Fig. 16.
Now suppose that in [Fig. 16] a is a wheel of twice the diameter of b; that the two are free to revolve about their fixed centres, but that there is frictional contact between their perimeters at the line of centres sufficient to cause the motion of one to be imparted to the other without slip or lost motion, and that a point be marked on both wheels at the point of contact d. Now let motion be communicated to a until the mark that was made at d has moved one-eighth of a revolution and it will have moved through an eighth of a circle, or 45°. But during this motion the mark on b will have moved a quarter of a revolution, or through an angle of 90° (which is one quarter of the 360° that there are in the whole circle). The angular velocities of the two are, therefore, in the same ratio as their diameters, or two to one, and the velocity ratio is also two to one. The angular velocity of each is therefore the number of degrees of angle that it moves through in a certain portion of a revolution, or during the period that the other wheel of the pair makes a certain portion of a revolution, while the velocity ratio is the proportion existing between the velocity of one wheel and that of the other; hence if the diameter of one only of the wheels be changed, its angular velocity will be changed and the velocity ratio of the pair will be changed. The velocity ratio may be obtained by dividing either the radius, pitch, diameter, or number of teeth of one wheel into that of the other.
Conversely, if a given velocity ratio is to be obtained, the radius, diameter, or number of teeth of the driver must bear the same relation to the radius, diameter, or number of teeth of the follower, as the velocity of the follower is desired to bear to that of the driver.
Fig. 17.
If a pair of wheels have an equal number of teeth, the same pairs of teeth will come into action at every revolution; but if of two wheels one is twice as large as the other, each tooth on the small wheel will come into action twice during each revolution of the large one, and will work during each successive revolution with the same two teeth on the large wheel; and an application of the principle of the hunting tooth is sometimes employed in clocks to prevent the overwinding of their springs, the device being shown in [Fig. 17], which is from “Willis’ Principles of Mechanism.”
For this purpose the winding arbor c has a pinion a of 19 teeth fixed to it close to the front plate. A pinion b of 18 teeth is mounted on a stud so as to be in gear with the former. A radial plate c d is fixed to the face of the upper wheel a, and a similar plate f e to the lower wheel b. These plates terminate outward in semicircular noses d, e, so proportioned as to cause their extremities to abut against each other, as shown in the figure, when the motion given to the upper arbor by the winding has brought them into the position of contact. The clock being now wound up, the winding arbor and wheel a will begin to turn in the opposite direction. When its first complete rotation is effected the wheel b will have gained one tooth distance from the line of centres, so as to place the stop d in advance of e and thus avoid a contact with e, which would stop the motion. As each turn of the upper wheel increases the distance of the stops, it follows from the principle of the hunting cog, that after eighteen revolutions of a and nineteen of b the stops will come together again and the clock be prevented from running down too far. The winding key being applied, the upper wheel a will be rotated in the opposite direction, and the winding repeated as above.