Formula for Radius of Turn
 

Does anyone have the formula for radius of turn for an airplane (with everything in metric units)???

I went to Wikepedia and found that the formula posted on their website is:

r = v^2 / g (tan bank ang)

Banked turn - Wikipedia, the free encyclopedia

However, Wikepdia does not give the units for r (radius) or speed of the airplane.

I think there is some type of correction/conversion factor to keep everything in metric units but I forgot the conversion factor.

Anyway, it would be great if you could help me out :D

aviationluver, no formula but a thumb rule:

1% of the ground speed. So for 250 KTS the turn radius is approximately 2,5 NM.

Very handy to quickly decide when to commence the turn.

Cheerio.:ok:

It all depends on what units you use for V and for g.
If you use knots and NM per squared hour you get radius in nm.
If you use meters per second and 9.8 meters per squared second, the radius is in meters.

9.8 m/s^2 = 9.8 x 3600 x 3600 / 1852 = 68578.83369 NM per squared hour

For a 250 kt with a 25º bank angle:

r= 250^2/68578.83369 x tan 25º= 1.954 NM

If you use NM/min, g = 9.8 x 3600 / 1852 = 19.04967603 NM/min^2
tan 25º= 0.466307658
tan 30º= 0.577350269
g x tan 25º = 8.883009818
g x tan 30º = 10.99833558

So if you find your speed (GS) in NM/min you can use, as a rule of thumb, that speed squared and divided by ten. This figure is approximately your radius of turn for a typical 25º-30º bank angle turn.

examples:

180 kt, 3 nm/min, 0.9 nm radius
300 kt, 5 nm/min, 2.5 nm radius
420 kt, 7 nm/min, 4.9 nm radius

I use it to join, or exit, DME arcs and it works really well. It is useful too when you have to initiate a turn to intercept a radial, or to intercept the LOC if you have distance and a bearing pointer. Sometimes if you wait until the CDI moves it is too late already. No need for the FMS!

So you see that 10% ground speed in knots is good enough and dead easy to calculate when there's plenty of other stuff screaming for brain space. :ok:

mustafagander, my calculation says it's 1% not 10% unless I've had too much of 12 years old Chivas. :}

Careful chaps!!

Surely the 1% equals the DIAMETER of the turn - at least, that's the way it's worked for me for a few decades!

Cheers
mcdhu

Aviationluver,
Wikipaedia is correct, as is Microburst. You just have to be consistent with the units.
I.e. if you want metres for the radius you must use metres per second for v and metres per second squared for g.
If you want the radius in Nautical Miles you must use Nm per second (!) for v and Nm per second squared (!!) for g.
If you really have to use Kts for speed (Nm per hour) then you'll also have to use Nm per hours squared for g (!!!)
As I say, you MUST be consistent with the units in the formula.
Good luck :ok:

As 28L said.

Use TAS (not IAS) for V, in m/s

g=9.81 m/s/s

r will then be in m


Or if you like old imperial units, use TAS in fps, and g in f/s/s, and you'll get an answer in ft.

G

standard rate (360 degree) turn at 180KTAS --> 6nm (11 112m) in 2min.

circumference (circle) = 2 x Pi x radius

11 112 = 2 x 3.14 x r

r = 1769m

If you want the answer, not the formula, try this..
Aircraft Turn Information Calculator

Careful when you use the term Standard Rate. Your AIP may state that turns be made at Standard Rate (= 2min) or 25 deg AoB, whichever is lesser. It says "whichever is lesser" because 2min turn ≠ 25 AoB turn.

Skunkworks' formula works when variable is timing = 2min.

Wiki formula works when the variable is AoB.

They may be equal under certain conditions, but not always.

The following may be helpful... from "Aviation Formulary V1.24" by Ed Williams

Metric? Just work out a factor!

Turns and pivotal altitude

In a steady turn, in no wind, with bank angle, b at an airspeed v

tan(b)= v^2/(R g)
v= w R

where g is the acceleration due to gravity, R is the radius of turn and w is
the rate of turn.

Pivotal altitude h_p is given by

h = v^2/g

With R in feet, v in knots, b in degrees and w in degrees/sec (inconsistent
units!), numerical constants are introduced:

R =v^2/(11.23*tan(0.01745*b))

(Example) At 100 knots, with a 45 degree bank, the radius of turn is
100^2/(11.23*tan(0.01745*45))= 891 feet.

The rate of turn w is given by:

w = 96.7*v/R

(Example) = 96.7*100/891= 10.9 degs/sec

The bank angle b_s for a standard rate turn is given by:

b_s = 57.3*atan(v/362.1)

(Example) for 100 knots, b_s = 57.3*atan(100/362.1) = 15.4 degrees
A useful rule-of-thumb, accurate to ~1 degree for speeds up to 250
knots, is b_s= v/7 (v in knots).


