Rolling Resistance - proportional to speed?

Kuklish

Hi all you physics people.... we are building a high-speed robotic camera dolly here in LA for a film, and are trying to make sure we calculate our torque requirements correctly.

The maximum torque required to propel a wheeled vehicle up an incline at a given acceleration is fairly easy to understand. But how in the heck do you figure the rolling resistance of the wheels into that basic equation?

I do know the basic equations to calculate the rolling resistance force, based on a coefficient at 3 mph... but does this stay the same no matter what the speed?

I've read conflicting accounts that the rolling resistance force is linearly proportional to speed... but then have also heard and seen in numerous bicycle charts, that it is a relative constant???

It makes a HUGE difference if is linearly proportional, and we will need huge motors.

Really need to figure this out as we have a lot of wheels on the dolly that are guide wheels and drive wheels that do not support the load, but are spring loaded with lots of force.

We are using this formula for our basic torque calculation :

Where:
T= torque at the drivewheel
a=desired acceleration
g=acceleration of gravity
θ= incline of ramp
m=mass of vehicle
R=radius of drive wheel

Many thanks to you all!

rcgldr

It's possible that what you've read about rolling resistance is related to power. If rolling resistance remains about the same regardless of speed, then the required force or torque remains about the same, but the required power will increase linearly with speed.

What type of wheels are being used? Rubber tires will have more rolling resistance than metal wheels. To avoid issues at very slow speeds (you mentioned 3 mph), metal wheels may be a better choice, but you'll need a very smooth track and very smooth wheels.

Kuklish

The top wheels are polyurethane Rollerblade style wheels 52mm in diameter. There are 16 of them riding on an aluminum beam. Soft wheels like this are industry standard for us as the provide smooth, vibration free, and above all, quiet, dolly travel.

The drive wheels are 4.5" diameter also with polyurethane treads of around 78 durometer. They do not take the load of the dolly, but are mounted perpendicularly and are spring loaded into the web of the I-beam.

There are also side and bottom mounted guide wheels.

I found a source that listed the coefficient of rolling friction for this durometer of poly wheels on a steel surface at .030 (inches at 3mph.) Along with the equation:

Where:
F=the force required to overcome the rolling friction
f=the coefficient of rolling friction (units must match same units as r
w=load on the wheels
r=radius of the wheels with friction

I then saw another source that said rolling resistance is proportional to speed so I modified the equation to be:

Where:
T= torque at the drivewheel (just from the rr of course)
s=speed of vehicle in mph
R=radius of the drivewheel

I sincerely hope this last equation is incorrect, because it seems astronomically high, and like I said, I also saw other sources that show the rolling resistance force as constant.

rcgldr

I looked at that source. Those equations are for power:

equation 6.2:

P_drag = ρ A c_d v³ / 2

equation 6.3:

P_rolling = c_r m g v

Kuklish

I'm not too clear on what solving for Power is, or if I even need to, but its seems then this would be a good thing then... meaning my last equation was incorrect for figuring torque? I should leave it as a constant force?

So I would just use this:

Instead of this:

And simply add this torque to the torque number I already got from calculating the mass going up the incline?

rcgldr

From this wiki article:

Rolling_resistance_coefficient.htm

equation for force, C_rr is coefficient of rolling resistance, which has no units.

F = C_rr m g

equation for torque (R is radius of wheel)

T = F R = C_rr m g R

C_rr could be defined as a function of radius. Your first source includes a table that defines C_rf to represent the table values and an equation where C_rf is defined in unit of inches (so radius would need to be defined in inches), to end up with the wiki style equations:

F = (C_rf / R) m g
T = F R = (C_rf/R) m g R = C_rf m g

or you could define C_rr as a function of radius, using units of inches for radius from that table:

C_rr = C_rf / R
F = C_rr m g
T = F R = C_rr m g R

Kuklish

Thanks for your replies,

I have two radii to be concerned with. One is the radius of the drivewheel (R) that i am trying to determine the overall torque on. The other is the radius of any additional wheel (r) that I am trying to determine the rolling resistance of.

So I can be very clear... should your equation

actually read...

Where:

R=radius of the drive wheel
r=radius of extra wheel that is contributing to the rolling resistance force

That way I can play with the diameter of these extra wheels till the overall torque is within acceptable limits.

I didn't see anything in the wiki article about speed, so I guess I just leave it constant force no matter what the speed.

Lastly, I am confused why, if the coefficient is a dimensionless number... the first article I referenced expresses it in "inches@3mph"?

rcgldr

The wiki article uses a dimensionless coefficient for rolling resistance, where the effect of the radius of a wheel is already taken into account. The first article you referenced has a table of coefficients based on the materials, but where the radius is not already taken into account. The coefficients in that table are stated in units of inches, and need to be divided by the radius of a wheel, also in units of inches, to end up with the dimensionless coefficient that the wiki article uses.

Kuklish

Man... you guys are awesome.... Huge thanks RC Glider for all your help. I will try to post a pic here at some point for you to see... its a pretty cool looking rig.