Rolling Resistance: A Trickier Term Than You Might Think

[Cyrus]

I came across an interesting little problem doing some work that I though I'd share. Typically, we think of three types of resistance for a wheel:

• Static Friction
• Kinetic Friction
• Rolling Resistance

I'm modeling the wheel of an airplane for a flight simulator. When the airplane is sitting still, it has some resistive force before it starts rolling. So I just found this force using the maximum coefficient of static friction for a tire (obtained from manufacturer). Turned out I got an insanely large number. Then I realized, <duh> that would be like having the airplane on full throttle (which it nearly turned out to require) and dragging the wheels across the runway.

So the only option left was to model the friction using rolling resistance (which I am already doing when the wheels are in motion). My understanding was that rolling resistance was defined for rotation of the wheel (because my basic books on physics all just assume the wheel is turning). I looked at my vehicle dynamics book and realized the curves for rolling resistance as a function speed go all the way down to zero, and have a power relationship with speed f0+ 3.24*fs(V/100)^2.5. Anyways, the point is that "rolling resistance" is a bit of a tricky term to use. After I plugged in some values I got a frictional force of 25.5 lbs for a 2550lb aircraft. It seems like a very reasonable number.

As a side note: All tires slip in the real world when you accelerate/break and that the coefficient of friction goes down as you increase the load.

The point is, tires do a lot of non-intuitive stuff and the basic presentation of friction in most books is woeful. You have been warned.

 

[Bob S]

Your number of 25.5 pounds force for pushing a 2550 pound aircraft (rolling resistance coefficient of 0.01) is similar to recommended low-loss tires for automobiles. The assumption for automobiles is that for an automobile of mass M going at V meters per sec, the tire power dissipation rate is about (1/2)R_tiresM V². Outside of engine (in)efficiency and air drag, tire rolling resistance is the biggest power loss.

 

[Cyrus]

I got the numbers for the rolling resistance using the equations in Gillespies Fundamentals of Vehicle Dynamics using the Stuttgart equation for automobiles. The weigh is nearly the same, and the tires are very close too. The tire pressure is 45PSI nose, 38PSI main. I still need to get the exact rolling resistance values from Goodyear on Monday, so in the mean time I'm using the value for auto tires. The aircraft tires are 6 ply, so I don't think they will be that far off from a cars value.

The interesting point was that I've never seen something describing the resistance to get a tire to just start rolling, and I'm not sure if rolling resistance is the best way to model this - but appears to be close enough.

Typically rolling resistance means the tire is already moving. In general though, I view modeling friction as more of a "good enough" approach so long as the numbers are reasonable since it is so highly dependent on the two surface conditions which vary drastically all the time. Ideally, you would need some correlation of resistance to some RMS value of the overall surface roughness of the entire airport runways/taxiways, repeated for every airport.

It's all really just guesswork, IMO.

The second interesting point is that rolling resistance is the predominant form of resistance for cars up to about 60MPH. Aerodynamic drag is not as important. It's even more predominant for aircraft because they are highly streamlined. So the aerodynamic drag on an airplane will be much less than a car on takeoff. (Meaning: Rolling resistance is a big deal!)

I'm going to plot the drag vs rolling resistance all the way up to the rotation speed of the airplane and I'll plot it for comparison and post it here.

 

[Bob S]

I got the numbers for the rolling resistance using the equations in Gillespies Fundamentals of Vehicle Dynamics using the Stuttgart equation for automobiles. The weigh is nearly the same, and the tires are very close too. The tire pressure is 45PSI nose, 38PSI main. I still need to get the exact rolling resistance values from Goodyear on Monday, so in the mean time I'm using the value for auto tires. The aircraft tires are 6 ply, so I don't think they will be that far off from a cars value..

The rolling resistance coefficient of tires varies, depending on many factors, including 'squirming", road smoothness, and sidewall design. See this pdf for a list of tires with low rolling resistance
http://greenseal.org/resources/reports/CGR_tire_rollingresistance.pdf

The interesting point was that I've never seen something describing the resistance to get a tire to just start rolling, and I'm not sure if rolling resistance is the best way to model this - but appears to be close enough..

You probably never heard of morning thump, (from the 1960's). Parking cars with (I think) nylon ply developed a flat spot when parked overnight. For a few miles in the morning, tires would go thump thump. For most tires, the force to push a car at a low speed is about the RRC times vehicle weight, independent of speed. Pushing a 1000 Kg car requires about 98 Newtons.

