Design the initial layout of a double$-$reduction gearbox for a e$-$bike. Input for the gearbox is a battery powered motor producing $350$ W at $2000$rpm$.$ The gearbox will connect to the bicycle's crank, with an output of $80$ rpm $.$ A photograph of a typical mid$-$motor mount for a bicycle is shown in Figure $1.$

Your layout should include:

$1.$ Specification of all gears, including gear types, materials, modules, number of teeth and relevant dimensions.

$2.$ Nominal materials and locations for all shafts, including the input shaft of the electric motor, the output shaft $-$ i$.$e$.,$ the bicycle$$s crankshaft that is attached to the pedals, and any shafts carrying intermediate gears.

$3.$ Location and specifications for required bearings.

Your design of the gears, shafts and bearings should be justified by the following:

$1.$ Analysis of torques and forces acting upon gears, shafts and bearings resulting from input torque of the electric motor and pedaling forces from the rider.

$2.$ Analysis of gear tooth stresses based upon transmitted loads.

$3.$ Computation of required load ratings for all bearings.

Additional considerations and desired characteristics:

$-$ The width of the final assembly must not exceed $125$mm$,$ so as not to interfere with the pedals. Otherwise, the assembly should be as compact as possible.

$-$ The motor body is $90$mm in diameter, $60$mm wide, and already includes acceptable bearings. The diameter and length of the shaft extending from the motor body can be customized according to your design.

$-$ Weight is important $$ lower is better $$ but should not be so low as to compromise mechanical integrity.

$-$ Your market research team has identified that noise is not an important design consideration. They believe customers will tolerate noise if price can be reduced.

$-$ You do NOT need to design a housing for the gear reducer or worry about how it will be attached to the bicycle frame at this time. You may assume that the housing will be strong enough to support loads from bearings and other sources as required.