Calculating Turn Radius Change During Bank Angle Change

[Badmachine]

Is there a standard formula for determing the rate of change for an aircraft turn radius, while the aircraft angle of bank (AoB) is also changing? (from 0 degrees AoB - 40 degrees AoB during a turn for example)

[FullWings]

Something like this?

Wolfram Alpha (http://www.wolframalpha.com/input/?i=d%2Fdt%28%28v*v%29%2F%28g*tan%28a%29%29%29)

Although I wouldn't really call it a "standard" formula...

[Checkboard]

I don't know why you want it, but you can just differentiate the radius of turn equation. :8

[CaptB737]

Just a quick calculation for radius:


R= 1/2 x TAS/100 (no wind)

D(Diameter) = TAS/100 (no wind)

Off course it is depending on bank angle as well !!!

Hope that helps ;-)

[werbil]

Radius of turn (in meters) equals 0.026978 multiplied by TAS in knots multiplied by TAS in knots divided by the TAN of the angle of bank.

[lovezzin]

You should post this in the Engineers Forum!! We deal with these kind of equations everyday in Design work. Are we assuming a constant pitch angle? Otherwise we need an additional term.

[Keith.Williams.]

Assuming that it is a balanced level turn, then
Radius of turn = V squared / g Tan AOB.

So if the V and the g stay constant and all we do is to change the AOB,
then radius is proportional to 1/Tan AOB.

[sycamore]

You will not have a balanced turn if you increase the bank and not increase the `g` at the same time ...

edit.try `googling .`nomogram of turning performance`.or csgnetwork.com/aircraft turn performance

[Basil]

For personal amusement in spreadsheet, I've used:

Rate in deg/sec = 1091*(Tan(Pi*a/180))/v

which, at 30deg 150kn gives 4.2 deg/sec

where
a = bank angle deg.
v = TAS kn.

No idea how accurate it is :confused:

[Keith.Williams.]

SYCAMORE

g is the acceleration due to gravity. We have no control over that.

If you mean that we also need to increase pitch to increase lift to maintain altitude in the turn, then you are of course correct. But that does not invaildate the equation.

[Badmachine]

Thanks for the replies.

Would this be the correct formula for calculating a bank angle:

Bank Angle = arctan(v^2/11.26r)

[Keith.Williams.]

The general shape of your equation is OK.

I am a less sure about your constant 11.26.

If the speed is in knots, and we use 32 ft/sec/sec for g, then to get the radius in feet we need to use something closer to 11.226.

[Capt Fathom]

Online calculator (http://www.luizmonteiro.com/Misc.aspx#LoadFactor) here....

[Oktas8]

Bank Angle = arctan(v^2/11.26r)

If you are interested in rate of change, the units don't matter.

The formula in SI units is (Bank Angle) = arctan (v^2 / g r)
or r = g / v^2 tan BA

Where A = bank angle and 1/tanA = cotA, and assuming that v is constant:

dr/dA = d/dA (g/v^2tanA)
= g/v^2 dcotA / dA
= g/v^2 (-cosec^2 A)
= g/v^2 (-1/sin^2 A)

If you plot radius (vertical axis) against bank angle (horizontal), at about 100kTAS, the gradient of the line at each value of bank angle is as follows. The "-" just means that as bank increases, radius decreases. The smaller the number, the less the radius will change for each 1° increase in bank.
5° -0.53
10 -0.13
15 -0.06
20 -0.03
25 -0.022
30 -0.016
35 -0.012
40 -0.010

Been years since I've done that sort of thing. Someone more engineering than I have, show me where I went wrong :)

If you want absolute values, the above formula uses radius in meters, v in m/s, g in m/s/s (9.81). Then you can convert into whatever units your bugsmasher is calibrated for...

[Badmachine]

Thanks for replies.

Regarding descending turns, what affect will changes in angle of descent have upon a change in turn radius (if any)? Online turn calculators specify they cannot calculate a turn radius for descending turns.