A few months back, Deepak Malani asked me about the torque and power requirements of a bicycle so as to get an idea of the required ratings of an electric motor that can be retrofitted in the bicycle. So I went about doing some back of the envelope calculations taking into account rolling friction, aerodynamic drag and inclines. Here's a simple MATLAB/octave script I wrote to carry out the calculations.

The following assumptions were made:

The following assumptions were made:

- The transmission (pedal-sprocket-chain assembly) is 100% efficient. Note that practical efficiencies are in the range of 96-98%
- The rider plus bike system is moving at a constant velocity
- There is no headwind, tailwind or crosswind

Plugging in some typical values for the parameters, a 65 kg rider on a 15 kg bike moving at 15 kmph on a level road (no incline), does mechanical work at a rate of 60 Watts. He/she would need to put in a torque of about 10 Nm into the pedals (assuming a gear ratio of 2).

You can use the script for plugging in various values of the parameters and compute the torque and power requirements under those conditions. Note that at even a couple of degrees of upward incline, the torque required to overcome gravity dominates over that required to overcome rolling friction and aerodynamic drag.

The computations in the script are for a constant speed. However, the rating of the motor needs to take into account the worst case scenario which would typically be an acceleration or even a constant speed climb up a specified incline.

A few points to note:

You can use the script for plugging in various values of the parameters and compute the torque and power requirements under those conditions. Note that at even a couple of degrees of upward incline, the torque required to overcome gravity dominates over that required to overcome rolling friction and aerodynamic drag.

The computations in the script are for a constant speed. However, the rating of the motor needs to take into account the worst case scenario which would typically be an acceleration or even a constant speed climb up a specified incline.

A few points to note:

- Coefficient of rolling friction (Crr) could vary between 0.0021 to 0.017 [1]
- Aerodynamic drag coefficient (Cdr) could vary between 0.65 to 1.1 [1]
- Gear ratio (GR) could be anywhere between 1.75 to 4 [2]
- The gear ratio (GR) and wheel radius (Rw) effect only the torque requirement but not the power requirement
- For electric bikes with hub motors, GR=1

**References:**

[1] Stetler, Energy Efficiency of Bicycle Transportation

[2] V. Ryan, The Basics - Gear Ratio

[3] Ashburn, The Physics of Moving a Bike

[4] Cycling Aerodynamics & CdA - A Primer

Hi Lalit,

ReplyDeleteAppreciate the simplicity with which you have described your calculations.