 

[RonL]

The numbers in post #1 compare about the same as my Womack Design Manual, it shows drawbar pull/1000 pounds to be...(depending on surface)

Concrete...10 to 20 lbs.
Asphalt... 12 to 22 lbs.

The formula for air resistance is...
F=FA x .0025 x MPH^2

F is additional drawbar pull needed to overcome air resistance
FA is frontal area in square feet
MPH is vehicle speed, miles per hour

I'm sure you have this in metric, but if not maybe it can help a little.

Ron

 

[Bob S]

The numbers in post #1 compare about the same as my Womack Design Manual, it shows drawbar pull/1000 pounds to be...(depending on surface)

Concrete...10 to 20 lbs.
Asphalt... 12 to 22 lbs.

The formula for air resistance is...
F=FA x .0025 x MPH^2

F is additional drawbar pull needed to overcome air resistance
FA is frontal area in square feet
MPH is vehicle speed, miles per hour

I'm sure you have this in metric, but if not maybe it can help a little.

Ron

The force is F =(1/2) C_pρAv² Newtons at velocity v
(Note that this has units Kg-m/sec² = Newtons)
where C_p is drag coeff (usually between 0.26 and 0.36)
ρ is density of air (about 1.2 kilograms per cubic meter)
A is frontal area of car (m²)

α β γ δ ε ζ η θ ι κ λ μ ν ξ ο π ρ ς σ τ υ φ χ ψ ω

 

[Cyrus]

Don't worry about the Aerodynamics, I have actual performance data I'm using from Cessna and NASA, which are a bit (a lot) more sophisticated than what's been posted.

I'm strictly looking at rolling resistance on the takeoff roll and taxi.

I measured the wheel base today, so I'm going to calculate the weight distribution on the main and nose gear. My guess is that about 60% is on the mains.

Weight transfer between the nose and main gear due to acceleration is going to be assumed negligible based on vehicle dynamics, which make that same assumption and undergo higher accelerations.

 

[RonL]

Thanks Bob S,
As an old truck and equipment operator, I learned that temperature plays a big part in rubber on hot surfaces, if there was a choice I would buy tires in winter so that they would be better tempered by summertime, this would extend wear quite a bit.

Do you think Cyrus should put any heat factors in his program?

Ron

 

[Cyrus]

Thanks Bob S,
As an old truck and equipment operator, I learned that temperature plays a big part in rubber on hot surfaces, if there was a choice I would buy tires in winter so that they would be better tempered by summertime, this would extend wear quite a bit.

Do you think Cyrus should put any heat factors in his program?

Ron

Aircraft tires spend too little time on the ground for heat to be a problem. Tires heat up after about 20 mins of use and become their optimal 'stickiness'. This just isn't true for airplanes. The wheels don't spend enough time turning to heat up and change their properties to any significant degree.

 

[RonL]

Aircraft tires spend too little time on the ground for heat to be a problem. Tires heat up after about 20 mins of use and become their optimal 'stickiness'. This just isn't true for airplanes. The wheels don't spend enough time turning to heat up and change their properties to any significant degree.

Sorry, I thought you were looking for accuracy, I guess you havn't walked on asphalt in the heat of the day, it can be pretty soft. I think you might find temperature near 160 F degrees, maybe that's just here in Texas.

Well, best wishes your project does sound interesting.

Ron

 

[Cyrus]

Sorry, I thought you were looking for accuracy, I guess you havn't walked on asphalt in the heat of the day, it can be pretty soft. I think you might find temperature near 160 F degrees, maybe that's just here in Texas.

Well, best wishes your project does sound interesting.

Ron

I think you and I are talking about apples and oranges. I'm talking about the tire heating up as it rolls, whereas you are talking about the tire properties because of baking in the hot sun.

That is a good point you brought up, and I will ask the manufacturer if they can give me such data on the tire. If not, I'll have to assume it's constant unless someone has an equation that gives a rough approximation of this effect.

 

[Cyrus]

I worked out some numbers for a normal CG location at 2421lbs (nearly at Gross Takeoff weight).

The front wheel holds up 500lbs, and each main wheel has a load of 960lbs. Pretty interesting.

I was close concerning the mass fraction. It's 80% main wheels, 20% nose wheel.