Two kinds of curves fulfill the requirement for gear teeth, that the common normal shall pass through the pitch point. These are the cycloidal and involute curves. The latter curve, for many reasons, has almost entirely displaced the former. The cycloidal curve is useful in special cases, and is still adhered to by its few advocates, as having peculiar merit, even for standard work. The general and best standard practice, however, is unalterably committed to the involute system, and experience has shown the reasons therefor to be sound ones.
The student can best approach the subject of the design of gear teeth through a study of the cycloidal system, the principles being capable of clearer illustration. Hence this system will be first presented.
The method of drawing the cycloidal curves by the use of rolling circles is illustrated in Fig. 113. The accurate curve, having been developed, may be transferred by the tracing-cloth method, as in cams, to each individual tooth; or arcs may be found by trial which approximate to the true curve; or a templet of stiff paper or cardboard may be made.
Fig. 114 shows a pair of epi-cycloidal gears designed to run together. The centers of the gears are at B and A; the pitch circles are shown in dot-and-dash, and are in contact at the pitch point P. The circle whose center is C, shown dotted, by rolling on the inside of the pitch circle of the gear B, generates the hypocycloid PE, which forms the flanks of the teeth on gear B; and by rolling on the outside of pitch circle of gear A, generates the epicycloid PF, which forms the faces of the teeth on gear A. In like manner the circle whose center is D, by rolling on the inside of the pitch circle of gear A, generates the hypocycloid PG, which forms the flanks of the teeth on A; and by rolling on the outside of the pitch circle of B, generates the epicycloid PH, which forms the faces of the teeth on B. The circles C and D are called the describing circles. If the gear B is the driver and is turning in the direction shown by the arrow, the flanks of its teeth act on the faces of the teeth on A from the point where they first come in contact until the point of contact reaches the pitch point; and from the pitch point on until the contact ceases, the faces of the teeth on B act on the flanks of the teeth on A. In other words, the hypo-cycloidal part of the tooth curve on one gear is generated by the same describing circle that generates the epicycloidal part of the tooth on the other gear with which it is in contact. This must always hold true, in order to have the gears run properly. The arc IP of the describing circle C, together with the arc JP of the describing circle D, forms what is called the path of contact; that is, the point of contact between the teeth is always somewhere on the line IPJ. If the gear A were the driver, the direction of rotation remaining the same, the path of contact would be LPK.
Fig. 113. Methods of Drawing Cycloidal Curves.
Fig. 114. Construction of Epicycloidal Gears.
To design a pair of epicycloidal spur gears, we must have given the pitch (either diametral or circular), the diameters of the pitch circles, or the number of teeth, and something to determine the size of the describing circles. Manufacturers have found by experience what are the best ratios of describing circles to pitch circles, and gears are designed according to those ratios. It is not well to have the diameter of the describing circle greater than 5/8 the diameter of the pitch circle, and it is better to have it smaller. If a set of gears is to be made, any one of which is likely to run with any other one, the same size describing circle must be used for the faces and flanks of all the gears; and this describing circle is often taken 5/8 the diameter of the smallest gear of the set. Sometimes when two gears are not part of an interchangeable set, but are designed to run with each other only, the diameter for the describing circle for the flanks of each gear is made equal to the radius of that gear; and when this is the case, the flanks are radial straight lines; or, as it is usually stated, the gears have radial flanks.
In Fig. 114, the two describing circles are of the same size and equal to the radius of the smaller gear, thus giving radial flanks on this gear. Let us proceed with the design of this pair of gears, given dimensions as follows: Gears to be 4 pitch (that is, as explained previously, 4 teeth per inch of pitch diameter); gear A to have 12 teeth; gear B 16 teeth; addendum equal to the diametral pitch; clearance equal to \ the addendum; describing circles each equal to radius of gear A.