Formula for Radius of Turn

[aviationluver]

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

[9.G]

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.

[Microburst2002]

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!

[mustafagander]

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.

[9.G]

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

[mcdhu]

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

[28L]

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

[Genghis the Engineer]

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

[Skunkworks]

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

[ITCZ]

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

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.

[BOAC]

How's about for pilots we keep it simple? We ALWAYS turn procedurally at Rate 1, don't we? 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...............]

:

[Keith.Williams.]

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!

[flybywire380]

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

I would agree with mcdhu.

[Microburst2002]

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

[mm43]

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

[BOAC]

Originally Posted by me

We ALWAYS turn procedurally at Rate 1, don't we?

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

[Artisan]

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.

[ITCZ]

How's about for pilots we keep it simple?

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.

We ALWAYS turn procedurally at Rate 1, don't we? FORGET angle of bank

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.

This IS techlog, is it not?

[BOAC]

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.

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

[mustafagander]

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!!