Prob $3.(35$ points$)$

A pair of spur gears with a pressure angle of $\backslash (25^\{\backslash$circ$\}\backslash )$ transmits $2\mathrm{.}5$ hp while the pinion rotates at $680$ rpm $.$ The diametral pitch is $10,$ The pinion has $24$ teeth, and the gear has $62$ teeth. The face width is $1\mathrm{.}25$ in $.$

Assume the overload factor is $\backslash ($ K$\_\{0\}=1\mathrm{.}75\backslash ).$ And the drive is a commercial enclosed gear unit. The quality number of the gear is A$11.$

$($A$)$ Calculate the bending stress number, $\backslash (\backslash$mathrm$\{$s$\}\_\{\backslash$mathrm$\{$t$\}\}\backslash )$ for the pinion and gear, and the contact stress number, $\backslash (\backslash$mathrm$\{$s$\}\_\{\backslash$mathrm$\{$c$\}\}\backslash ).$ Assume that the gear blank is solid $($thickness factor $\backslash (\backslash$mathrm$\{$K$\}\_\{$B$\}=1\mathrm{.}0\backslash )),$ and the gears are made from through$-$hardened, grade $1$ steel.

$($B$)$ Compute the required allowable bending stress number, $\backslash (\backslash$mathrm$\{$s$\}\_\{\backslash$mathrm$\{$at$\}\}\backslash )$ for gear and pinion, the required contact stress number, $\backslash (\backslash$mathrm$\{$s$\}\_\{\backslash$mathrm$\{$c$\}\}\backslash ).$ Design for a reliability of $0\mathrm{.}90,$ a design life of $2000$ hours, and a safety factor of $1\mathrm{.}25.$

$($C$)$ Determine required hardness for the pinion and the gear based on the contact stress.

Submit excel spreadsheet that you create for the problem as well.