Pointed Teeth

The previous sections have shown how gear teeth generated by a specific basic rack can have different tooth thicknesses, and that the outside diameters of such gears are altered from standard as a function of the change in tooth thickness. In practice, these tooth pro-files are achieved by “feeding in” or “holding out” the gear-generating tools. It is custom-ary to feed a specific cutter to a definite depth into the gear blank that has a greater than standard outside diameter. The cutter, when working at its full cutting depth, will still be held out from the standard N/P_d pitch circle.

Teeth generated thicker than standard will have tips of a width less than standard since the cutter must be “held out.” For any given number of teeth, the tooth thickness can be increased such that the tip will become pointed at the outside diameter circle. In Figure 3.7, four gears all having the same number of teeth, diametral pitch, and pressure

angle are shown. In Figure 3.7a, the cutter shown by a rack has been fed to the standard depth. The outside diameter of this gear is standard, (N + 2)P_d. Note the tooth thickness at the tip. In Figure 3.7b, the outside diameter was somewhat enlarged and the cutter fed in the standard whole-depth distance starting from the enlarged outside diameter. This results in the thicker tooth (which would operate correctly with a standard gear on an enlarged center distance) and a somewhat thinner tip. In Figure 3.7c, the outside diameter has been established at a maximum value at which it is still possible to achieve a standard whole-depth tooth; the tooth thus generated is pointed. Figure 3.7d shows what happens if the maximum outside diameter and tooth thickness are exceeded. The resulting tooth does not have the correct whole depth because the involute curves cross over below the expected outside diameter. This tooth is similar to the one shown on the pinion member in Figure 3.6g.

The amount that the outside diameter of a gear is to be modified is usually a function of the tooth thickness desired. Equation 3.18 below gives the relationship usually employed.

The maximum amount that the tooth thickness of a gear can be increased over stan-dard to just achieve a pointed tooth can be calculated from the simultaneous* solution of Equations 3.16 and 3.17.

*Equations 3.16 and 3.17 are solved by assuming a series of values for ϕ″. Curves are plotted for ∆Do^max versus ϕ″. The point where the curves cross indicate a simultaneous solution for the two equations. For instance, a 20-tooth pinion of 1 pitch having a 20° pressure angle and cut with a standard cutter has a crossing point for ϕ″

of 39.75° and ∆Do^max of 2.44 in. It should be kept in mind that these equations solve problems for one pitch only (divided by pitch to adjust answer for other pitches) and make no allowance for backlash. In the metric system the solution is in millimeters and for one module. (For other modules, multiply the answer by the module.)

Standard pitch line of cutter

Standard pitch

line of cutter Standard pitch

line of cutter

circle of gear Standard pitch

circle of gear

Standard pitch

circle of gear Tooth thickness

Outside diameter circle

Cutting long-addendum teeth. (a) Standard tooth; (b) long-addendum tooth; (c) long-addendum (pointed) tooth;

(d) long-addendum (over-enlarged) tooth.

∆Domax N cos

= cosφ − −

φ 1 2 (3.16)

∆D N

o

inv inv

max

=

(

)

φ 2 (3.17)

The equivalent amount that the tooth thickness must be increased above the standard to achieve this increase in outside diameter is given by Equation 3.18.

ΔT = d (inv ϕ″ − inv ϕ) −T (3.18)

where D_o^max is the maximum outside diameter at which a tooth having full working depth will come to a point, as follows

D N

P D

omax= + o

2+∆ (3.19)

where

T—tooth thickness at standard pitch diameter ϕ—standard pressure angle of hob or rack ϕ″—pressure angle at tip of tooth

3.1.2.6  Undercut

An undercut tooth is one in which a portion of the profile in the active zone has been removed by secondary cutting action. Under certain conditions, the path swept out by the tip of a generating-type cutter will intersect the involute active profile at a diameter greater than the limit diameter. Such a tooth is said to be undercut since no contact with the mat-ing gear can take place between the undercut and the limit circles.

The amount of undercut will depend on the type of tool used to cut the gears; in gen-eral, a hob will undercut a gear to a greater degree than will a circular, shaper-type cutter.

Form-type cutters do not normally produce undercut. In calculating the undercut diam-eter, it is customary to use an equation based on the type of tooling that will produce the greatest undercut, usually a hob. This is done to give the shop the greatest freedom of choice in selecting tools. If the most adverse choice will produce a satisfactory tooth profile, then all other types of cutter that might be chosen by the shop would prove to be satisfactory, except in those cases where undercut is intended to provide clearance for the tips of the mating teeth.

In general, spur gears of 20° pressure angle having 18 or more teeth and made to stan-dard- or long-addendum tooth proportions will not be undercut. In each of the “standard systems” for different types of gear, the minimum number of teeth recommended for each pressure angle is usually based on undercut.

Helical gears can usually be made with fewer teeth than can spur gears without getting into problems of undercut.

Figure 3.8 shows three gears having the same number of teeth, each produced by gen-erating-type cutters. Figure 3.8a is the tooth form of a gear having standard addendum.

Figure 3.8b shows the same shape of the tooth that results when the tip of the rack-type cutter is operating at a depth exactly passing through the point of intersection of the line of action and the base circle. Figure 3.8c shows the shape of a tooth with results when the cutter works at a depth somewhat below the intersection of the line of action and the base circle.

Undercutting is a product of generating-type cutters. Gears cut by forming-type cutters do not usually have undercut. In certain cases, undercut may ensure tip clearance in mat-ing gears. Racks operatmat-ing with gears havmat-ing a few teeth may show tip interference unless the pinions were generated with special shaper-type cutters or hobs.