Turn Rate Formula

[Badboy]

Originally posted by clerick
When i was unpacking some books i ran across an old one that came with Falcon Gold WAY back in the 90's.  It's called "Art of the Kill", interesting book but in it the author mentions a formula for turn rate

TR = K G/V

G = the g's the aircraft is pulling in the turn
V = is velocity

but K is only listed as a constant.  I would assume that this constant varies from plane to plane.  Could any of you explain what K represents and how it is derived?

Hi clerick

That formula is perfectly correct. Pete Bonanni wouldn't get that sort of thing wrong. He is simply using a slightly different definition of the terms than those normally used in order to keep the expression simple in order to focus on the concepts. The value you are looking for is K = 1257 when the speed is in miles per hour.

So if you are turning at 4g and 200mph your turn rate will be:

1257*4/200 = 25 degrees per second (dps)

The more complex formula others have posted would yield 24.3dps. The difference is due to the fact that Bonanni is using Radial G whereas the formula others have posted are actually using the Body axis normal load factor, and not the radial G.

If I use typical spread sheet math symbols and use n for the body axis normal load factor, the radial G can be found by using Pythagoras's theorem as G = SQRT(n^2 - 1).  that's where that term under the square root comes from in those other equations.

Hope that helps.

Badboy




Edit, forgot to mention where the 1257 came from:

It's just gravity times a conversion factor to get miles per hour to ft/sec and another conversion factor to get radians per second to degrees per second. Using spread sheet math again it looks like this:

=32.185/(5280/(60*60))*(180/PI()) = 1257.3168

You should be able to see what each of those numbers represents.

[dtango]

Ah, thanks for clarifying Badboy.  Radial G vs. normal load factor.

Tango, XO
412th FS Braunco Mustangs

[Gianlupo]

Badboy, question: what do we read on the g-meter? The radial G or the body normal axis load factor?

[Badboy]

Originally posted by Gianlupo
Badboy, question: what do we read on the g-meter? The radial G or the body normal axis load factor?

The g-meter will read the body axis normal load factor, because that’s the way it is orientated, which will be slightly less than the load factor due to lift (L/W) which in turn will be slightly more than the radial G if the turn is carried out at constant altitude.

Badboy

[MWL]

Greetings,

  My brain hurts.......where's the asprin.

Regards,

[clerick]

thanks for the help

[Gianlupo]

Originally posted by Badboy
The g-meter will read the body axis normal load factor, because that’s the way it is orientated, which will be slightly less than the load factor due to lift (L/W) which in turn will be slightly more than the radial G if the turn is carried out at constant altitude.

Badboy

Ok, thank you, but... another question arises. I thought the body axis normal factor was L/W, clearly it's not. So, what is the body axis normal factor? And how is it mathematically defined? Thanks for the patience, Badboy!

[Scherf]

I wanted to do AeroEng at one point.

I can only say, in hindsight, that it's a darn good thing for the air-going public that I didn't.

[Badboy]

Originally posted by Gianlupo
Ok, thank you, but... another question arises. I thought the body axis normal factor was L/W, clearly it's not. So, what is the body axis normal factor? And how is it mathematically defined? Thanks for the patience, Badboy!

They are almost the same, but not quite, because the lift acts normal to the free airstream, and the aircraft axis is tilted back relative to that due to angle of attack. Because the AoA is normally small, the difference is generally less than about 4%. So the G force acting normal to the aircraft's longitudinal body axis, is (L/W)*Cos(Alpha). Here is a diagram:



EDIT:

Sorry about the F-16s, I originally drew the diagram for a discussion about Jets...  Also just to correct myself, what Bonanni has done in his simplified equation is to assume that the AoA is negligible and can be ignored, and that the aircraft is banked at 90 degrees, so that the one vertical G isn't needed.  You get, a slightly better turn rate by doing that in practice, at the expense of loss in altitude.

Hope that helps.

Badboy

[Bruv119]

hmmm interesting thread.  Shame Gian's avatar is distracting my line of thought.

I'd hit that !

[Gianlupo]

Got it, Badboy, thank you again.

Bruv, the purpose of my avatar is just to show my actual location on an innovative kind of geographic map.

[clerick]

Maybe i can help explain the difference between "axis body normal" and "radial" g in a different way.

     In the fantastic multi media diagram i "made" the aircraft is pulling 5 "axis body normal" g's, the numbers i put on the diagram show the radial g's.




     On the entrance and exit the ABN g's are 5 but the Gr is 4.  So in affect the pilot feels the full 5 g's but the aircraft is only pulling 4 "effective" or "radial" g's because gravity is pulling down the ABN.

     When perfectly vertical the pilot still feels 5 g's and now the aircraft is pulling 5 effective g's because gravity is pulling perpendicular to the ABN.

     At the top the pilot is still feeling 5 g's but now the aircraft is pulling 6 radial g's because gravity is pulling up along the ABN.

As i understand it, and i am open to correction, the radial g's are what actually turn the aircraft, the ABN g's are what affect the pilot.