We have seen in the preceding pages how the outlines of cycloidal gear teeth are generated by a point in a circle rolling on the pitch line. We have noted that the point of contact between the teeth is always somewhere on the describing circles, drawn tangent at the pitch point. The outlines of involute gear teeth, which are far more common than cycloidal teeth, are generated by a somewhat similar process. In the case of the involute, however, the describing point is located on a straight line, rolling, not on the pitch circle, but on another circle inside the pitch line, known as the base circle. The result of rolling a straight line, as noted above, is the same as if we stand up on the drawing board a small cylinder of diameter equal to the base circle, fasten one end of a string to some point in its circumference, and then allow the string to unwrap from the cylinder, a pencil point at the free end of the string marking on the paper below it the involute curve.
The method of drawing the involute curve is shown in Fig. 117; and it is obvious from this figure that the curve can never extend inside the base circle, although it may go any distance above it.
Fig. 118 shows a pair of gears with involute teeth, drawn according to the principles stated below. The circular pitch and diameters of pitch circles are calculated in the same way as described for cycloidal gears. The centers A and B are chosen, and the pitch circles drawn tangent at the pitch point P, as before. In involute gears, the point of contact between the teeth is always somewhere on an inclined line, CD, passing through the pitch point. The angle which this line makes with the tangent XY, is called the angle of obliquity (equal also to PBD). Its size has an important bearing on the action of gear teeth; and there are special conditions which, for the best tooth action, would call for widely different angles. It is not well, however, to have the angle of obliquity of different values, as it would then be impossible for any two gears to run together, except those based on the same angle. The angle of obliquity which has been quite generally adopted and which seems to fulfill the average conditions best, is 15°. In the present case, therefore, draw the line CD at an angle of 15° with the tangent XY; with A and B as centers, draw circles tangent to CD; these circles are called the base circles. The addendum, dedendum, and root circles are then drawn at the same relative distance from the pitch circles as in the case of cycloidal gears. The spacing of the teeth is now accomplished by stepping the dividers, set to the circular pitch, around the pitch circle. At any convenient points on the base circle, as G and E, generate the involutes in accordance with the method of Fig. 117, or as explained in Mechanical Drawing, Part II. Then, by the tracing-cloth method, or by the use of a templet fitted to this curve, draw in the tooth curves at points R, S, T, etc., on the pitch circles. This gives us the working part of the teeth, and the remainder of the tooth to the root circle consists of a radial line. Fillets are put in at the bottom of the teeth, as in the case of cycloidal gears.
Fig. 117. Method of Drawing Involute Curves.
As has been stated above, the point of contact between the teeth is always somewhere on the line CD; it is therefore obvious that, if the circle struck through the top of the tooth on one gear cuts the base circle of the other gear at a point outside of point C, there can be no true contact at the top of the tooth. Instead of there being true contact, the top of the tooth will actually dig into the lower portion of the tooth of the other gear. This is known as interference, and is overcome by slightly rounding off the top of the tooth down to the circle through point C, so that it will clear. Since the path of the point of contact is along the line CD, this line also represents the common normal to any pair of teeth in contact, and therefore is the line of pressure between the teeth. The obliquity of this line of pressure to the line of centers AB causes a thrust between these centers, tending to force the gears apart; and this has been considered an objection to the use of involute gears. With the standard 15° involute, however, experience has shown that this thrust is ordinarily of small importance. A similar thrust exists in cycloidal gears, but is constantly changing in value, being a maximum at the beginning and end of contact of a pair of teeth, and zero when the pair of teeth are in contact at the pitch point. It will be noted that the involute tooth is of simpler outline than the cycloidal, being a single curve instead of a reverse curve. If the exact distance between the centers A and B of a pair of involute gears be not maintained, owing to wear or to some other cause, the gears will still continue to run perfectly together; whereas in the case of cycloidal gears the action is seriously impaired by such a condition.
Fig. 118. Diagram Showing Pair of Involute Gears in Mesh.