Question $3[8$ Marks$]$

A shaft is designed to transfer $25$ kW of power from a pulley to a compressor. The shaft operates at a speed of $1750$ rpm and is made of AISI $1030$ steel $($annealed$,$ ground surface finish$)$ with the following properties: ultimate tensile strength, $\backslash ($ S$\_\{$u t$\}=430\backslash$mathrm$\{$MPa$\}\backslash )$ and yield strength, $\backslash ($ S$\_\{$y$\}=317\backslash$mathrm$\{$MPa$\}\backslash ).$ A pulley with a diameter of $0\mathrm{.}45$ m has been selected, along with specific axial dimensions for the shaft. Due to the possibility of moderate shock and slight overload, a design safety factor of $2\mathrm{.}2$ is recommended. The pulley is to be mounted using a standard key. The shaft diameters are to be designed according to the DE$-$ Goodman criteria. The requirement is to design the shaft diameters $(\backslash (\backslash$boldsymbol$\{$d$\}\backslash )$ and $\backslash ($ D $\backslash )),$ ensuring that the larger diameter $"\backslash ($ D $\backslash )"$ is $12$ mm greater than diameter $"\backslash ($ d $\backslash )".$ Assume that the end bearings do not produce reaction moments and that the reaction forces act through the center of the bearings. All dimensions are in mm$.$

Note $1$: In your design and analysis, use fatigue stress concentration factors for normal and shear stresses of $2\mathrm{.}14$ and $3\mathrm{.}0,$ respectively, at the critical$-$location. These account for the shoulder and key slot.

Note $2$: Perform one complete design iteration. If the design is found unsatisfactory after the first iteration, identify necessary changes to achieve adequate strength without performing additional calculations.

$\backslash [$

k$+=2\mathrm{.}14$

$\backslash ]$