15% is pretty steep, I think? If that means it's 15% of 90 degrees, it's ... 13.5 degrees!
My driveway is 16 degrees, which would be, what 18% grade? It looks fierce, and only on a good day can you ride even a mountainbike up that. (Some people seem to have a lot of good days). I have ridden a road bike up it too, but I dont think it's kind on the equipment.
There are calculations you can do, to figure out what sort of torque and forces you're dealing with on gradients like that.
If you don't mind, I'll stick with my 16 degree grade? Let's say local gravity is roughly 9.81 N/kg (or equivalent to m/s/s if you like). Let's say you weigh a decent sort of 60kg, plus your bike and whatnot is another 20, so there's 80kg. For the sake of these calculations, you're not pedalling...
The force of gravity on this setup is therefore 9.81* 80 = 785 N
On an 18% grade most of that force is directed into the driveway surface, a smaller amount is directed down the driveway. The down-driveway component is 785N * sin(18degrees) = 216N
If you want to accelerate up the hill, that adds to this force, but let's say you're happy with a constant speed, and we can just run with this force of 108N pushing you back down the hill. This is what the motor has to overcome to keep constant speed.
If you held your hand against the top of the tyre, hard enough to stop the bike going backwards, there would be a force of 216N applied to your hand. If your hand was instead a friction drive, the force would be applied to ... the friction drive. Let's say the contact area between friction drive and tyre is 1 square centimetre. I have no idea if that's realistic? If it is, then your force per area is 216N per .0001 square metres, or 2 160 000 N/m^2, which is 313 psi or 22 bar. If you don't provide that much pressure, the roller will slip.
How much force is that actually? Well the coefficient of static friction tells you how much force you need to press the two surfaces together with, to sustain the tangential force of 216 N. The coefficient between concrete and rubber is 1.0, which tells you about the tyre on teh ground, but the coefficient between sandpaper and rubber might be different. Ballpark though, you'll need to press the roller against the tyre with about 216 N, or about 20kg of load.
If that's not true, then this next bit might still be.
You;ve got your 216N force, and at the perimeter of the wheel, where it's touching the friction drive, that acts as a torque on the friction drive. Where two gears touch (in this case the wheel and the friction drive), the tangential force they each experience is the same, but the torque on each is different, according to their radius. So the torque on the wheel (let's say 26 inch? 66cm diameter) is 216 * 0.3 = 64 Nm. This torque transferred to the wee motor is 216 * .025 = 5.4 Nm. Great! Less torque on the motor, which makes sense cause it's going at higher speed. But can the motor deliver this much torque? Hmm, how to know.
You migth be lucky and strike a manufacturer that publishes torque specs, but few seem to. They do claim Kv values though, which can be fiddled to say somethign about torque. Kv is in rpm / volt, but if you express it in (rad/s)/volt, then it's inverse is the torque constant, Kt. A motor I've been looking at has a Kv of 270, in (rad/s)/volt this is 270*2*pi = 28.27. The inverse of that is 0.03537, which I'd guess is in Nm per amp, as I=T/Kt.
So can the motor deliver the 1Nm we were asking of it? Well, the current required for that torque is:
I = 5.4 / 0.03537
= 153 amps.
No worries! Let's say you're running 5s, for a voltage of ~20v, so your power will be: 153*20 = 3053 w.
Awesome. How fast will 3kw get a person up a hill?
Gravitional potential energy is mgh, so change in gravitational energy is mg delta h. Our change in energy is 3053 w, our mass and gravity stay the same so the
delta h = 3053/(mg)
= 3053/(80*9.81)
=3.9 m/s vertically!
Right on
And the vertical component is like a quarter of our total velocity, so we'll be going over ground at four times this, or 16 m/s, which is 50km/h.
Rock on!
Eric
