For the eggheads: The math behind an uncoordinated turn

[PosClimb]

I'm trying to determine, mathematically, if the aircraft is in a Skid or Slip condition while turning.

Does anyone know the math?

[HazelNuts39]

In a coördinated turn at true airspeed V the bank angle φ is given by:

tan φ = V * ω /g

where ω is the rate of turn and g is acceleration of gravity (in consistent units).

[DaveReidUK]

Centrifugal force is the "equal and opposite reaction" of the airplane to the change in direction and acts equal and opposite to the horizontal component of lift.

In the days when lecturers were prone to administer a rap over the knuckles with a ruler to encourage learning, I seem to recall that mention of "centrifugal force" was a sure way to get one. :O

[photofly]

That was when lecturers were sufficiently unsophisticated in their understanding of physics to be unable correctly to perform an analysis of forces in a rotating frame. Thankfully things have improved since then.

[awblain]

Mr Fly, in order to avoid a rap across the rudder in the rotating frame, rudder must surely be applied to keep the aircraft pointing in the tangent direction, which I guess the tendency to depart from, the ruler-wielder would attribute to a coriolis force.

[EEngr]

xkcd: Centrifugal Force (http://xkcd.com/123/)

[bubbers44]

Remember the days this was so simple? Keep the ball in the middle and you were in a coordinated turn. No formulas required. I saw a skidding turn in a Bonanza out of Burbank, Ca. one day and it ended up in a fiery crash with 4 dead from 300 feet on take off with an engine malfunction. The pilot didn't keep the ball in the middle, simple as that. He rolled over when he stalled and went straight into the ground at the departure end.

[DaveReidUK]

That was when lecturers were sufficiently unsophisticated in their understanding of physics to be unable correctly to perform an analysis of forces in a rotating frame. Thankfully things have improved since then.

Yep, we couldn't imagine non-inertial frames of reference in those days, all we knew was that you had to step on the ball.

[deltahotel]

Must be missing something here! Where does the 'centrifugal force' come from? The only forces are weight and the total aerodynamic reaction. The latter is at right angles to the wing and has two components - vertical, which in a level turn equals and opposes weight and horizontal towards the centre of the turn which is the centripetal force which makes the ac turn.

[FlightPathOBN]

so after all of these years.. the dynamics of an aircraft in a turn...wow...:ugh:


this discussion should be coupled with the theory of lift discussion...

I will throw in another mf to the discussion. A turn is a rate of change. As the ac goes from level, to the bank required to make the desired turn, the rate of change is constantly changing, hence the definition of a spiral.

I have had to deal with this on a frequent basis designing RNP turns. As one notes, RNP procedure turns are a simple radius. If one looks at turn on final, the transition between level and full bank limited by the ac or criteria, is a spiral. One has to be very careful with that transition, hence the slip takes the ac out of containment.

In the days when lecturers were prone to administer a rap over the knuckles with a ruler to encourage learning, I seem to recall that mention of "centrifugal force" was a sure way to get one.
havent 'those days been a while! :hmm:
http://wright.nasa.gov/airplane/Images/turns.gif

[john_tullamarine]

Where does the 'centrifugal force' come from?

Probably better to keep oneself nice by thinking of centripetal acceleration (and force) with an opposite centrifugal reaction ?

[deltahotel]

Where does 'centrifugal force' come from? wasn't a real question.

[bubbers44]

Wasn't that relevant really.

[barit1]

bubbers44:I saw a skidding turn in a Bonanza out of Burbank, Ca. one day and it ended up in a fiery crash with 4 dead from 300 feet on take off with an engine malfunction. The pilot didn't keep the ball in the middle, simple as that. He rolled over when he stalled...

As a recent PPL holder 48 years ago, I was taught that lesson in an old Ryan PT-22. My instructor (Purdue's Irwin Treager) talked me through a couple routine stalls and slowflight maneuvers, then suggested I hold a little rudder (cross-control) to see what goes.

Well, about 1-2 mph above the expected stall speed, we did the prettiest snap roll! Happened so fast that by the time I could respond, we were shiny-side up again. A lesson not soon forgotten. :=

[Lightning Mate]

Don't try that in a high wing-loading swept wing jet mate - it's called departure.

[barit1]

Don't try that in a high wing-loading swept wing jet mate - it's called departure.

Yes, and in fact the PT-22 has 2 or 3 degrees sweepback, which undoubtedly aggravated this behavior.

One question comes to mind though - 23 July 1983, Air Canada Flight 143 (The Gimli glider) - The captain had glider/sailplane experience, found himself a bit high on approach, and dared to slip that 767 on final.

