Pedal Force to Rear Wheel

The following diagrams the transformation of pedal force to the rear wheel.

There are four forces and four radii. The radii are the crank shaft, the front chainwheel, the rear cassette, and the rear wheel. Again note that the front wheel plays no part. The four forces are the pedal force, the force created on the chain, the chain force on the rear cassette, and the force generated by the rear wheel on the road.

From the Pedal Force to Chain Force

Let’s first focus on the force migration to the chain via the chainwheel. Notice we have a combination of forces and torques to contend with.

The pedal force is a true force, but it is applied a crankshaft distance away from the hub. It therefore acts as a turning force on the hub in a similar fashion to a wrench.

T₁ = Pedal Force * Crank Radius = F₁ * R₁

We next want to compute the force on the chain and here is the only real subtle point. Torques are derived from forces so we cannot simply assume a torque applied to the hub gets converted to a force at the chain. We need to ask the question as to what force at the chain would produce the pedal torque on the hub.

T₁ = T₂

T₂ = Chain Force *Front Crank Ring Radius = F₂ * R₂

F₁ * R₁ = F₂ * R₂

 F₂ = Chain Force = (R1/R2) * Pedal Force

What does this tell us so far? The Crank Length is larger than the Front Chainwheel radius.  So we see the Chain Force is larger than the pedal force by a factor of R1/R2. We therefore have the first piece of the second part of our discussion.

Chain Force to RearWheel Force

The Chain force is the same at the front chainwheel and at the rear cassette. So we have,

F₂ = F₃

Now we repeat our first calculation. We have a chainwheel pulling on the rear hub producing a Torque. We then look to determine what force on the Rear Wheel would produce an equivalent Torque.

T₃ = Chain Force *Rear Crank Radius = F₃ * R₃

T₄ = Rear Wheel Force *Tire Radius = F₄ * R₄

T₃ = T₄

Setting the two torques equal to each other we get the following:

F₃ * R₃ = F₄ * R₄

F₂ = F₃

(R_1//R₂) * Pedal Force = Rear Tire Force * (R_4//R₃)

Putting it Together: Pedal to RearWheel Force

Here is the key result. If you did not follow the Force this and Torque that discussions, this is what you want to understand. RearTire Force is related to Pedal Force by the current bicycle architecture as defined through two fixed and two gear radii. This result can be written equivalently in two ways:

           Rear Tire Force = Pedal Force * (R₁ R_3)//(R₂ R₄)

            Rear Tire Force = φ * Pedal Force

                        Where φ = (R₁ R_3)//(R₂ R₄) = (R₁/R4) * (R_3//R₂₎

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