The results required from Example Example Problems 9–1, 9–2, 9–3 are:
\begin{array}{lllc}N_{P}=20 & N_{G}=70 & P_{d}=6 & F=2.00 \text { in } \\n_{P}=1750 \mathrm{rpm} & n_{G}=500 \mathrm{rpm} & D_{P}=3.333 \text { in } & D_{G}=11.667 \text { in } \\s_{C P}=s_{C G}=100769 \mathrm{psi} & K_{R}=1.25 & N_{C P}=2.1 \times 10^{9} \text { cycles } & N_{c G}=6.0 \times 10^{9} \text { cycles }\end{array}
The equation relating the contact stress number with the adjusted allowable contact stress number is:
s_{c}<s_{a c}^{\prime}=s_{a t} \frac{Z_{N}}{S F \cdot K_{R}}
Use this equation to solve for the safety factor, SF:
S F=\frac{S_{a c}}{S_{c}} \cdot \frac{Z_{N}}{K_{R}}
From Figure 9-22, we find pitting resistance stress cycle factors:
\begin{aligned}&Z_{N P}=0.88 \\&Z_{N G}=0.91\end{aligned}
Part (a) Let’s consider the first proposed material, ductile iron ASTM A536 type 80-55-06. Using Table 9-9, the allowable contact stress number for this material is
s_{a c}=77000 \mathrm{psi}
Solving for the safety factor S F of the pinion and gear:
\begin{aligned}&S F_{p}=\frac{S_{a c}}{S_{c P}} \cdot \frac{Z_{N P}}{K_{R}}=\frac{77000 \mathrm{psi}}{100769 \mathrm{psi}} \cdot \frac{0.88}{1.25}=0.54 \\&S F_{G}=\frac{S_{a c}}{S_{c G}} \cdot \frac{Z_{N G}}{K_{R}}=\frac{77000 \mathrm{psi}}{100769 \mathrm{psi}} \cdot \frac{0.91}{1.25}=0.56\end{aligned}
Both safety factors are below the minimum value of 1.00, as recommended by AGMA and are therefore considered to be unsatisfactory for this application.

Both safety factors are below the minimum value of 1.00, as recommended by AGMA and are therefore
considered to be unsatisfactory for this application.
Part (b) considers a through hardened steel SAE 4340 OQT 1000. Using Appendix 3, the properties of the material are:
s_u = 171 000 psi, s_y = 158 000 psi, HB = 363, 17% elongation
Using Figure 9–19, along with a Brinell hardness of HB = 363 for though-hardened steel gears material, the allowable contact stress number is:

s_ac = 146 000 psi
The safety factors, SF, of the pinion and the gear are:

\begin{aligned}S F_{p} &=\frac{S_{a c}}{S_{C P}} \cdot \frac{Z_{N P}}{K_{R}}=\frac{146000 \mathrm{psi}}{100769 \mathrm{psi}} \cdot \frac{0.88}{1.25}=1.0 \\S F_{G} &=\frac{S_{a c}}{S_{C G}} \cdot \frac{Z_{N P}}{K_{R}}=\frac{146000 \mathrm{psi}}{100769 \mathrm{psi}} \cdot \frac{0.91}{1.25}=1.1\end{aligned}
The safety factors for the pinion and gear are both within the range of 1.0 to 1.5, as recommended by AGMA, with the pinion’s factor at the lowest limit.
Part (c) considers SAE 8620 steel, case hardened by carburizing to a Rc 60-64, with a minimum effective case depth of 0.025 in. Using Table 9-9, the allowable contact stress number is:
s_{a c}=180000 \mathrm{psi}
The safety factors S F for the pinion and the gear are:
\begin{aligned}S F &=\frac{S_{a c}}{S_{C P}} \cdot \frac{Z_{N P}}{K_{R}}=\frac{180000 \mathrm{psi}}{100769 \mathrm{psi}} \cdot \frac{0.88}{1.25}=1.26 \\S F &=\frac{S_{a C}}{S_{C P}} \cdot \frac{Z_{N P}}{K_{R}}=\frac{180000 \mathrm{psi}}{100769 \mathrm{psi}} \cdot \frac{0.91}{1.25}=1.30\end{aligned}
The safety factors for both the pinion and gear are well within the range of 1.0 to 1.5, as recommended by AGMA.