Given that an aircraft would normally start a turn from straight and level, how would one go about calculating the rate/radius of turn since there would be a delay before the aircraft can get to full bank etc. I am talking airliners.
Thank you.
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$\begingroup$ You might find this answer helpful. $\endgroup$– Peter KämpfJul 18, 2021 at 14:38
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$\begingroup$ @Peter, Although the answer you reference was extremely informative, it is about roll rate and how it increases due to aileron deflection until reaching a steady stable rate. Isn't the OP's question more related to simply calculating how far the aircraft moves laterally during the delay while he/she is establishing the desired bank angle? $\endgroup$– Charles BretanaJul 18, 2021 at 14:56
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$\begingroup$ @CharlesBretana. Yes, you are right. Yo need to add knowledge contained in this answer and maybe this answer, too. $\endgroup$– Peter KämpfJul 18, 2021 at 15:07
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$\begingroup$ @Peter, another topic related to roll rates, (but also not related to this OPs question) that is very interesting is Inertial Roll Coupling. $\endgroup$– Charles BretanaJul 19, 2021 at 21:02
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$\begingroup$ @CharlesBretana Yes, and you thought I wouldn't know? $\endgroup$– Peter KämpfJul 20, 2021 at 5:25
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2 Answers
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As it takes only a few seconds for an airliner to roll into a turn, the act of rolling in will not significantly affect the calculation of turn radius.
"Full bank", generally not exceeding 20-30 degrees with passengers (who wish to be comfortable), will an importantant factor. Even more critical will be airspeed, which for an airliner, will be roughly in the range of 150 - 250 knots on approach to landing or departure.
The formula for Radius is:
R = V$^2$/(g x tangent bank angle)
Tangent bank angle is close to linear up to 30 degrees, so one may calculate (and plot) a "correction factor" based on a forward speed of 80 - 100 meters per second. Notice the turn could also be "shaped" by reducing speed or increasing bank angle (within safe limits).
And since you have your radius of turn and velocity, rate of turn in degrees per second is derived as:
Rate = 360 degrees/[(2 × radius × Pi)/True Air Speed]second
An airliner at altitude flying at full tilt (TAS > 450 knots) will usually not need maneuvering of this type.
answered Jul 19, 2021 at 3:25
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$\begingroup$ Thank you for your response. I have been using the formulas mentioned by you, however the delay in the bank (from 0 to 25 max) is causing the radius formula to not work and adding a time or distance based correction is not producing consistent results as the aircraft performance in turn seems to be heavily dependent on how wide or narrow the turn is. Thereby I was hoping for some formula to correct for this. Thanks. $\endgroup$– V3ERJul 19, 2021 at 4:43
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$\begingroup$ V3ER there is hope using one type of airliner at a given approach speed. Keep in mind wind will figure greatly as well. The pilots will have to adjust anyways to keep a standard ground track, which is why ground reference points, such as a church steeple, large building, lake, or sports field, can be very helpful in flying a pattern. $\endgroup$ Jul 19, 2021 at 5:02
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In order to calculate the radius, assuming an immediate change to 30° bank is a good first order assumption. Robert is right.
If you want to be more precise, you need to model the initial roll movement. Aileron deflection will happen within 1 second and rolling might maybe take another 2 seconds. Next, shortly before the desired roll angle is reached, ailerons are moved back to neutral. During this time the radius will shrink from infinite to the final value of the 30° bank turn. Due to the time it takes to move the ailerons, the plot of radius over time will have no sharp corners but will quickly and smoothly go from infinite to the final value where it will stay.
Again, assuming an immediate change to 30° bank does not incur a large mistake.
answered Jul 21, 2021 at 15:53