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I've been designing an electric skateboard which is driven by only one of the rear wheels (for the moment). As I was picking out motors for the board, I soon realized I would need to find a balance between torque needed to climb hills, and RPM to get a good top speed.

So I'm searching for the best KV (RPM/v) motor/gearing ratio combination.

For all calculations I've assumed the following:

  • Total mass (board + rider) = 85 kg
  • Voltage = 12 Cells * 3.2V (LiFePo4) = 38.4 V
  • Max Speed = 40 km/h = 11.11 m/s
  • Hill Grade = 15%
  • Coefficient of friction between rubber and asphalt: 0.65
  • Diameter of wheel = 80 mm

I tried assuming maximum kinetic energy as being equal to maximum gravitational potential energy at the top of the hill.

$$E_{k,max} = \frac{1}{2}mv_{max}^2 = E_g = mgh_{max}$$

Then solving for $h$.

I'd take that and use $y/x = 0.2$, solve for $x$, then use some trigonometry to setup a Newton's Second Law problem and solve for force needed to overcome friction and gravity (Equilibrium problem).

Then using $T = F \times r$, find torque needed at the wheels, assume some gearing ratio that keeps that and the $V_{max}$. Then solve for RPM and divide by total voltage for motor KV.

My mechanics are rather rusty and I could use some help.

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  • $\begingroup$ you can use your formula to figure out how much energy you are going to pull from the battery. For force&power look at answers below. $\endgroup$ – jumpjack Mar 6 at 14:13
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I think you're overcomplicating things. To push 85 kg up a 15% slope against gravity of 9.8 m/s2 requires a force of

$$ \sin (\arctan (0.15) ) = 0.1483 \approx 0.15 $$

$$F = 85 kg \cdot 9.8 \frac{m}{s^2} \cdot 0.15 = 125 N$$

With an 80 mm wheel, this requires a torque of

$$T = 125 N \cdot 0.04 m = 5 Nm$$

To do this at a forward speed of 11.11 m/s requires a fair amount of power:

$$P = Force \cdot Velocity = 125 N \cdot 11.11 \frac{m}{s} = 1400 W$$

which is about 36 A from your 38.4 V battery pack, and your wheels will be turning at

$$\frac{11.11 \frac{m}{s}}{\pi\cdot 0.08 \frac{m}{rev}} \cdot 60 \frac{sec}{min} = 2600 RPM$$

Does any of this help you move forward?

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  • $\begingroup$ Thanks a bunch, I think I got on one pattern of thinking and found it hard to go back! This is what I'm looking for. $\endgroup$ – Satchel Jan 4 '16 at 5:39
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    $\begingroup$ The calculation of force should be $F=m \, g\, \sin{\theta}$. In this case, $\sin(\tan^{-1}0.15) \approx 0.15$, so it works, but that's not the general case. $\endgroup$ – Carlton Jan 4 '16 at 13:18
  • $\begingroup$ @Carlton: Do you understand that we're just doing some ballpark estimation here? Small-angle approximations are perfectly acceptable. $\endgroup$ – Dave Tweed Jan 4 '16 at 13:46
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    $\begingroup$ @DaveTweed - While I would agree that small-angle approximations are acceptable, OP's "mechanics are rather rusty", and you didn't state/show that you were using any small-angle approximations. $\endgroup$ – Chuck Jan 4 '16 at 13:48
  • $\begingroup$ @Chuck thanks for the concern, I did pick up on that however I just assumed that he did that calculation and rounded off to 0.15 $\endgroup$ – Satchel Jan 4 '16 at 19:24
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The answer provided by Dave is correct but it covers only a very specific case of no-friction and constant speed. There are three forces acting on a vechicle which must be accelerated from 0 to v speed over a slope:

enter image description here

Cr is the rolling friction coefficient, around 0.01 for cars on asphalt, so probably something less for bikes. Don't know for skateboards, but they have forul large&flat wheels, so maybe it's more similar to car coefficient than bike coefficient.

Being it a skateboard over a slope, it has a very slow speed, so we can neglect air drag, but for a generic vehicle you must also take it into account:

$$ F_D = \frac 1 2 \rho C_D A v^2 $$

$$ \rho = air density = 1.22 kg/m3 $$ $$ C_D = \text{drag coefficient (around 0.8 for bike/human, 0.2-0.3 for cars)}$$ $$ A = \text{Section area or Frontal area, m2}$$ $$v = \text{speed, m/s}$$

Inertia force resists to acceleration.

I found a calculator: it's designed for robots, so it disregards both wheel and air drags:

https://www.robotshop.com/community/blog/show/drive-motor-sizing-tool

This is a much more complex&complete simulator, but I don't understand how to input custom data for motor:

https://vesc-project.com/calculators

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