Bevel gears are used to connect shafts whose axes intersect. The angle between the shafts is not necessarily a right angle, but this is the most common angle used. Fig. 119 shows a pair of bevel gears connecting two shafts whose axes intersect at a right angle.
The cones OPA and OPB are called pitch cones; the cones CPB and DPA, normal cones, and it is on these normal cones that the outlines of the teeth are laid out; BP and AP are the pitch diameters of the gears, and are found from the pitch and number of teeth just as the pitch diameters of spur gears are found.
To draw such a pair of gears, we must have given the angle between the shafts, the pitch and number of teeth in each gear, and the face of the tooth PE. The outlines of the teeth may be either involute or cycloidal; the addendum, deden-dum, and clearance are determined by the same empirical rules as were applied to the other gears which have been discussed.
Referring to Fig. 119, the gears shown are 2-pitch, 16 and 20 teeth, respectively, with face PE equal to 2 inches.
According to previous understanding, the addendum or the dedendum for a standard tooth is the reciprocal of the diametral pitch - or, in this case, ½". Making the clearance 1/8 of the addendum, would give 1/8 of ½" = 1/16". The teeth are of the involute form, with an angle of obliquity of 15°. Choosing point 0, draw the lines OC and OD, making an angle of 90° with each other; calculate the pitch diameters of the gears; lay off on OC the distance OH, equal to ½ the pitch diameter of the smaller gear; and through H draw a line perpendicular to OC. In like manner lay off on 0D the distance OJ, equal to ½ the pitch diameter of the larger gear; through J draw a line perpendicular to OD, meeting the perpendicular which is drawn through H at P; and make HB equal to HP, and JA equal to JP. From A, P, and B, draw lines to 0, producing the pitch cones; through P draw CD perpendicular to OP, meeting OC and OD in C and D, respectively. Join CB and DA, and we have the normal cones. Through C, P, and D, draw perpendiculars. Draw LMK parallel to CPD at any convenient distance. Draw arcs of circles tangent at the point M. These arcs are now to be treated as pitch circles on which to design the tooth curves, in exactly similar fashion to the method already outlined for spur gears.
Through point M draw the line of obliquity SR, and draw the base circles tangent to this line. With the addendum chosen as above, equal to ½", it will be found that the addendum circle of the larger gear will cut the line of obliquity beyond the point R, where SR is tangent to the base circle of the pinion. This means that true contact cannot occur at the top of the gear tooth, so the tooth should be slightly rounded off, to prevent interference with the flank of the pinion. The limit of this rounding-off of the point of the tooth is determined by striking a circle with center L through the point R, as it is obvious that below this point on the tooth of the gear there will be true involute contact. The root circles are drawn by setting off the clearance, as in the preceding cases. One tooth on each gear is drawn on the development of the pitch circle, and the completion of the drawing of the teeth in the several views of the gears is merely a problem in projection.
Fig. 120. Skeleton Diagram for Bevel Gears not at Right Angles.
With L as a center, draw a series of arcs (shown dotted) cutting the tooth which was drawn on the pitch circle, and the line LMK, at 2, 3, 4, etc.; from 2, 3, 4, etc., draw lines perpendicular to CD, cutting CD at 5, 6, 7, etc.; from these points draw lines to 0; along P0 lay off PE; through E draw a line perpendicular to P0, cutting 60, 60, etc.; and from the points of intersection draw other lines parallel to PB. With center O', taken at any convenient place on CO prolonged, and with radii equal in turn to a5, b6, etc., draw circles as shown. On the circle which is drawn with HP as a radius (marked "pitch circle"), space off the circular pitch; and on each of the circles in turn, lay off the teeth of the same width as they are on the corresponding circles drawn through 1,2,8, 4, etc. The rest of the construction can be understood by a careful study of the figure. The other gear is drawn in the same way.
The drawing of the teeth for bevel gears whose shafts intersect at another angle than a right angle, is accomplished by following out the same principles as noted in the case at hand. The skeleton outline of such a pair of gears is shown in Fig. 120, the angle between the axes being 60°. These gears are 2-pitch, 16 and 20 teeth, respectively, the same as in the previous case; and their construction affords an interesting comparison therewith.
To draw the teeth on a pair of bevel gears as described in Fig. 119, is a tedious process and requires considerable patience and drafting skill. It is really little more than an exercise in advanced projection drawing, but, as such, is valuable to the student. It must not be thought, however, that to detail a pair of bevel gears for manufacture, such a drawing is necessary. Usually, standard proportions of teeth are specified, and the detail of the gears is comparatively simple. An illustration of a pair of bevel gears of standard proportions of teeth, detailed ready for the workman's use, is shown in Fig. 39, Machine Drawing, Part I, and it is seldom necessary to show more.
The foregoing study of the outlines of gear teeth is given in brief and elementary form. The student cannot hope to gain a familiar comprehension of the action going on between the teeth of gears, without going more deeply into the subject than is possible in these pages. The action of gear teeth is one of the most complicated subjects to investigate and understand, as with each new condition of number and type of teeth, new points of action are developed.
A good practical article on gear teeth is "A Treatise on Gear Wheels" by George B. Grant; and the student is referred to this book for a further study of the subject.
There are many special points to be observed in designing the outlines of gear teeth, in order to insure the best operation of the gears. These points cannot be well explained without the actual undertaking of the design of the teeth. If the student wishes to familiarize himself with tooth action, he cannot do better than to choose a variety of cases, and lay out each one, studying the several points as they come up.
It should be remembered that the action of a small pinion, meshing into a large gear is considerably different from that of two large gears meshing into each other. With certain relative numbers of teeth of gear and pinion, as many as three pair of teeth may be in contact at all times; while, in certain other combinations, but two are in contact at all times, and in certain others only one. Changes in the tooth dimensions, diameters of describing circles, angles of obliquity, etc., alter all these conditions, so that there is an endless variety of combinations, each of which presents some new feature only to be understood by actual layout of the particular case.
In gear-tooth work, the student will often find it an advantage to make the layouts to double the actual size, and sometimes larger. A fine, hard pencil must be used, and extreme accuracy in determining the points must be adhered to. The layout of gear teeth is one of the severest tests of the draftsman's ability in line work.