Connection between bank, radius, airspeed and rate of turn

[photofly]

Does anyone have a simple and correct way to explain all the connections between bank angle, radius of turn, rate of turn, and airspeed?

This is something TC says needs to be known.

[Schooner69A]

Turning is accomplished by banking and allowing excess lift to generate a turn. Now, disregarding ‘g’ limitations on the wing, the faster you go, the more lift you can generate towards the center of the turn (because you can then bank some more) which leads to reduced radius of turn and an increased rate of turn. In theory, you should be able to disappear up your own backside. However, at some point, a shockwave will begin to form on the wing which will slowly increase your radius of turn due to the reduction in lift but the increasing speed will allow the rate of turn to continue to increase. At some point, the circle is so big that even the rate of turn will start to fall off. Again, I reiterate, the preceding disregards ‘g’ limitations on the wing. In the real world, increasing speed would result in decreasing radii and increasing turn rates up to the point the ‘g’ limit was reached. At that point, increasing airspeed will result in an increasing radius of turn because you can't generate anymore lift towards the center of turn. ('g' limitation)

[RyanHealy29]

Turning is accomplished by banking and allowing excess lift to generate a turn. Now, disregarding ‘g’ limitations on the wing, the faster you go, the more lift you can generate towards the center of the turn (because you can then bank some more) which leads to reduced radius of turn and an increased rate of turn. In theory, you should be able to disappear up your own backside. However, at some point, a shockwave will begin to form on the wing which will slowly increase your radius of turn due to the reduction in lift but the increasing speed will allow the rate of turn to continue to increase. At some point, the circle is so big that even the rate of turn will start to fall off. Again, I reiterate, the preceding disregards ‘g’ limitations on the wing. In the real world, increasing speed would result in decreasing radii and increasing turn rates up to the point the ‘g’ limit was reached. At that point, increasing airspeed will result in an increasing radius of turn because you can't generate anymore lift towards the center of turn. ('g' limitation)

Assuming you're flying at a safe enough airspeed to make a turn, doesn't increased airspeed essentially always result in increased turn radius?

[CpnCrunch]

Turning is accomplished by banking and allowing excess lift to generate a turn. Now, disregarding ‘g’ limitations on the wing, the faster you go, the more lift you can generate towards the center of the turn (because you can then bank some more) which leads to reduced radius of turn and an increased rate of turn. In theory, you should be able to disappear up your own backside. However, at some point, a shockwave will begin to form on the wing which will slowly increase your radius of turn due to the reduction in lift but the increasing speed will allow the rate of turn to continue to increase. At some point, the circle is so big that even the rate of turn will start to fall off. Again, I reiterate, the preceding disregards ‘g’ limitations on the wing. In the real world, increasing speed would result in decreasing radii and increasing turn rates up to the point the ‘g’ limit was reached. At that point, increasing airspeed will result in an increasing radius of turn because you can't generate anymore lift towards the center of turn. ('g' limitation)

Assuming you're flying at a safe enough airspeed to make a turn, doesn't increased airspeed essentially always result in increased turn radius?

That would be true if you were using the same bank angle, but at higher speeds you can have a higher bank angle without stalling.

[RyanHealy29]

Turning is accomplished by banking and allowing excess lift to generate a turn. Now, disregarding ‘g’ limitations on the wing, the faster you go, the more lift you can generate towards the center of the turn (because you can then bank some more) which leads to reduced radius of turn and an increased rate of turn. In theory, you should be able to disappear up your own backside. However, at some point, a shockwave will begin to form on the wing which will slowly increase your radius of turn due to the reduction in lift but the increasing speed will allow the rate of turn to continue to increase. At some point, the circle is so big that even the rate of turn will start to fall off. Again, I reiterate, the preceding disregards ‘g’ limitations on the wing. In the real world, increasing speed would result in decreasing radii and increasing turn rates up to the point the ‘g’ limit was reached. At that point, increasing airspeed will result in an increasing radius of turn because you can't generate anymore lift towards the center of turn. ('g' limitation)

Assuming you're flying at a safe enough airspeed to make a turn, doesn't increased airspeed essentially always result in increased turn radius?

