Need help with real life math problem

[thetexan]

I am a pilot with a problem. I love math and study it as a hobby but this problem seems to be beyond my abilities. I need to have help in actually solving it and would also like to know how you solved it please.

Below is a diagram of a MD88. It has a wheelbase of 72ft 5 in and a wheeltrack of 16ft 8in.

Given that the aircraft starts at a parked position (position A) that is 90 degrees (perpendicular) to the taxi line with the nosewheel on the T-Bar on the taxi line. As the aircraft begins to taxi the nosewheel is tillered full left to 90 degrees. The nosewheel is kept on the centerline during the entire event. As the aircraft moves, the main gear, of course, will follow and at some point the aircraft will be perfectly lined up on the centerline with the main gear symetrically straddling the centerline (position B). I suspect the track of the main gear will follow a sort of parabolic curve but I don't really have a clue.

The question is... what distance will the nosegear have to travel along that taxi line before the aircraft finally becomes perfectly lined up and the main gear evenly straddling the line? Assume no slippage of the tires. This is a real world problem requiring calculus I suspect and it is beyond me. Please help.

http://www.photomiracles.com/md88.jpg

 

[phyzguy]

No calculus needed. Draw a circle with the nose wheel in the center and the two main gear on the perimeter of the circle. The radius of the circle will be a little longer than the wheebase, and should be (assuming wheeltrack is the distance between the main gear):

R = sqrt(wheelbase^2 + (wheeltrack/2)^2)

Now roll this circle to the left 90 degrees. The nose wheel will track in a straight line, and after rolling 90 degrees, the main gear will straddle the centerline. The distance traveled by the nose wheel will be 1/4 the circumference of the circle or (pi*R)/2.

 

[AlephZero]

The answer in #2 is wrong, because it ignores the fact that the main gear wheels can only roll in one direction. if you draw a picture of the circular motion in #2 you wlll see the main gear is moving sideways relative to the aircraft.

The correct solution is in http://en.wikipedia.org/wiki/Tractrix. The aircraft will never line up perfectly, unless the nose wheel crosses the center line and then moves back onto it when the main wheels are symmetrical.

You can line up the aircraft perfectly in its own length. First do a 180 degree turn with the nose wheel at 90 degrees, then back up till the main gear is on the center line, then turn back 90 degrees with the nose wheel at 90.

 

[phyzguy]

I didn't know about this constraint on the main gear wheels. Thanks for the correction.

 

[thetexan]

Very interesting.

I can assure you the main gear WILL eventually precisely track behind the nose gear and equally spaced on either side of the line AND WITHOUT driving the nose gear past and then back onto the line. So it is not true, in real life, that the mains will never line up. They eventually will. And in a relatively short distance. It is that distance that I want to precisely (as near as possible) determine.

To me this will involve a limit or series. The mains may not mathematically ever line up but in reality they eventually will. So, I believe, the curve of the main gear will converge to a limit. And it will be that limit that is the distance I am looking for. It's similar to the paradox that states that before you can run a 100 yard dash you must first run 50 yards of it. But before you run 50 yards you must first run 25 and so on. That series seems to indicate that you can never get to 100 yards, but, in fact you do. So the series converges to some number and that is the distance I am looking for.

I really appreciate your all's help. However, if someone can actually do the math I would appreciate it since that is beyond my knowledge so far.

thanks,
tex

 

[phyzguy]

You'd have to tell us the constraint on the main wheels before anyone can solve the problem. Assuming the main wheels move along frictionlessly, I think my answer in post #2 is the right one. If the main wheels are "dragged" along sideways, and this is a constraint on the problem, then I don't think it can be solved unless you can quantify what the constraint on the main wheels is.

 

[AlephZero]

The plane will line up to wiithin some practical tolerance in a finite distance. Nobody is likely to be checking this to fractions of an inch.

Your assumption of no tire slippage is physically impossible since the plane is turning and there is a finite area of tire in contact with the ground.

In real life all the landing gear components have some flexibility in them so the plane body won't move "perfectly above" the wheels all the time. The main wheels may be intentionally toed out or in slightly (like cars) rather than parallel.

In other words, to make a more realistic model of this needs a lot more information than we have available.

FWIW I smiled that the wheeltrack was the nice round number of 200 inches. Boeing design their fuselages in 100 inch long sections. I guess there's no point in using more significant figures than you need

 

[thetexan]

All six wheels are freewheeling, free to spin forward or backward and independent of each other. There is no axle interconectivity between tires.

