VII. In a pair of pieces in rolling contact every straight line traversing the line of contact is a line of connexion.
§ 40. Cylindrical Wheels and Smooth Racks.—In designing cylindrical wheels and smooth racks, and determining their comparative motion, it is sufficient to consider a section of the pair of pieces made by a plane perpendicular to the axis or axes.
The points where axes intersect the plane of section are called centres; the point where the line of contact intersects it, the point of contact, or pitch-point; and the wheels are described as circular, elliptical, &c., according to the forms of their sections made by that plane.
When the point of contact of two wheels lies between their centres, they are said to be in outside gearing; when beyond their centres, in inside gearing, because the rolling surface of the larger wheel must in this case be turned inward or towards its centre.
From Principle III. of § 39 it appears that the angular velocity-ratio of a pair of wheels is the inverse ratio of the distances of the point of contact from the centres respectively.
| Fig. 96. |
For outside gearing that ratio is negative, because the wheels turn contrary ways; for inside gearing it is positive, because they turn the same way.
If the velocity ratio is to be constant, as in Willis’s Class A, the wheels must be circular; and this is the most common form for wheels.
If the velocity ratio is to be variable, as in Willis’s Class B, the figures of the wheels are a pair of rolling curves, subject to the condition that the distance between their poles (which are the centres of rotation) shall be constant.
The following is the geometrical relation which must exist between such a pair of curves:—