Was he merely lucky not to depart the regime of positive roll control, or did he have some insight beyond his seat-of-the-pants about how far to push the beast? :uhoh:

[bubbers44]

You can slip, either forward or side slip in any airliner as long as you have the proper speed. All my crosswind landings in airliners were side slips to compensate for the drift, the forward slip is the same except the nose is not positioned lined up with the runway. The aerodynamics are exactly the same, one is used to compensate for crosswind and land straight, the forward slip is to cause fuselage drag to increase drag and descent rate.

I heard a story of a 737 operator doing it when idle power didn't work and landed just fine. I wouldn't advertise I did it however.

For us that use the side slip to have the plane land straight and not in a crab to not abuse the landing gear on touchdown we also know we need to have the proper speed. It is quite simple.

[denniscrilly]

Listen to bubbers he is right on the money with this stuff.

[Lightning Mate]

Yes, and in fact the PT-22 has 2 or 3 degrees sweepback, which undoubtedly
aggravated this behavior.

I can't believe that, because it has a relatively high thickness/chord ratio.

[owen meaney]

Where does the 'centrifugal force' come from?Conservation of angular momentum was how was explained to me.

[barit1]

Listen to bubbers he is right on the money with this stuff.

So the safe IAS for a slip is a documented number, not merely seat of pants. I am impressed! :ok:

[bubbers44]

We all know a side slip is required any time you use the side slip to land so the actual added speed is dependant on how much side slip you use. Sometimes you use a little, sometimes a lot, so adjust for conditions on short final. Don't get any charts out just land and use your eyeballs. Always worked for me. Flying isn't that hard if you keep it simple.

Simple situations can be made hard by making it complicated as we are here.

[barit1]

Well, bubbers44, that's what I sorta expected.

Almost.

In my T-craft, runway length is hardly a problem, but obstacle clearance at local fields can be a challenge. So I slip with the nose well down to kill some L/D - I couldn't tell you my IAS if I had to, my eyeballs are outside the plane. Gotta do it that way, got no RA, and I un-slip it at about 10 feet.

Many years ago my dad test-flew a rebuilt J-3 for a friend. After a half-hour airwork, he came back and landed first time using no more than 500 feet of runway. Although he had plenty of prior J-3 time, that had been 20 years earlier.

But thanks for the "big iron" perspective. :)

[rudderrudderrat]

Hi barit1,
Was he merely lucky not to depart the regime of positive roll control, or did he have some insight beyond his seat-of-the-pants about how far to push the beast?
He could "push the beast's" rudder until he was about to run out of aileron control, which is easy to feel in a Boeing through the control wheel displacement.

If he were trying to do the same thing in an Airbus FBW - then he'd have to be looking at the Flight Control System Page to see how much aileron was being applied for him.

[bubbers44]

I flew the T craft and taught my FAA medical examiner how to fly in it and when I got really bored one day found I had flown over 75 types of aircraft going through my log book including experimental biplanes which I certified for aerobatics for the FAA. I was lucky when aviation was not so restrictive we could do this. My FAA friend had a one of a kind mini biplane that was a blast but the only one built on some pilots garage with drawings on the floor. It was called the Bailey Bitty Bipe.

Doing slips in any airplane is easy, just push rudder one way and aileron the other. You can call it a forward slip or a side slip depending what you are using it for. We learned to do it before solo 50 years ago.

It still works today. I feel sorry for the new guys who are taught automation and didn't get to experience what I did.

[PantLoad]

A skid will kill you; a slip will save your life....

Profound words.


Fly safe,


PantLoad

[cwatters]

With reference to Superpilots diagrams..

For a correct turn with no loss of height..

Vertically..

mg = LCos(x)

Where

m=mass of aircraft
g = gravity
L = Lift
x = bank angle

Solve for L
L = mg/Cos(x) ...................... (1)

Horizontally..

mv^2/r = LSin(x)

where
v = velocity
r = radius of turn

Solve for L
L = mv^2/rSin(x) ......................(2)

Equate (1) and (2)

mg/Cos(x) = mv^2/rSin(x)

Mass cancels

rearrange

Sin(x)/Cos(x) = v^2/rg

Sin(x)/Cos(x) = Tan(x)

so I think in a normal turn

Tan(x) = v^2/rg

If Tan(x) > v^2/rg then it's a slip
If Tan(x) < v^2/rg then it's a skid

Someone check the maths :-)

[HazelNuts39]

Impeccable maths. But where do you get r = radius of turn? You can get it from:

V = ω * r
where ω = turn rate. With that, you get:

Tan(x) = v * ω /g

[cwatters]

Makes sense.