The pivotal altitude is given by:

h_p = v^2/11.23

(Example) At 100 knots groundspeed the pivotal altitude is 100^2/11.23 = 890
feet.

How's about for pilots we keep it simple? We ALWAYS turn procedurally at Rate 1, don't we?:rolleyes: FORGET angle of bank.

Basic rotational algebra tell us that R=v(TAS)/'omega'(Rate of turn) where omega = radians per whatever (not degrees/???).

Rate 1 is PI*60 radians per hour. So, if you choose to work in nm (who on earth would want to use something else...:)), at 240kts TAS:-

R(nm)=240/3.142xxx*60 =1.27nm - close enough for government work Even easier is TAS/180. That I could handle in the cockpit. Most of the above is just TOO much:) [I never could roll out on a radial, anyway...............]

:

For JAR exams use

R in meters = (0.51 x TAS) ^2 / g (tan bank ang)

Where TAS is in knots and the 0.51 converts this into m/sec.

Or use 0.515 if you want to be a bit more accurate.

Remember to multiply the TAS by 0.51 before you square it. The mathematically challenged often square the TAS then multiply by 0.51. This does not work!

I would agree with mcdhu.

Hi BOAC

Rate one turns can be done at low speeds (for airliners) but at high speeds we would exceed the 30º bank angle. In these occasions we have to calculate in a different manner. And in some airplanes there is not bat and ball
anymore.
I remember when I had a sim test for an airline in a 727 simulator. We had to do the "B figure", I don´t know if in UK you are familiar with it. Well it is like a series of racetracks and procedure turns, which legs have to be timed. Turns are supposed to be made at 3º/sec.
During the procedure we had to use three speeds: hi, med, low. 300 kt, 250 kt, 200 kt.
Timing was impossible, of course, so it was impossible to make a good B figure! The sim was a complete wreck, so it was a nightmare...

cheers

The same spreadsheet formula using 180/PI() for MicroSoft and RADIANS for Open Office:-

=ROUNDUP(POWER(TAS,2)/((TAN(AoB/(180/PI())))*68625),3) MicroSoft EXCEL
=ROUNDUP(POWER(TAS;2)/((TAN(RADIANS(AoB)))*68625);3) Open Office CALC

TAS = KNOTS
AoB = DEGREES

RESULT = Turn Radius in Nautical Miles

Example TAS 240KTS AoB 25 degrees = 1.8NM (radius)

and if you REALLY want to use metric units, enter TAS in KMH and divide answer by 1.852, e.g.

=ROUNDUP(POWER(TASkmh,2)/((TAN(AoB/(180/PI())))*68625),3)/1.852

NOTE: BOAC's "back of the fag packet" method gives about the same result!

mm43

- Microburst - you telling me you are flying procedures at 300kts:eek:/ That's how you hit cumulo granitus. If you are talking about cruise speeds - why are you bothered about 'radius of turn'?

If you want to join a DME arc or intercept a LLZ course etc, the following method has worked well for me;

1. For a rate one turn, use a Bank Angle of =(TAS/10)+7 (Degrees)
2. Turn radius for a rate one turn is approx = 0.9% of G/S (Nm)

Note the use of Ground Speed rather than airspeed, if you are interested in your ground track.

I would violently agree with you, if it were for use by pilots in the cockpit, with a groundspeed > 0kt!

But our friend did not specify in his original post, whether he sought a rule of thumb for inflight use, or a formula for his latest iPhone or Excel spreadsheet project.

The clue for me was that he posted in TechLog (bing!), a geometric expression (bing bing!), but could someone help this USA resident with one that used metric units (bing bing bing!). Us rule-of-thumb pilots only work on a tan when we are on a tropical layover, and metric when forced to fly through foreign airspace!

If our friend is doing some research, or aims to produce some kind of calculator (javascript, iPhone, spreadsheet or otherwise) then why use a Rule of Thumb when a floating point arithmetic gadget is doing the computations? That's sloppy.

If our friend is attempting to work out his radius or diameter of turn to write an Ops manual or a Procedure, then likewise, why not use a formula and apply factors for wind and varying pilot skill?

If our friend is looking for a rule of thumb to see how wide he will fly in a holding pattern he is assigned tomorrow, then formula? Forgetaboudit! Use a rule of thumb as above.

No, that is not correct.

I am permitted to turn at 30ºAoB, it is normal for the category of aircraft, and do so when circling.

"Procedurally"?

I am instructed to turn at the lesser of Rate 1 or 25ºAoB in departure and approach procedures. "The lesser of", because Rate 1 and 25ºAoB are rarely coincident.

I am instructed to make turns at 15ºAoB in an EOSID procedure.

