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I have been looking for a formula for two days now to calculate the diameter and how many teeth my driven gear needs.

The driver gear is 14 teeth and RPM will be 7104 (Motor KV x Batt Voltage). This driver gear will have a timing belt attached to the driven gear. How do I work out diameter and teeth needed for driven gear?

  • Are you trying to work out what size rear sprocket you need ? 7104rpm sounds insanely high for an e-bike – Dan K Jul 15 '20 at 11:27
  • Hi, yeah I'm trying to work out the driven pulley which will be attached at the rear wheel. I expect this pulley to have a very large diameter. I'm using 72 x 21700 cells in 12s6p which is 44.4 V at 3.7V each with a 160kV motor – Saxasianguy Jul 15 '20 at 11:34
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    You need to know what size the wheel is, what speed range you want and hence the wheel rotational speed. The speed ratio of two gears, toothed pulleys or sprockets is simply the ratio of number of teeth – Argenti Apparatus Jul 15 '20 at 12:22
  • Once you have all the parameters it's simple mathematics. – Daniel R Hicks Jul 15 '20 at 12:30
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In most jurisdictions, e-bikes must not provide assistance above a certain speed. One way to achieve this is by using a freewheel mechanism and configuring the controller to never exceed the motor's max RPM.

In the EU the max assisted speed is 25 km/h, which is about 7 m/s.

A typical hybrid bike might have a 700c rim with a 1.5" tyre, giving an overall diameter of about 700mm, so a circumference of 2.2m.

This means your target wheel speed is 3.18 Hz (revs per second).

7104 rpm = 118.4 Hz

Your ideal ratio would be 118.3/3.18 = 37.2327/1

There is no way you're using a 555 tooth wheel gear. (37 x 15 = 555)

At this point you have two options and may need to use both:

  1. Use a planetary gearset to implement a 2-stage reduction, a 9:1 gearbox would allow you to hit 37.2 overall with a 62 tooth gear at the wheel (9 x 62/15 = 37.2). If you're using a standard size RC or powertool motor (e.g 700 series), you can probably buy off-the-shelf gearboxes that just bolt on.
  2. Use a different motor with a lower Kv.

The calculations above assume you want the lowest possible gearing for maximum assistance at low speeds. This is most helpful in an e-bike. I haven't made any consideration of the motor's torque curve or efficiency curve.

Most commercially sold e-bikes use a mid-drive arrangement: the motor is connected to the crank, so the motor gets the same gear reduction as the rider. This is beneficial because the motor would only need to be efficient across a narrower range of speeds, e.g 50-100rpm vs 5-25km/h. With a brushless motor your controller can apply field weakening to continue delivering max power beyond the motor's base speed.

  • Thank you for this! You've explained everything really well. 7104 RPM is my max RPM, don't think the bike will be able to sustain that. Maybe I'll work backwards and start off with a max speed and work out pulley diameter from there This is for a project (I'm not building it in real life) and the guideline says I can't use gears so I'm using pulley and timing belt. – Saxasianguy Jul 15 '20 at 22:22
  • Do you mean kW for the motor? kV is kilovolts. – Michael Harvey Jul 16 '20 at 16:42
  • I got the casing wrong, it's Kv (capital K, subscript v) which is the motor velocity constant, revs per volt(peak) for an unloaded brushless motor. – Emyr Jul 20 '20 at 16:11

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