IBL $($Chapter $14-$covered using problem based learning method$)$ME $302-$ Machine Design IIGear Design Chapters $13$ and $14,$ from textbookProblem StatementThe Figure $1$ below shows an electric motor driving a machine by means of three straight tooth spurgears required to have $16,32,$ and $24$ teeth with a diametral pitch of $8.$ An idler gear is used tofacilitate the smooth transmission of motion between gears and ensure that the output shaft rotatesin the desired direction without changing the speed ratio. The idler shaft is supported by bearings Aand B as shown. The input shaft experiences a maximum $($operational$)$ torque of $100$ lb$.$in andangular speed of $1700$ rpm$.$ The gearset should have a $10-$year lifespan with $1-$shift operation.During operations, the gear temperature could reach maximum of $350$F$.$ Utilize AGMA standardfull$-$depth teeth to design a spur gearset. You are required to select suitable material and full$-$depthteeth parameters to ensure a minimum safety factor of $2$ and $2$ against bending and pitting failure,respectively.Figure $1$ illusterates the setup, where the electric motor is used to drive the machine through theidler gear.MotorDriverFigure $1$: Electric motor, idler gear, bearings, and machine shaftsThe following assumptions and limitations must be applied during the design process$\mathrm{.}1.$ The gear set is made of spur gears of commercial quality level $($the designer should take intoconsideration operating speed$)\mathrm{.}2.$ The gear set output and input nature of load should be carefully considered based onoperation$\mathrm{.}3.4.6.$ Assume $20$ pressure angle and diametral pitch of $8.$Both pinion and gear are made from the same material and hardness$\mathrm{.}5.$ Interference must be avoided.The minimum safety factor against bending and pitting should be $2$ and $2,$ respectively$\mathrm{.}7.$ The minimum reliability is $0\mathrm{.}99,$ and the gearset should have a $10-$year lifespan with $1-$shiftoperation$\mathrm{.}8.$ During operations, the gear temperature could reach maximum of $350$F$.$Steps to solution:

a$.$ List the design constraints provided.

b$.$ List all assumptions or relevant information that must be identified to start the solution.

c$.$ Locate and indicate the input and output shaft gears in Figure $1.$

d$.$ From Figure $1,$ which one of the gears is the idler? Indicate on the sketch in c$.$

e$.$ What are the benefits of using an idler in gear design? What are the limitations$?$

f$.$ Draw the free body diagram showing forces acting on the pinion, idler and gear. $\{$Hint: Refer to Shigley Ex$.13-7$ for typical sketches$\}.$

g$.$ Estimate the pinion, idler and gear diameters.

h$.$ Estimate the required gear unit enclosure spacing by calculating center distance, volume, and deciding accordingly.

i$.$ Determine the no$.$ of teeth on the pinion, idler and gear to achieve the required gear ratio, with no interferance. Double check if the diameters obtained in $($ g $)$ need to be modified to meet the design requirements.

j$.$ Calculate the angular speed and pitch$-$line velocities of the idler and gear using the gear ratio. Are the angular speeds and pitch$-$line velocities same or different for the $3$ gears? Explain your anwsers.

k$.$ Calculate the maximum tangential loads and max. horse power to be transmitted from the pinion$-$to$-$idler and idler$-$to$-$gear. Are there any power losses in the system?

I. Develop an excel sheet containing all design relations and parameters needed for the calculation of bending and pitting safety factor.

m $.$ Determine all the gear teeth design parameters needed to calculate the bending and contact stresses.

n$.$ Determine the maximum gear teeth bending and contact stresses.

o$.$ Select a suitable material for the gear and pinion ensuring that the minimum safety factor against pitting and bending is satisfied. Also, in case of anticipated failure scenario, identify the source of the threat.

p$.$ According to your findings in $o,$ what would you recommend to the designers regarding the potential threat of failure in the gear? Provide details.