My aircraft AFGS commands 25ºAoB turns <=FL200 and 20ºAoB turns >FL200 if GCP selected. If in full autoflight, the Honeywell Pegasus FMS commands a far lower AoB if at high TAS/high altitude.

Less "procedurally"... If I'm avoiding Wx that is a fair distance ahead, I might choose to turn at only 5º or 10º AoB by use of the AoB limiter on my GCP.

So no, "we" pilots don't "always" turn at Rate 1 "procedurally". Plenty of exceptions to that rule.

Thats where a Rate of Turn calculator based on AoB might come in handy for those pilots and others that design procedures.:ok:

This IS techlog, is it not?

Well, ITCZ, I suppose it depends on how anally retentive any particular pilot wishes to be? a.l. already has the equation and his answer (see Genghis the Engineer's post) if he wishes 8 decimal point precision in metric bits.

Your post yesterday was more than sufficient to cure a.l. of insomnia. :)

If I wished to avoid weather at 5 or 10 AoB I would probably use the 'TLAR' rule of thumb.:ok:

I trust you spotted the carefully hidden :rolleyes: in my previous post?

A bit late to apologise for getting it wrong, but I did, didn't I?

1% works fine, 10% is a bit generous :}

I use it for turning onto DME arcs etc.

I must re read my posts!!

This chart may be of use to some. It was knocked up many years ago for a mate who ran an I/F procedural trainer school and wanted several charts to simplify briefings. He's long out of the workforce and I have no continuing interest in copyright so I've deidentified it and you are welcome to use it if it is of any value to you.

As I recall it made life in the Link Trainer a bit easier but, as we all know, rule of thumb and watching the needles tends to be the go in the air.

I note that a few posters have been at pains to show concern about consistency of units.

That's all fine but, at the end of the day, you can use whatever units you choose just so long as you do the conversion calculations to make the equations correct ... I think that speeds in megafurlongs/fortnight or deci-lightyears/weekend would be rather cute ... but, somehow, I don't think that will take off ...
http://img.photobucket.com/albums/v3...mancenilID.jpg

I've never seen so much discussion over what should be a relatively straight-forward matter:ugh:

The application of the various simplifications (e.g. 1% of TAS) is good in practice. Perhaps the original poster needed something more accurate, as may be required when working out OEI Escape Routes etc.

The two formulae given below apply 2 different constants, one for entry speed in Knots, the other for entry speed in Km/Hr.

If Entry speed is in Knots :-

R = TAS^2 X .02698711 / Tan AoB

If Entry speed is in Km/Hr :-

R = TAS^2 X .00786818 / Tan AoB

Example : TAS = 200 Kt (370.4 Km/Hr) at Bank angle 25 degrees.

R = 200^2 X .02698711 / Tan 25 = 2314.96 M....... OR

R = 370.4^2 X .00786818 / Tan 25 = 2314.96 M

If you want the answer in Km, divide by 1000:ok:

Loke Old Smokey said

R = TAS^2 X .02698711 / Tan AoB

or

R= TAS^2/(37*tan AoB)

which is more or less the same.

Will give you the radius of the turn in meters.

For those of us who have an engineering or similar background, unit conversion is routine bread and butter stuff. For those who are not comfortable with such antics, you might like to see how it is done -

The sequence requires that you keep rigorous track of the variables, constants and units .. which I have separated below to make it a bit easier to follow.

Explanation -

Using speed, V, in kts, g in ft/sec^2 and I want the answer to be radius in nm. You might just as easily have started with some other units - makes no difference to the technique.

To change units, we use the "trick" of multiplying by unity ("1") which doesn't change the value of an expression. We can extend this by noting that dividing something by itself is "1" eg 4/4 = 1. The secret, then, lies in further extending this to account for different units which represent the same quantity. So, for instance, we can say that 1nm/1nm = 1 which is the same as saying that 1nm/6080ft = 1. (or you might chose to use 6076.131 as the conversion - depends on what reference you look up). This then allows us to cancel out unwanted units and we just carry the resulting numbers into the main calculation. This last bit is very important and the source of much error when folk start learning about unit conversions.

At the second line we need to get rid of the hour, second, and feet units. To make it easier to follow we can do it in two stages.

To get rid of the hours and seconds, we note that

1 hour = 60 x 60 seconds = 3600 seconds

as we have to get rid of hours^2 and seconds^2 (whatever those units might represent physically is not a concern) we can square the conversion units to come up with

1 hour^2 = 3600^2 seconds^2 which gives the unity expression

1/3600^2 with units hr^2/sec^2. Whether you put the hours or seconds on the top line is determined by the original equation. In this case we want to get rid of hr^2 on the bottom and sec^2 on the top so it makes sense to insert hr^2/sec^2 as in the graphic.

