Designing and estimating.

Assuming the range of material strength, the first analysis was focused on the shear stress when a standard cassette gear attaches to pedaling and motoring chains. By the maximum shear stress under the real cases, compare this number to the yield strength of a commercial gear in order to ensure there will not be any deformation occurred. 

Stress = T r / (polar moment of inertia)

α ( t ) = 1 I τ ( t )

Δ ω = ∫  α ( t ) d t

v = R ω

Velocity = radius of wheel * time contacting * π * radius of cassette * yield strength / mass of cassette