First things first, some measures of energy, power and rotation.

Disclaimer: I'm european so I do 0,5 instead of 0.5, and SI units.

Energy stored in kinetic energy: 0,5*mass [kg]*velocity^2 [m/s] (^2 means squared).

Rotational power: Nm*Omega (angular speed). [Rad/s] (often confused with angular frequency).

RPM=Omega[Rad/s]*60/2*Pi.

Omega[Rad/s] = RPM*2*Pi/60

Then, imaginary motor data:

Motor resistance: 100mOhm.

Torque per amp: 1,1 Nm/A

Max Rpm: 500

Max Rad/s = 52,3

Motor losses:

Parasitic losses approximated to 2W per rad/s. (Bearing and Hysteresis losses).

Copper losses: I^2*R (R i is motor resistance).

Parasitic losses: Taking these into account is in disfavor of the gearbox connected motor. They won't make a huge difference, but taking all things into concideration, they need to be taken into account.

Gearbox:

Three speed.

1:1 70km/h at 500RPM

1:1,50 46,6km/h at 500RPM

1:3 23,3km/h

This was 'designed' by taking the whole gear range and dividing them by three. There might be more efficient ways to do this, taking into account kinetic energy scales by the square.

Bikes:

With gearbox: 110kg

Without gearbox: 100kg

And so it begins, gogo: lets see who reaches 70km/h first.

Energy @ 70km/h gearbox bike: 20700J

Energy @ 70km/h non gearbox bike: 18800J

Challenge: accelerate as fast as possible to 70 km/h. 70km/h = 19,4 m/s. We run 50A to the motor, giving us a steady 250W losses and 55Nm from the motor. This is the result:

In red: gearbox.

X-Axis: Wheel RPM

Y-Axis: Power

For those extra interested, power calculations are done in Openoffice Calc - the file is added. Look through all the sheets. I've modified torque for the calculations, but modifying rad/s would net the same result.

Power is one thing, what is more interesting is acceleration. To do this we find the mean power value, based on this - we can take energy and divide by mean power - giving time. The non- gearbox solution is easy. Half of 2880W is 1440W. 18800J/1440W equals

**13 seconds.**Parasitic losses (mean): 50W*13s=650J. Active losses (mean): 250W Total losses: 3900J. Total efficiency:18800J/18800J+3900J = 82,8% Total energy: 22700J

The gearbox solution is harder, mean is found by taking the average of all the samples and dividing by the sample amount. When I do this in Calc, I get 1980W average. 20700J/1980W equals

**10,5 seconds**. Parasitic losses (mean): (1,6s*50W+3.16s*77W+5,74s*86W)= 828J. Active losses: 250W. Total losses: 3330J Total efficiency: 20700/20700+3330=86,1% Total energy: 24030J

Calculations for individual gears:

Energy @ 23,3km/h: 2302J Mean power: 1440W. Time in gear: 1,6s

Energy @ 46,6km/h: 9215J Mean power: 2190W. Time in gear: (9215J-2302J)/2190W = 3,16s

That was the boring half of the story - here is the fun part that is closer to reality aswell. How long does the bikes use to 46,6km/h? (making calcs easier )

Gearbox model:

**(1,6+3,16)**

__4,76s__Non-gearbox model: Mean power: 975W. Total energy: 8380J. Total energy/power =

**8,6s**Now look at that difference. The gearbox model clearly offers much better performance.

Observations regarding this model:

1. Gearbox bike is disfavoured when it comes to weight.

2. Gearbox bike gearbox efficiency is not calculated in.

3. Power graph gear changes is a bit sloppy modeled.

4. Total energy to get both bikes up to speed is lower in the gearless bike, but this is much due to nr 1.

5. Gearbox bike absolutely owns gearless bike in acceleration test.

6. The imaginary motor model is very real, but the whole system with inverter is kept out of the equation. If added, gearbox bike would benefit.

7. Both motors are run under peak efficiency point (@88%)(witch is about 16A phase for just under 18Nm).

8. A gearbox solution greatly improves efficiency.

Observations regarding gearboxes:

1. If you have too much torque off the line, a gearbox will be no good.

2. An induction motor is, due to non-fixed V/f characteristic, not similar to a permanent magnet motor(hereafter PMM). (Tesla motors anectdote).

3. "Electric torque" is less "peaky" than ICE torque.

4. Several speed gearboxes on ebikes are very rare (is there any purpose built?).

5. If you had "infinite gears", mean power would be doubled from a PMM.

6. A gearbox allows the motor to spin faster so that the motor does not need to have same phase amps (biggest source of loss during accel).

7. A gearbox solution greatly increases low range performance.

Please don't turn this post into another gearbox shitfest. I might have commited blunders and I'd ideally like to use more time on this post, but I don't have the time to(many calculations is missing - but I've tried adding them where there could be confusion).

About the overly arrogant title, yep - that's typically me (I've been wrong more than one time, it's nothing new - but self-confidence in the field is just a bit too high ): Just know I'm sincere when I say I believe they've earned the title

Edit: Fixed image (X-axis was off.)