Radius for turn Formulae
 

Here is the formulae I have for radius of turn:

r=V^2/(11.26*tanTHETA)
question: the value for V in V^2; is that to be in ft. per second or Knots per hour?

The second formula:
r=V^2/(32.2*tanTHETA)

I tried both of these and came up with diferent answers with each. Can anyone shed some light, perhaps with a full explanation of units required or a different formula?

Thanks

Radius of turn = V²/(g.tan(bank angle))

I would normally use V in m/s, radius in metres and g in m/s/s. But your second formula is identical to mine for V in fps and radius of turn in ft (g is 32.2 fps/s).

It's important to be self-consistent with units. These formulae generally use either m, s, kg, N; or ft, s, lb, lbf. Don't mix them, and don't use mph, kph or knots which will only work with added conversion factors.

Also bear in mind that V is TAS, not EAS. To convert from EAS (or CAS, or roughly IAS) to TAS at subsonic speeds, divide by SQRT(sigma) where sigma is the relative density.

G

At lower altitudes and moderate speeds, I find that 1% g/s gives pretty good results for things like turning onto DME arcs. It's simple and easy to work out quickly and accurately.
For me, K.I.S.S. every time! :)

Would you mind giving an example for your formulae. Like: with IAS 210 @ 5000ft (TAS 230) bank 25, and than how your formula works.
Thanks

230 kn TAS = 118.22m/s (x 0.514)
tan (25°) = 0.4663
g = 9.80665 (universal(ish) constant)

Thus..

R = V²/g.tan(bank)

= 118.22² / (9.80665 x 0.4663)
= 3056.29m
= 1.65 nm (divide metres by 1850 for nm)

Thus at 5000ft sHp, 210 kn EAS (giving 230 kn TAS), a 25° banked turn will give a 1.65nm radius of turn.

If you want to turn this into turn rate, work out the circumference (2.Pi.R) = 10.37nm. At 230 kn TAS you would fly around this in (230 / 10.37 = ) 0.045 hours, or (x 60 x 60 = ) 162 seconds. That's 360° in 162 seconds, or (360 / 162 = ) 2.2°/s.

I'm sure that there are rules of thumb for all this, but being an engineer I live with a calculator in my hand anyway - and tend to use it for my flying too (I confess to struggling in CAA / JAA exams because I'm totally unused to the Dalton computer, and suchlike tools).

G

Or my way:-
Assume nil wind
TAS=G/S=230kt
1%=2.3nm
Now, if you wait till 2.3nm prior to start your turn onto the DME arc, you'll be as near as damn it due to inertia of the a/c.
In any event you'll be within the tracking tolerances (2nm) by a good margin.
K.I.S.S. :)

mustaf:spoton,why do we make it harder than KISS?,believe me fellas,1% nails it every time.
cheers

Musta, I do prefer to use the 1% rule, particularly when trying to train a new sprog.

Genghis, I must admit that I do appreciate the detail you go into with some of your posts. I have "plaigurised" (spelling) some of your material for the benefit of training also. Hope you don't mind!

It is also handy to know how many track miles to run once we are established on an arc. Remember your basic trig. from school? I think someone touched on it previously. For example,with 30 degrees to run on a 10 mile arc, you have approximately 5.5nm to the inbound track. (2 pie R divided by 360 degrees multiplied by the degrees to run). The only benefit I guess from knowing this is to help you plan the gear and flap selection. Nothing worse than dragging it in from too far out with gear and flap hanging out.

Ballpark figures:
On an 8nm arc, multiply the degrees to run by .140: 10nm arc by .174: 12nm arc by .209: and 15nm arc by .262. Makes for easy planning!

Happy new year!

Shakespeare, it's a pleasure to know that occasionally somebody finds my ramblings useful - plagiarise away.

G

Chaps, may I suggest you get out more!

What's this 1% rule crap. I think all pilots should memorise this radius of turn formula immediately, carry scientific claculators at all times and always crunch the numbers prior to turning!

hi every one,

try these,

First of all i'll do the bank angle formulae. See if it make sense.

Bank angle for rate 1 turn is approximately ((TAS*0.1)+7)=Bank Angle.BUt these limited to about 25Deg.

