The action of the teeth in internal wheels is less a sliding and more a rolling one than that in any other form of toothed gearing. This may be shown as follows: In [Fig. 73] let a a represent the pitch circle of an external pinion, and b b that of an internal one, and p p the pitch circle of an external wheel for a a or an internal one for b b, the point of contact at the line of centres being at c, and the direction of rotation p p being as denoted by the arrow; the two pinions being driven, we suppose a point at c, on the pitch circle p p, to be coincident with a point on each of the two pinions at the line of centres. If p p be rotated so as to bring this point to the position denoted by d, the point on the external pinion having moved to e, while that on the internal pinion has moved to f, both having moved through an arc equal to c d, then the distance from e to d being greater than from d to f, more sliding motion must have accompanied the contact of the teeth at the point e than at the point f; and the difference in the length of the arc e d and that of f d, may be taken to represent the excess of sliding action for the teeth on e; for whatever, under any given condition, the amount of sliding contact may be, it will be in the proportion of the length of e d to that of f d. Presuming, then, that the amount of power transmitted be equal for the two pinions, and the friction of all other things being equal—being in proportion to the space passed (or in this case slid) over—it is obvious that the internal pinion has the least friction.
Chapter II.—THE TEETH OF GEAR-WHEELS.—CAMS.
Wheel and Tangent Screw or Worm and Worm Gear.
In [Fig. 74] are shown a worm and worm gear partly in section on the line of centres. The worm or tangent screw w is simply one long tooth wound around a cylinder, and its form may be determined by the rules laid down for a rack and pinion, the tangent screw or worm being considered as a rack and the wheel as an ordinary spur-wheel.
Fig. 74.
Worm gearing is employed for transmitting motion at a right angle, while greatly reducing the motion. Thus one rotation of the screw will rotate the wheel to the amount of the pitch of its teeth only. Worm gearing possesses the qualification that, unless of very coarse pitch, the worm locks the wheel in any position in which the two may come to a state of rest, while at the same time the excess of movement of the worm over that of the wheel enables the movement of the latter, through a very minute portion of a revolution. And it is evident that, when the plane of rotation of the worm is at a right angle to that of the wheel, the contact of the teeth is wholly a sliding one. The wear of the worm is greater than that of the wheel, because its teeth are in continuous contact, whereas the wheel teeth are in contact only when passing through the angle of action. It may be noted, however, that each tooth upon the worm is longer than the teeth on the wheel in proportion as the circumference of the worm is to the length of wheel tooth.