Chapter 15, Problem 11P

Apply the relations of Prob. 15-7 to Ex. 15-1 and find the Brinell case hardness of the gears for equal allowable load W^t in bending and wear. Check your work by reworking Ex. 15-1 to see if you are correct. How would you go about the heat treatment of the gears?

EXAMPLE 15-1 A pair of identical straight-tooth miter gears listed in a catalog has a diametral pitch of 5 at the large end, 25 teeth, a 1.10-in face width, and a 20° normal pressure angle; the gears are grade 1 steel through-hardened with a core and case hardness of 180 Brinell. The gears are uncrowned and intended for general industrial use. They have a quality number of Q_v = 7. It is likely that the application intended will require outboard mounting of the gears. Use a safety factor of 1, a 10⁷ cycle life, and a 0.99 reliability.

1. (a)   For a speed of 600 rev/min find the power rating of this gearset based on AGMA bending strength.
2. (b)   For the same conditions as in part (a) find the power rating of this gearset based on AGMA wear strength.
3. (c)    For a reliability of 0.995, a gear life of 10⁹ revolutions, and a safety factor of S_F = S_H = 1.5, find the power rating for this gearset using AGMA strengths.

15-7 In straight-bevel gearing, there are some analogs to Eqs. (14-44) and (14-45) pp. 766 and 767, respectively. If we have a pinion core with a hardness of (H_B)₁₁ and we try equal power ratings, the transmitted load W^t can be made equal in all four cases. It is possible to find these relations:

	Core	Case
Pinion	(HB)11	(HB)12
Gear	(HB)21	(HB)22

1. (a)      For carburized case-hardened gear steel with core AGMA bending strength (s_at)_G and pinion core strength (s_at)_P, show that the relationship is

${\left(s_{at}\right)}_G=\left(s_{at}\right)P\frac{J_p}{J_G}{m}_G^{-0.0323}$

This allows (H_B)₂₁ to be related to (H_B)₁₁.

1. (b)      Show that the AGMA contact strength of the gear case (s_ac)_G can be related to the AGMA core bending strength of the pinion core (s_at)_P by

${\left(s_{ac}\right)}_G=\frac{C_p}{{(C_L)}_G{C}_H}\sqrt{\frac{S_H^2{(s_{at})}_P{(K_L)}_P{K}_x{J}_P{K}_T{C}_s{C}_{xc}}{N_{P}I{K}_s}}$

If factors of safety are applied to the transmitted load W_t, then S_H = $\sqrt{S_F}$ and $S_H^2/S_F$ is unity. The result allows (H_B)₂₂ to be related to (H_B)₁₁.

1. (c)       Show that the AGMA contact strength of the gear (s_ac)_G is related to the contact strength of the pinion (s_ac)_P by

${(s_{ac})}_P={(s_{ac})}_G{m}_G^{0.0602}{C}_H$