Machinery's Handbook, 31st Edition
Spur Gearing
2235 (5) (6)
N n a φ
tan tan B
tan tan − ^
h
= −
φ
cos cos D P B N c φ =
Contact Ratio.— The contact ratio of a pair of mating spur gears must be well over 1.0 to assure a smooth transfer of load from one pair of teeth to the next pair as the two gears rotate under load. Because of a reduction in contact ratio due to such factors as tooth deflection, tooth spacing errors, tooth tip breakage, and outside diameter and center distance tolerances, the contact ratio of gears for power transmission as a general rule should not be less than about 1.4. A contact ratio of as low as 1.15 may be used in extreme cases, provided the tolerance effects mentioned above are accounted for in the calcula- tion. The formula for determining the contact ratio, m f , using the nomenclature in the previous section is: (7a) or (7b) or (7c) where R 0 = outside radius of first gear; R B = base radius of first gear; r 0 = outside radius of second gear; r B = base radius of second gear; C = center distance; I = pressure angle; and, p = circular pitch. Both formulas, Equations (7a) and (7b), should give the same answer. It is good practice to use both formulas as a check on the previous calculations. cos P R R r r C Lowest Point of Single Tooth Contact.— This diameter on the pinion (sometimes referred to as LPSTC is used to find the maximum contact compressive stress (sometimes called the Hertz Stress) of a pair of mating spur gears. The two-step formulas for determining this pinion diameter, d L , using the same nomenclature as in the previous sections with c and C as unnamed angles used only in the calculations are: . 628318 tan tan − m N f = A B ^ h . 628318 tan tan a − m N f = b ^ h sin m f B B 0 2 2 0 2 2 θ θ = − + − −
. n 628318 = − cos cos d P c n L φ =
tan tan c a
(8) (9)
In some cases it is necessary to have a plot of the compressive stress over the whole cycle of contact; in this case the LPSTC for the gear is required also. The similar two-step formulas for this gear diameter are:
. tan tan C A N 628318 = − cos cos D P C N L φ =
(10) (11)
Maximum Hob Tip Radius.— The standard gear tooth proportions given by the formulas in Table 2 on page 2210 provide a specified size for the rack fillet radius in the general form of (a constant) × (pitch). For any given standard this constant may vary up to a maximum
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