Question 14–3: The pinion of Examples 14–1 and 14–2 is to be mated with a 5...

From Table A–5 we find the elastic constants to be E_{P}=30 Mpsi , \nu_{P}=0.292, E_{G}=14.5 Mpsi , \nu_{G}=0.211 We substitute these in Eq. (14–13) to get the elastic coefficient as 

C_{p}=\left[\frac{1}{\pi\left(\frac{1-\nu_{P}^{2}}{E_{P}}+\frac{1-\nu_{G}^{2}}{E_{G}}\right)}\right]^{1 / 2}     (14–13)

From Example 14–1, the pinion pitch diameter is d_{P}=2 \text { in }. The value for the gear is d_{G}=50 / 8=6.25 \text { in }. Then Eq. (14–12) is used to obtain the radii of curvature at the pitch points. Thus

r_{1}=\frac{d_{P} \sin \phi}{2} \quad r_{2}=\frac{d_{G} \sin \phi}{2}          (14–12)

The face width is given as F=1.5 \text { in. Use } K_{v}=1.52 from Example 14–1. Substituting all these values in Eq. (14–14) with \phi=20^{\circ} gives the contact stress as

\sigma_{C}=-C_{p}\left[\frac{K_{v} W^{t}}{F \cos \phi}\left(\frac{1}{r_{1}}+\frac{1}{r_{2}}\right)\right]^{1 / 2}         (14–14)

Table A–24 gives H_{B}=262 for ASTM No. 50 cast iron. Therefore S_{C}=0.32(262)= 83.8 kpsi. Contact stress is not linear with respect to the transmitted load [see Eq. (14–14)]. If the factor of safety is defined as the loss-of-function load divided by the imposed load, then the ratio of loads is the ratio of stresses squared. In other words,

One is free to define factor of safety as S_{C}/σ_{C}. Awkwardness comes when one compares the factor of safety in bending fatigue with the factor of safety in surface fatigue for a particular gear. Suppose the factor of safety of this gear in bending fatigue is 1.20 and the factor of safety in surface fatigue is 1.34 as above. The threat, since 1.34 is greater than 1.20, is in bending fatigue since both numbers are based on load ratios.If the factor of safety in surface fatigue is based on S_{C} / \sigma_{C}=\sqrt{1.34}=1.16, then 1.20 is greater than 1.16, but the threat is not from surface fatigue. The surface fatigue factor of safety can be defined either way. One way has the burden of requiring a squared number before numbers that instinctively seem comparable can be compared.