§ 45. Sliding Contact: Forms of the Teeth of Spur-wheels and Racks.—A line of connexion of two pieces in sliding contact is a line perpendicular to their surfaces at a point where they touch. Bearing this in mind, the principle of the comparative motion of a pair of teeth belonging to a pair of spur-wheels, or to a spur-wheel and a rack, is found by applying the principles stated generally in §§ 36 and 37 to the case of parallel axes for a pair of spur-wheels, and to the case of an axis perpendicular to the direction of shifting for a wheel and a rack.
In fig. 101, let C1, C2 be the centres of a pair of spur-wheels; B1IB1′, B2IB2′ portions of their pitch-circles, touching at I, the pitch-point. Let the wheel 1 be the driver, and the wheel 2 the follower.
| Fig. 101. |
Let D1TB1A1, D2TB2A2 be the positions, at a given instant, of the acting surfaces of a pair of teeth in the driver and follower respectively, touching each other at T; the line of connexion of those teeth is P1P2, perpendicular to their surfaces at T. Let C1P1, C2P2 be perpendiculars let fall from the centres of the wheels on the line of contact. Then, by § 36, the angular velocity-ratio is
α2/α1 = C1P1/C2P2.
(23)
The following principles regulate the forms of the teeth and their relative motions:—
I. The angular velocity ratio due to the sliding contact of the teeth will be the same with that due to the rolling contact of the pitch-circles, if the line of connexion of the teeth cuts the line of centres at the pitch-point.
For, let P1P2 cut the line of centres at I; then, by similar triangles,