in the figure. The pitch circles must still be regarded as the rolling circles rolling upon each other. Suppose b a tracing point on b, then as b rolls on a it will describe the epicycloid a b. A parallel line c d will work at a constant distance as at c d from a b, and this distance may be the radius of that part of d that is within the pitch line, the same process being applied to the teeth on both wheels. Each tooth is thus composed of a spur based upon a half cylinder.
[1] From an article by Professor Robinson.
Comparing [Figs. 53] and [54], we see that the bases in [53] are flattest, and that the contact of faces upon them must range nearer the pitch line than in [54]. Hence, [53] presents a more favorable obliquity of the line of direction of the pressures of tooth upon tooth. In seeking a still more favorable direction by going outside for the point of contact, we see by simply recalling the method of generating the tooth curves, that tooth contacts outside the pitch lines have no possible existence; and hence, [Fig. 53] may be regarded as representing that form of toothed gear which will operate with less friction than any other known form.
This statement is intended to cover fixed teeth only, and not that complicated form of the trundle wheel in which the cylinder teeth are friction rollers. No doubt such would run still easier, even with their necessary one-sided contacts. Also, the statement is supposed to be confined to such forms of teeth as have good practical contacts at and near the line of centres.