To get rid of feet, we note that

1 nm = 6080 feet (or some similar conversion value) so the unity expression is

1 = 6080/1 with units ft/nm as we need to cancel out ft on the bottom in the original expression

Notice that I now end up with nm as the only unit on the RHS of the equation and this is what I needed. Note also that I have taken the conversion units (3600^2 and 6080) into the constants expression so that I don't lose track of them.

If we do the arithmetic to simplify the numbers we get 0.0000145694 as the conversion constant. As that is a dreadful number to work with, I prefer to replace it by the reciprocal on the bottom line. All this means is I note that

1/2 = 0.5/1, so

0.0000145694/1 = 1/68637

To convert to metres, we note that

1 = 1852/1 with units m/nm so that we can cancel out the nm and to get to Old Smokey's version we do the reciprocal trick.

Note that the small difference in constants is a consequence of which values go into the intermediate steps. It may be important to the academic purist but, functionally, the end result is sufficiently similar not to worry too much about it .. all depends on the reference table from which you pick your conversion constants.


http://img.photobucket.com/albums/v3...2/Graphic1.jpg

Jeez J_T, multiplication is much more fun than division! Just ask any rabbit (or poodle) :ok:

Now to work out a formula for Pugilistic Animus' asymptotic turn.:)

You nerds :8

You nerds

C'mon, now .. don't be like that .... OS and I can't help ourselves.

J_T despite my humor I think it's nice to see folks [in the technical fraternity] who still know what they are doing and what they talking about and don't consider pedantry at all; I have received innumerable educational benefits from you all :ok: from a humble student of the art,...

May you find marry a gorgeous AND experienced computer to write your polar plots:}

May you find marry a gorgeous AND experienced computer to write your polar plots

love it ...

is there an IOS application that covers these kind of formulas?
like sort of an advanced E6B

thanks

.. at the risk of making myself look a bit of a goose ... what's an IOS ?

ha ha, sorry John, its iPhone Operating System for us the lazy guys.

ah .. par for the course for us old pharts ...

NOMOGRAM OF TURNING PERFORMANCE.
 

In the halcyon days when Bett Windsor's Flying Club (aka the RAF) flew real aeroplanes, without all these clever wiggly amps and gigabytes to help, we were issued with a small card/piece of paper which if I recall correctly had three/four lines of figures on it.

Angle of bank was one, TAS was another, radius of turn was the third and possibly "g" force was the fourth.

All one had to do was to put a straight edge along 2 of the values and read off the other(s) where the edge intersected the line of values.

All this tan theta and mental maths was obviously too hard for the top guns of the period and it sounds like a prime example of KEEP IT SIMPLE.

Has the K I S S principle been abandoned forever in favour of geeks and shiny touchscreens?

To the OP, I appreciate this cribsheet is not a formula but it seemed to be worth that fine institutions efforts to educate their pilot and navs.

Can any ex RAF/FAA exiles out there recall the nomogram and put what's left of my mind at ease, please?! :)

K.I.S.S.
 

φ is a Bank °

Turn Radius (r) in NM = TAS^2 / 68649 / tg φ °
Turn Radius (r) in sec=TAS / 19.07 / tg φ °

for calculatons on mobile phone calculator

Radius in NM for (φ 30°) = TAS*TAS /39683
(φ 15°) = TAS*TAS/ 18382
(φ 10°) = TAS*TAS/ 12107

for calculations from one's "top of head"

TAS 199 kt φ30 r=1 NM
φ16 r=2 NM
φ11 r=3 NM

Radius in SECONDS (φ 28°) ~ TAS / 10
(φ 15°) ~ TAS / 5
(φ 10°) ~3 xTAS /10

Turning Point= Radius (NM or KM) * tg (( next CRS – present CRS) /2) ;)

http://www.tscm.com/maneuver.pdf is one source

There is of course a very accurate formula to get the answer.

Holy Guacamole, they're even resorting to using Greek letters now!

What is the context of the OP need for turn radius?

R= NM

Figure your true airspeed TAS, tailwind component TW, and FAA std bank angle of 18 degrees;

R= (V KTAS + V KTW)^2 * 0.0000449

Non-standard bank angle;

R= [(V KTAS + V KTW)^2 * (1.4589*10^-5)] / tan(desired bank angle)

Given a turn, what is your bank angle;

Bank Angle= tan^-1 {[(V KTAS + V KTW)^2 * (1.4589*10^-5)] / R}

Rule of thumb for turning radius
 

Hi,

I was told in training as a general rule of thumb, 1% of G/S for 25-30 degree AOB for a 90 degree turn.

For example joining a DME arc at 90 deg with a G/S of 200kt, you need to initiate the turn 2nm before the arc so as not to overshoot the arc.

Of course we don't always join the DME arc at 90deg. Say if you join at 45deg, you need only 1nm in the above example.

Same applies for QDR intercepts if you are told to fly over a VOR to intercept a radial outbound.