For example in the above scenario, at first, in the turn, the right side gear roll forward and the left gear turns backwards until there is forward motion of the main gear.

When I stated no slippage I mean that both main trucks follow the nose track freely and do not 'bind' because of radius of turn.

I don't know what more is needed. The tractrix curve mentioned earlier seems like the answer but I don't know how to do the math. Should I put this question in the Calculus forum since the tractrix site indicated that differential equations and intregals were used?

tex

 

[coolul007]

Are any brakes applied to the rear wheels to help the track and turning of the plane?

 

[uart]

Very interesting.

I can assure you the main gear WILL eventually precisely track behind the nose gear and equally spaced on either side of the line AND WITHOUT driving the nose gear past and then back onto the line. So it is not true, in real life, that the mains will never line up. They eventually will. And in a relatively short distance. It is that distance that I want to precisely (as near as possible) determine.
tex

No I think you'll find that it's asymptotic, it gets closer and closer to the center line but never quite gets there. Of course in practice all measurements have a tolerance and you can never keep your front wheel on the center line to within 0.01" (or something ridiculous like that) anyway. So in reality this point is moot.

I could plot the equation for you with the parameters of the plane. You find that at the main landing gear is centered to within about 11" of the center line after 300 feet, and to within about 2.5" after 400 feet.

What tolerance to you realistically require for this alignment, do you want it within an foot? Or within 1"? We don't know. You a have to specify a certain allowed tolerance here before you can reasonably solve this problem.

 

[AlephZero]

Should I put this question in the Calculus forum since the tractrix site indicated that differential equations and intregals were used?

I think most "question answers" would know enough calculus to handle this without moving it from here, but it you want to move it just sent a PM to one of the mentors (see the bottom of the page listing the threads in the forum).

To repeat, the math solution of the problem as you posed it says it approaches the center line closer and closer but never actually reaches it. So either you have to define some tolerance for what you mean by "lined up" (to within a foot? or an incl?), or we need a more detailed math model.

I would guess the likely answer is that the tow truck driver "overshoots" a bit (maybe only by a foot or two) and then pulls the nose wheel back onto the center line when the main wheels are lined up.

 

[coolul007]

The air plane tracks through 90 degrees so when inline it has gone twice its wheelbase length.

 

[thetexan]

Yes, in a turn like that it would require braking the left mains. The reason is that so much power is used on the right engine to start the fuselage rotating left that it would cause the nose gear (in a 90 degree position) to skid unless you created the pivot point on the left mains by braking them.

This aircraft is considered centerline thrust because the engines are so close to the longitudinal axis of the plane. There is very little asymetric thrust as a result. This would not be the case in a 737 or 767.

I would like to know 2 things...

1. Could not the curve be calculated as a infinite series converging on some value that could be considered the distance?

2. I would like a rough calculation (to withing 6 inches of symetrical) based on the graph of the curve please.

Thank you all,

Tex

 

[thetexan]

coolu007,

According to just a guestimate of the tractrix curve it appears, in all cases, that the distance will be somewhere in the neighborhood of 4 times the wheelbase. How did you arrive at 2 times the wheelbase?

tex

 

[uart]

coolu007,

According to just a guestimate of the tractrix curve it appears, in all cases, that the distance will be somewhere in the neighborhood of 4 times the wheelbase. How did you arrive at 2 times the wheelbase?

tex

Yes. If it followed the tractrix curve exactly it would line up to within 6" at 338' (about 4.6 times the wheelbase).

 

[coolul007]

Yes. If it followed the tractrix curve exactly it would line up to within 6" at 338' (about 4.6 times the wheelbase).

In a perfect world, The wheelbase length forms a square, where the nose wheel tracks on one side. Since this is a fixed length, rigid distance the nose wheel cannot move the length of the wheel base without the rear wheels being a fixed distance behind. I look at this in 2 ways, 1) a piston on a train wheel, or 2) a fixed rudder(left rear wheel) on a boat in a river with the nose anchored in the middle of a moving stream. Either way the rear wheels, through a lot of friction, should track a 90 degree arc to the line. Without the rear wheel friction(this is the variable in the track) the plane would never turn to be inline on the pull path. There is my thinking, reality may be different.

 

[thetexan]

338'. That's what I was looking for! Thank you uart and everyone else. Uart, did you calculate this (if so what formula) or did you just approximate it by looking at the tractrix curve?

tex

 

[uart]

I programmed the tractrix equation into MATLAB and got the result from there.