Fig. 90.
The depth below pitch line or the length of flank is, therefore, the distance between the pitch circle and the base circle. Now even supposing a straight line to be a portion of the circumference of a circle of infinite diameter or radius, the conditions would here appear to be imperfect, because the generating circle is not rolled upon the pitch circle but upon a circle of lesser diameter. But it can be shown that the requirements of a proper velocity ratio will be met, notwithstanding the employment of the base instead of the pitch circle. Thus, in [Fig. 90], let a and b represent the respective centres of the two pitch circles, marked in dotted lines. Draw the base circle for b as e q, which may be of any radius less than that of the pitch circle of b. Draw the straight line q d r touching this base circle at its perimeter and passing through the point of contact on the pitch circles as at d. Draw the circle whose radius is a r forming the base circle for wheel a. Thus the line r p q will meet the perimeters of the two circles while passing through the point of contact d at the line of centres (a condition which the relative diameters of the base circles must always be so proportioned as to attain).
If now we take any point on r q, as p in the figure, as a tracing point, and suppose the radius or distance p q to represent the steel spring shown in [Fig. 89], and move the tracing point back to the base circle of b, it will trace the involute e p. Again we may take the tracing point p (supposing the line p r to represent the steel spring), and trace the involute p f, and these two involutes represent each one side of the teeth on the respective wheels.
Fig. 91.
The line r p q is at a right angle to the curves p e and p f, at their point of contact, and, therefore, fills the conditions referred to in [Fig. 41]. Now the line r p q denotes the path of contact of tooth upon tooth as the wheels revolve; or, in other words, the point of contact between the side of a tooth on one wheel, and the side of a tooth on the other wheel, will always move along the line q r, or upon a similar line passing through d, but meeting the base circles upon the opposite sides of the line of centres, and since line q r always cuts the line of centres at the point of contact of the pitch circles, the conditions necessary to obtain a correct angular velocity are completely fulfilled. The velocity ratio is, therefore, as the length of b q is to that of a r, or, what is the same thing, as the radius of the base circle of one wheel is to that of the other. It is to be observed that the line q r will vary in its angle to the line of centres a b, according to the diameter of the base circle from which it is struck, and it becomes a consideration as to what is its most desirable angle to produce the least possible amount of thrust tending to separate the wheels, because this thrust (described in [Fig. 39]) tends to wear the journals and bearings carrying the wheel shafts, and thus to permit the pitch circles to separate. To avoid, as far as possible, this thrust the proportions between the diameters of the base circles d and e, [Fig. 91], must be such that the line d e passes through the point of contact on the line of centres, as at c, while the angles of the straight line d e should be as nearly 90° to a radial line, meeting it from the centres of the wheels (as shown in the figure, by the lines b e and d e), as is consistent with the length of d e, which in order to impart continuous motion must at least equal the pitch of the teeth. It is obvious, also, that, to give continuous motion, the length of d e must be more than the pitch in proportion, as the points of the teeth come short of passing through the base circles at d and e, as denoted by the dotted arcs, which should therefore represent the addendum circles. The least possible obliquity, or angle of d e, will be when the construction under any given conditions be made such by trial, that the base circles d and e coincide with the addendum circles on the line of centres, and thus, with a given depth of both beyond, the pitch circle, or addenda as it is termed, will cause the tooth contacts to extend over the greatest attainable length of line between the limits of the addendum circles, thus giving a maximum number of teeth in contact at any instant of time. These conditions are fulfilled in [Fig. 92],[3] the addendum on the small wheel being longer than the depth below pitch line, while the faces of the teeth are the narrowest.
[3] From an article by Prof. Robinson.
In seeking the minimum obliquity or angle of d e in the figure, it is to be observed that the less it is, the nearer the base circle approaches the pitch circle; hence, the shorter the operative length of tooth flank and the greater its wear.