The obliquity of the action in passing the line of centres is nothing; the maximum obliquity is the angle EID = E′ID; and the mean obliquity is one-half of that angle.
It appears from experience that the mean obliquity should not exceed 15°; therefore the maximum obliquity should be about 30°; therefore the equal arcs DI and ID′ should each be one-sixth of a circumference; therefore the circumference of the describing circle should be six times the pitch.
It follows that the smallest pinion of a set in which pinion the flanks are straight should have twelve teeth.
§ 50. Nearly Epicycloidal Teeth: Willis’s Method.—To facilitate the drawing of epicycloidal teeth in practice, Willis showed how to approximate to their figure by means of two circular arcs—one concave, for the flank, and the other convex, for the face—and each having for its radius the mean radius of curvature of the epicycloidal arc. Willis’s formulae are founded on the following properties of epicycloids:—
Let R be the radius of the pitch-circle; r that of the describing circle; θ the angle made by the normal TI to the epicycloid at a given point T, with a tangent to the circle at I—that is, the obliquity of the action at T.
Then the radius of curvature of the epicycloid at T is—
| For an internal epicycloid, ρ = 4r sin θ | R − r |
| R − 2r |
| For an external epicycloid, ρ′ = 4r sin θ | R + r |
| R + 2r |
(28)
Also, to find the position of the centres of curvature relatively to the pitch-circle, we have, denoting the chord of the describing circle TI by c, c = 2r sin θ; and therefore