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#45
03-07-2006, 09:31 PM
 Paradoxic Guest Join Date: Mar 2006 Posts: 7
The Plane Cannot Take Off

Quote:
 Originally Posted by Chronos Could you please give us the equation for frictional force as a function of relative speed? Because all of the models for dry friction with which I'm familiar have the force being approximately constant with speed. In other words, it's possible to construct a system such that the frictional force would be enough to prevent the plane from taking off, but such an airplane wouldn't be able to take off from a normal runway, either.
Well, I'm trying not to get too much into this without taking away from my present work (let's hear it for PowerPoint!). But O.K., without having to drag out my first year Engineering Statics and Dynamics text book, here's a little something to chew on, using a 747. Data is taken from the Boeing website, as well as Goodyear Aviation.

Using one common 747 configuration, the aircraft at takeoff is 130 tons. ~80% of that weight is distributed on the aircraft's 16 main gear tires, which are sizable, 49" tall and 19" wide - the designation is H49x19.0-22, for aircraft tires the second number is the nominal width, not aspect ratio as in passenger cars. (oh, each tire costs \$3,600).

Length of the tire's ground footprint under load is 20% of it's circumference.
49" x PI x 20% = 31"
footprint is 31" x 19.0" = 589inČ

With 65 tons of downward force on each of the main tires, there is

65/589 = 220lbs/inČ

on the contact surfaces. Now, at this point I haven't gotten the time to look up the coefficient of friction (”) between the tires and the tarmac, but it's pretty damn high... ” is the tangential force required to create slippage, divided by 220lbs/inČ. It's gonna be pretty damned high, high enough that there will be no slipping of the tires on the tarmac. So we know that the tires and the tarmac will roll together, with a resultant significant backwards force on the tires.. This means that a sufficiently powerful treadmill can provide enough backwards force to keep the plane on the ground, because the rolling resistance plus to friction between the tarmac and the treadmill is too significant to overcome.

Now, as far as the landing gear assembly is concerned, guess what? The frictional resistance of the tire assemblies (ball bearings, etc) is pretty considerable until the plane starts going down the runaway and some lift is provided by the wings. As the plane's wind speed increases, the plane's relative weight on the wheel assemblies decreases, and frictional resistance decreases, etc.

What does this all mean?

If the plane, from the very get-go, does not generate any air speed, the frictional resistance of the landing gear, added to the backwards force generated by the treadmill - it's possible that, on a treadmill, with the plane at full thrust, the plane will not only be stationary, but the wheels may be spinning at significantly less than they would at takeoff speed - maybe even a speed relative to the treadmill of less than 50mph, far less than the average takeoff speed of 155mph.

According to Goodyear aviation: "Heavy loads and high speeds cause the heat generation in tires to exceed that of all other tires and can have a very detrimental effect...rubber dissipates heat slowly...for this reason, aircraft tires and only be used intermittently.". So, rotate the tires too fast and they burn up. Think about that.

So, a 747 cannot take off, nor can any other commercial ariliner. But guess what? Start plugging in these figures for a light and powerful plane, and the problem appears differently.... a commerical airliner can't take off, but maybe a jet fighter can?

The moral of this story is that people that like non-engineering thought experiments shouldn't build airplanes.