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In some electric bikes, a motor is mounted to the hub of the front wheel. When turned on, this hub motor applies a torque, turning the wheel, reacting against the ground, with a resultant force acting against the inertia of bike / rider / air resistance etc.

How can I work out the force applied to the centre of the axle, given a motor acting with specific torque t and wheel or radius r, assuming the tyre does not overcome the friction force of the ground?

Lets say we fix the wheel, axle, and motor to a fork, that is anchored (so nothing can move). Lets assume the fork cannot flex.

Forgive my crude drawing, but this might look something like this:

enter image description here

The Force of the motor acts on the rim, then the reaction force from the road acts on the fork (I think). The net foce is to the right hand of the page (where the motor is at the centre of the wheel).

Assuming the motor has a Torque of 40Nm, what is the force on the fork centred at the axle?

(I'm trying to determine if my fork can support a front motor, so the next stage will be to look at whether this fork can support the force at that distance)

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Assuming the motor has a Torque of 40Nm, what is the force on the fork centred at the axle?

Since the system is fixed in place, there is no acceleration or rotation. Therefore any torque developed by the motor is completely countered by friction with the ground, and that force is transmitted to the fork. So this is just $\frac{40\rm{Nm}} {311\rm{mm}} = 129\rm{N}$. If the wheel could rotate and the frame could accelerate, that force would be reduced.

Presumably if instead of anchoring your fork you allowed it to move with minimal resistance, you could attach a scale to it and directly measure the force that the motor is creating.

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  • $\begingroup$ I gave your answer a plus, but you need to correct it by changing the denominator to 340mm+311mm=651mm, not 311mm. $\tau=F*r$, $F=\frac{\tau}{r}$ $\endgroup$
    – kamran
    Commented May 19, 2023 at 22:16
  • $\begingroup$ The torque rotates the wheel. Therefore the force it creates against the ground depends on the radius of the wheel. There will be a (different) torque on the fork's attachment to the frame, but that's not part of the calculation. $\endgroup$
    – BowlOfRed
    Commented May 19, 2023 at 22:38
  • $\begingroup$ Many thanks for your answer. So in the case of any actual bicycle, the peak force experienced at the axle (assuming a rigid fork) would be this max? But it would actually be lower because there'd be a movement force of the bike. $\endgroup$
    – Chuck
    Commented May 20, 2023 at 9:39
  • $\begingroup$ If that's the only force on the bike, yes. In reality a rolling bike will hit bumps and the wheel has brakes. Those forces are likely to be much greater than the acceleration forces from the motor. $\endgroup$
    – BowlOfRed
    Commented May 20, 2023 at 22:00

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