Where as if you use a moderate bank angle the approximation is:
Bank angle= (TAS * Turn rate)/7
Eg...TAS 350, Turn Rate 1/2(1.5Deg/sec),
Bank Angle= (350*1/2)/7=25Deg

Radius of turn; At rate 1 the Turn Radius =TAS/180
Eg...TAS120, Turn radius=120/180=0.7nm.

Dan, if people like me got out more, there'd be nobody designing aeroplanes - then where would you be?

I never suggested taking a calculator in the air as such (although I have been known to) any more than I'd suggest doing your flight planning after, rather than before, take-off....

G

[This message has been edited by Genghis the Engineer (edited 04 January 2001).]

G. thanks for the explanation. I think it's good knowing both formulae. Yours on the ground and the shorter one in the air.

Does anyone know the address for the publishers of De Principia Mathematica ?

A simple revision would, I am sure, have it selling like hot cakes among you mathematically inclined aviators.

If Sir Isaac Newton had had the foresight to express g in useful units, rather than this 32.2 ft/sec/sec; 9.81 metres/sec/sec nonsense, I am sure you would all go out right now and grab your copy of his book. And Mr Kermode would have a lot less to talk about on the subject of consistent units of measurement.

Ike should have used nautical miles/hour/hour, or Knots/hour. Then g is the easily remembered number 68600 (roughly). Just rolls off you tongue, doesn't it ?

If we bung that in the Genghis/gaga example, using PROPER measures of speed and distance, we get

tan(angle of bank) = TAS squared / (68600 * turn radius)
= (230 x 230)/(68600 x 1.65). That is, use knots and nm directly.

AOB = 25 degrees.

So everyone, repeat after me

"THE FLYING PERSONS GRAVITATIONAL CONSTANT IS 68,600 kt/hr (to about 3 places)"

And NO, I don't want to be nominated as the supplier of this week's most useless bit of information.

HAPPY NEW YEAR TO ALL

Not strictly on subject, chaps, but is not 'knots per hour' tautological?? http://www.pprune.org/ubb/NonCGI/frown.gif
I thought that a knot was a speed of 1 nautical mile per hour. http://www.pprune.org/ubb/NonCGI/confused.gif

No, knots per hour (admittely an unusual unit, but it works for echo tango) is a valid unit of deceleration - identical to nautical miles per hour per hour. Just as stall tests are normally done at 1 knot per second - or nautical miles per hour per second.

An aviators version of principia, an interesting idea, but I'm not convinced it would sell. Those of us whose prefered method of doing sums isn't using rules of thumb are probably in the minority.

G

When I needed to calculate the radius of a turn in order to fly a safe low level demo I divided the square of my indicated speed by 10 times the reading I intended to use on the g meter.

I offer four advantages for this method:

 The answer is in feet.

 No conversion of units is required

 You barely need to know how to spell maths.

 The answer is 13% pessimistic which keeps you out of trouble with the display line even with an on crowd wind (and or your off days)


Worked for me

JF

Excellent JF. Keep it simple for pilots!

mustaf, correct me if I am wrong but I thought the tracking tolerance on a DME arc was 1/2 nm and protection 2nm.
thanks for an answer.
pesket

John Farley,

I was curious to see why your V squared/10G formula works. I would have liked to use it on a C172, except I can't find the G meter.

So I tabulated theoretical and JF methods in the range 5 to 85 degrees bank, 100 to 450 knots, and calculated the differences.

Here are three cases

AoB 20, 1.064g, 200kt. Theoretical r 9743 ft, JF r 3759 ft.
AoB 40, 1.305g, 300kt. Theoretical r 9509 ft, JF r 6894 ft.
AoB 60, 2g, 150kt. Theoretical r 1152 ft, JF r 1125 ft.

For us mortals who get really worried above 60 AoB, your formula is not so good. At 30 AoB, your error is 43% the WRONG way.

The two methods break even a little over 60 degrees, then the JF approach is 11% safe at 80 degrees (5.76g), and 12.4% safe at 85. So we can see where you used to play.

Lastly, why is your formula so good in the 60+ range ?

(Dan Winterland, I promise I will go fishing immediately I have posted this).

Answer: The cosine of an angle asymptotically approaches 1/tan of the same angle as the angle approaches 90. And the g load is related to cos(AoB); the turn radius to tan(AoB)

Gone fishin.

