Machinery's Handbook, 31st Edition
2236 Spur Gearing which it is geometrically impossible to exceed; this maximum constant, r c (max), is found by the formula: (12) where b is the similar constant in the specified formula for the gear dedendum. The hob tip radius of any standard hob to finish cut any standard gear may vary from zero up to this limiting value. Undercut Limit for Hobbed Involute Gears.— It is well to avoid designing and specifying gears that will have a hobbed trochoidal fillet that undercuts the involute gear tooth pro- file. This should be avoided because it may cause the involute profile to be cut away up to a point above the required contact diameter with the mating gear so that involute action is lost and the contact ratio reduced to a level that may be too low for proper conjugate action. An undercut fillet will also weaken the beam strength and thus raise the fillet tensile stress of the gear tooth. To assure that the hobbed gear tooth will not have an undercut fillet, the following formula must be satisfied: (13) where b is the dedendum constant; r c is the hob or rack tip radius constant; n is the number of teeth in the gear; and φ is the gear and hob pressure angle. If the gear is not standard or the hob does not roll at the gear pitch diameter, this formula can not be applied and the determination of the expected existence of undercut becomes a considerably more complicated procedure. Highest Point of Single Tooth Contact.— This diameter is used to place the maximum op- erating load for the determination of the gear tooth fillet stress. The two-step formulas for determining this diameter, d H , of the pinion using the same nomenclature as in the pre vious sections with d and D as unnamed angles used only in the calculations are: ( max r c ) . 0785398 cos φ − b sin φ = 1 sin − φ . n 05 sin b r r c − + φ sin c # φ
. tan tan d b n 628318 = + cos cos d P d n H φ =
(14) (15)
Similarly for the gear:
. tan tan D B N 628318 = +
(16)
(17) True Involute Form Diameter.— The point on the gear tooth at which the fillet and the involute profile are tangent to each other should be determined to assure that it lies at a smaller diameter than the required contact diameter with the mating gear. If the TIF diameter is larger than the contact diameter, then fillet interference will occur with severe damage to the gear tooth profile and rough action of the gear set. This two-step calculation is made by using the following two formulas with e and E as unnamed angles used only in the calculations: cos cos D P D N H φ =
b r
r
n 4
− +
tan tan e
2 2 c φ
c
(18) (19)
c
m
φ = −
sin
cos
φ
cos cos d P e n TIF φ =
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