1. Calculate the diameters of the pitch circles.
2. Draw the center line X Y on the paper; and on this center line locate the centers A and B a distance apart equal to ½ the sum of the two pitch diameters. About these centers draw the pitch circles, of diameters as calculated. This will make the pitch circles tangent at the pitch point P.
3. Calculate the addendum and dedendum, adding this amount to and subtracting from the radii of the pitch circles. Then draw the addendum and dedendum circles with the radii thus found.
4. Draw the root circles with radii equal to the radii of the pitch circles minus an amount equal to the dedendum plus the clearance.
5. Draw the describing circles tangent to each other and to the pitch circles at the point P.
6. With the describing circle C rolling on the outside of the pitch circle of A, generate the epicycloid PF, continuing it until it meets the addendum circle of A. With the describing circle D rolling on the inside of the pitch circle of A, on the opposite side of line of centers from which the circle C rolled, generate the hypocy-cloid PG. Since the diameter of D is equal to the radius of the pitch circle of A, the hypocycloid PG will be a radial line; and consequently, after the student has become familiar with this fact, it will not be necessary actually to roll the circle to generate such a hypo-cycloid. The epicycloid PF and the hypocycloid PG together form one side of the tooth of gear A.
7. Divide the circumference of the pitch circle into as many equal parts as the gear has teeth, and through these points draw curves like the curve GPF. This may be done by making a templet of stiff paper that will just fit the curve GPF, and by means of this templet, transferring the curve to the points 1, 2,3, etc. Next find the points a, b, d, etc., half-way between 1 and 2, 2 and 3, etc., since there is to be no backlash, and through these points draw curves similar to GPF, but turned so as to curve the other way. Now, by filling in with full lines that part of the addendum circle between the points F and N, R and S, etc., and filling in the root circle between T and V, etc., we have the outline of the teeth on the gear A. In practice, instead of making sharp corners at T and V, as shown by the dotted lines, fillets are put in with arcs of circles, these fillets being made as large as possible and still allowing space so that the corner of the teeth on the other gear shall not strike.
8. Construct the teeth on the gear B in the same way as the teeth on A were constructed, the describing circle D generating the epicycloid PH by rolling on the outside of the pitch circle of B, and the describing circle C generating the hypocycloid PE by rolling on the inside of the pitch circle of B. The hypocycloid is not a straight line in this case, as the diameter of C is not equal to the radius of the pitch circle of B.
The calculations for the above case are as follows: 4 pitch means 4 teeth per inch of diameter; and as there are 16 teeth in B, its diameter will be 16 = 4"; 12 teeth in A will give 12 = 3" diameter.
The addendum for a standard machine-cut gear is usually made equal to the dedendum, and is equal to the reciprocal of the pitch.
Fig. 115. Layout for Annular Gears.
Hence, to find the addendum and dedendum in the present case, take the reciprocal of 4, which is ¼".
The clearance, being 1/8 the addendum, is equal to 1/8 of ¼ = 1/32". If the student tries to follow the above description by actually drawing these gears, it will be found necessary to draw them to about 3 times their actual size in order to bring out the points clearly.
That is to say, the pitch circles should be made 9" and 12"; the addendum and dedendum ¾"; the clearance 3/32"; the numbers of teeth, of course, remaining 12 and 16.