ET

pesket,
Here in Oz the tolerance on a DME arc is +/- 2nm of the nominal arc.
Refer Oz AIP ENR 1.5-13 para 1.12.1 note.

I can't help suspecting that when JF was using that particular formula he wasn't making turns with less than 60° of bank, or as slow as 150 knots all that often.

You may not know his name down under, and I hope John will forgive me blowing his trumpet for him. JF is a very distinguished test pilot, with a 20 year record of test flying British jet fighters (when we used to build them unaided) - and inevitably as a company pilot, displaying them as well.

This of-course assumes that he is who he says he is on Pprune, but I am happily convinced that that is the case.

G

EchoTango

You said:

Answer: The cosine of an angle asymptotically approaches 1/tan of the same angle as the angle approaches 90. And the g load is related to cos(AoB); the turn radius to tan(AoB)

However:

According to me g = 1/Cos AoB

Does that affect the price of fish when you are upside down?

BTW Hope you had a good holiday in the sun Ed

Regards

JF

Despite my flippant comment earlier in this thread, this is something we have to use doing the job I do - Air to Air Refeulling. We often have to set up RVs calculating the radius of turn to ensure we roll out exactly one NM ahead of the customer. However, a lot of this info is already calculated and tabulated.

When these tables aren't valid, do also have a natty gadget called a 'Turn Radius Nomogram' which is a graph with various parameters (Alt, IAS, g, Radius etc.) which does all this for us. I tried to copy it at put it in this thread, but failed on the technical side.

If anyone wants a copy, mail me with your snail mail address and I will send you one.

John Farley,

You are quite right. We have terrible problems down here knowing when to replace functions with their inverses when applying formulae out of english books. The local zoo had a similar problem when they brought out some pommie bats. Poor buggers wore their teeth out in a couple of months trying to hang on in the bat cave.

I was looking at something else with the cos; 1/tan thing.

JF's formula for r is V/10G, where G = 1/cos(a) (V in knots, r in ft)

The theoretical r is (V * 6080/3600)squared / 32.2 tan(a)

So the fractional error in the JF formula is (JF radius - Theoretical radius) / (Theoretical radius)

That error function reduces to

1 - (cos(a)/10) / (2.852 / 32.2 tan(a))

And as a approaches 90, cos(a) / tan(a) approaches 1. That was the cos; 1/tan matter I noted.

So as a approaches 90 degrees, the error approaches 1 - ( 0.1 / .08857), or 12.9%. That is the 13% limit you originally cited.

Dan Winterland

JF's formula is within 13% of correct in the range 51 to 90 degrees bank. It is spot on at 62.5.
Its a very good method if you are low, fast and busy showing off, or engaged in a tactical exercise. Like in JF's old playground.

Don't think it would suit you. Sounds like your nomogram and tables will be hard to beat in your case.

John, Had a great Xmas. Thank you.

ET

EchoTango

If only somebody could be found to put all Tech Forum threads to bed as comprehensively as that.

Much appreciated.

JF

Thanks everyone for the interest and help.

The formula r=V^2/11.26Tan(theta)
I discovered works with TAS in place of V,
Thus simplifying units. (just plug the variables in)

I found this formula usefull for teaching purposes.

Thanks again,

All this technical stuff.....in days of old when etc.......
Using quarter million topo Vampire rate 1 turn use a sixpence (6d), for Canberras some Belgium? coin with a hole in the middle.

TAS squared over g tan theta is a jolly useful formula! When I used to fly triangular aeroplanes, I really couldn't be @r$ed to draw a different track for different speeds at low level. The hole in the end of a standard RAF nav rule was always used for drawing the turns and I just varied the angle of bank to stay on track at different speeds!
It also enabled me to produce ALL the turn ranges for various AAR RV procedures for ATP56A back in 1991; didn't cost HM a cent as there was no such thing as GEMS back then! I evolved (and still have) the generic expressions for all RVs, although I could only solve the RV C by an iterative process. So if we suddenly decide to change any parameters (AoB, TAS split, roll-out range) then I can recalculate the turn ranges very quickly. But this time HM'll have to pay me for my troubles!!

what a forum ... what an amount of expertise ...

could'nt some of you gurus have a look at my topic 'V2 .....'

still longing for the answer that hits it on the head ...

thanx in advance
LF

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Let there be cold beers on a hot day !!!

Measure with a micrometer, cut with an axe.