A group of EBE students are designing a double$-$reduction spur gear train set as shown in Figure $2.$ The shaft $a$ is driven by a motor attached by a flexible coupling attached to the overhang. The gears have $20$ pressure angles, with the numbers of teeth as $N1=16,N2=40,N3=32,N4=96.$ The module for all gears is $4$ mm $.$

The motor rotates at a speed of $1200$ rpm $,$ and provides a torque with its value assigned by using the last $4$ digits of your sID divided by $10,$ taking a unit of Nm $($see an example below$).$ Assuming a $100\%$ efficiency for the power transmission.

Torque parameter determination example: Suppose your sID is S$1234567,$ then the torque parameter for you is: $\frac{4567}{10}=456\mathrm{.}7Nm.$

The input shaft $a$ is to be machined using an AISI $1020$ cold$-$drawn steel. A general shaft layout is shown in Figure $2($b$).$ A design factor of $1\mathrm{.}5$ is required.

$($a

$($b$)$

Figure $2.($a$)$ Sketch of a double$-$reduction spur gear train set; $($b$)$ Layout of the input shaft a$.($Dimensions in mm $)$

$(1)$ Determine the overall gear ratio of the train, and the power to be transmitted

$(2)$ Determine the pitch circle diameter, circular pitch, addendum, and dedendum for gear N$1.$

$(3)$ Determine the center distances between shafts a$-$b$,$ and shafts b$-$c$.$

$(4)$ Determine the torque on each shaft.

$(5)$ Determine the tangential force and radial force acting on gear N $1$ and gear N $4$ respectively. Draw a free$-$body diagram of gear N$4$ and show all the forces that act upon it$.$

$(6)$ Perform a force analysis on shaft a to find the bearing reaction forces, and generate shear, torque, and bending moment diagrams.

$(7)$ Estimate the endurance limit for the shaft a$,$ taking into account the surface factor, size factor, and the reliability factor. You may assume a shaft diameter of around $50$ mm $,$ and a reliability of $99\%.$

$(8)$ Determine potential critical locations for stress design on shaft $a,$ and provide brief justifications.

$(9)$ Determine the fatigue stress concentration factor and the fatigue shear stress concentration factor at the gear keyway on shaft a$.$ Assuming a typical radius at the bottom of the endmilled keyseat of $\frac{r}{d}=0\mathrm{.}02,$ and a shaft diameter of up to $75$ mm $.$

$(10)$ Determine critical diameter of the shaft a at the gear keyway based on fatigue and static stresses. Check yielding safety with this diameter.