help calculating velocity up a ramp

C.R.C,
I'm trying to estimate the speed of a LEGO robot traveling as fast as possible up a ramp of a given slope.... here is what I have:

Given data:
The robot has these characteristics: mass: 0.55 kilograms ramp angle a = 20 degrees = 0.349 radians (there will be 2 other ramp angles as well, but this will show the math) wheel radius = 0.04445 meters gear ratio: 25:1 (there will be 2 other gear ratios as well, but this will show the math) 1 drive motor
The motor has these specs: RPM: 340 = 35.6 Radians per second Noload current: 0.009 Amps Stall Torque: 0.055 Newton-Meters Stall Current: 0.34 Amps
Here is the procedure I have tried thus far, but I get bogged down in the final Step...
Step 1: Calculate the power available at the wheel
Applying the gear ratio: RPM: 35.6 / 25 = 1.424 Radians per second Stall Torque: 0.055 * 25 = 1.375 Newton-Meters
Applying the wheel radius: Force at wheel = stall Torque / wheel radius = 1.375 / 0.04445 = 30.933 Newtons
Is that right?
>From Inspiration to Implementation, page 217 I can calculate the theoretical maximum possible power:
Pm,max = 1/4 * (Stall Torque) * (radians per second) = 1/4 * (1.375 ) * (1.424) = 0.4895 Watts
So, Pm,max tells me how much power I have available at the wheel... 0.4895 Watts?
Step 2: Calculate how much force is needed on the ramp
Assuming constant velocity up the ramp, such that acceleration is ZERO, F = m * a = 0.55 * 0 = 0
So the net force must be zero, and the applied force "Framp" must be the sum of the frictional force "Ff" plus the force of the weight of the robot acting down "Fw"
Framp = Ff + Fw
Given that Ff = u * Fn where u = coefficient of friction, g = the normal gravitational force and (m*g) is just the weight of the robot, so that Ff = u * (m * g) * cos(ramp angle) = u * 0.55 * cos(20deg) = u * 0.55 * 0.9396 = u * 0.5167
Assuming the coefficient of friction u = 0.3 (from same book), then Ff = 0.3 * 0.5167 = 0.155
So we now have Framp = 0.155 + Fw
Given that Fw = m * g * sin(ramp angle) Fw = 0.55 * 9.8 * sin(20deg) = 0.55 * 9.8 * 0.3420 = 1.8434
So finally, the applied force is... Framp = 0.155 + 1.8434 = 1.9984
So, Framp tells me how much force is needed on a ramp of 20 degrees... 1.9984 (what are the units? Newtons or Newton-meters or Watts or kilograms?
Step 3: Calculate the speed up the ramp
Here is where I get uncertain. I need to apply these two formulas? 1) Power = Fapp * Velocity (Is Power for the available power.... 0.4895 Watts?)
2) angular Speed (a.k.a. rotational velocity) = Velocity / wheelRadius (Is rotational velocity just the RPMs at the wheel?)
I need to solve for Velocity, so rearranging them yields 1) Velocity = Power / Fapp 2) Velocity = (rotational velocity ) * wheelRadius
And here I am stuck, since I don't know: a) units for Framp b) is Power in equation 1 above the same as Pm,max? c) how to use equation 2 above...
Thanks for your help!
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nospam wrote:

When implementation is expensive, heavy design is a good idea. The whole purpose of legos is to make experimentation easy. I don't think they're built to that tight of tolerances... and don't get caught up in overdesign; it happens to the best of us. http://en.wikipedia.org/wiki/Leonhard_Euler#Berlin

Units propagate; trig functions are unitless... Given 1g=9.8 m/s^2, 1g*cos(60 deg) = 4.9 m/s^2.

Yep. If your wheel has a radius of 1, then it has a circumference of 6.28; for every full rotation, you move 6.28 units... angular velocity * 2 * pi * radius = linear velocity.

If you want to calculate the max speed, it is something like that.
Tweaks/comments on your equations:
<power> = <friction power> + <conversion to potential energy> Power is a change in mechanical energy per unit time. You are interested in a steady-state velocity (no acceleration), hence we can ignore the power needed to gain kinetic energy (from a standstill). Friction power is lost to heat. Potential energy is gained as you go up the slope.
<friction power> ~= <friction coefficient> * <velocity> note that the friction coefficient is *lost* energy; the wheel/ground interface may have a very high friction coefficient, but that doesn't matter since the wheel is rolling, not sliding. Major sources of friction in wheeled vehicles are - geartrain friction - axle bearing friction - wheels squishing - (at high speeds) air friction
<conversion to potential energy> ~= <mass> * <gravity> * <slope> * <velocity> Potential energy is ~ m*g*height; <slope>*<velocity> gives a change in height/unit time. e.g. a 60 degree slope results in m*g*v/2 (cos(60)=1/2).
When you substitute <power>=<max power>, solving should result in <velocity>=<max velocity>.
- Daniel
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OK,
I built a spreadsheet but the numbers just don't seem right. Is there a way I can post the spreadsheet or email to someone to take a look?
Thanks! Jeff
D Herring wrote:

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