That would be true if you were using the same bank angle, but at higher speeds you can have a higher bank angle without stalling.

Ah! Makes sense. Thanks for the clarification.

[DanWEC]

Just a note on this topic- the formula that TC had accepted, which is: Bank angle = TAS/10 +7 seems to be suddenly defunct, as it only is accurate within a relatively narrow speed range (I think around 130 kts... though I may stand to be corrected).

The new formula that they are pushing to be taught is Bank Angle = 15% of TAS.

15% is mentally easy enough for me (and not much is....), as you just take 10%, then add half of it on again for 15.

[Colonel Sanders]

Dan: I think you are describing a rate one turn.

I think PF was asking for the general case of any
level, co-ordinated turn of any given airspeed and
bank angle:

radius = velocity^2 / G tan bank angle

http://en.wikipedia.org/wiki/Banked_tur ... eronautics

Given the radius and the airspeed, it's not hard to
figure out how long it will take to go around a complete
circle of 360 degree (circumference 2 pie R) and
hence it's rate of turn:

Time x Speed = Distance
Time = Distance / Speed
Time = 2 Pie R / Speed

Example: let's say you calculate that it will take
45 seconds to complete a 360 degree turn of an
entire circle. That yields a rate of turn of:

360 degrees / 45 seconds = 8 degrees per second

Since a rate one turn is defined as 3 degrees per
second, that is 2.7 times a rate one turn.

PF is a really, really smart guy and knows all this.
I have a horrible feeling I have fallen into some kind
of trap ...

[photofly]

Absolutely no trap.

I can work out the maths, and get all the right answers. But I was having to teach it to some not very technical people - it's in the ground school syllabus under "Flight Operations" - and I'm sure that TC doesn't expect pilots to know all the formulae.

Normally it's possible to come up with an explanation that's basically correct, intuitive, and covers all (enough) of the possibilities to answer TC's exam questions without being too technical, but for this I just couldn't.

So I wondered what other people would suggest.

Actually the syllabus says "Bank/speed vs rate/radius of turn" - so what needs to be explained? As I remember, the way this tested is with a question like:

An aircraft in a turn increases its airspeed (*) without changing its (*) rate of turn.
A) the (*) turn radius increases and the (*) bank angle decreases
B) .... decreases .... increases....
C) ...
Etc etc

(*) permute at leisure

[Colonel Sanders]

I am relieved. You are simply discussing the proportional
relationships. Back to your original question:

explain all the connections between bank angle, radius of turn, rate of turn, and airspeed?

This is going to be very difficult for a non-pilot to
understand.

First, you need to explain that an airplane, like a bicycle,
is tilted during a turn, so that all the forces are vertical.
Not like a car, which turns flat and experiences forces
pushing you towards the outside, during the turn. There
is an inclinometer or "ball" which indicates this.

This is really important, and unless the person understands
this, everything that follows is confusing rubbish.

Second, you need to explain that you are only
looking at level turns, that are flown at a
constant airspeed. This is nearly always
an implicit and unstated assumption, but in real
life turns are rarely level, and they are rarely at
a constant airspeed - nearly everyone changes
their altitude and airspeed during a turn, even if
they don't intend to.

So, now that we have established some ground
rules (ball in the center, level, constant airspeed)
we can get back to your question:

bank angle, radius of turn, rate of turn, and airspeed

I would state that there are inputs (what the pilot
does), and there are outputs - what the aircraft
does, as a result.

The pilot chooses the bank angle and the airspeed,
and the aircraft as a result flies with a particular
radius and rate of turn.

You need to explain radius and rate of turn. Radius
is pretty simple - imagine looking at the circle the
aircraft is flying, from above - but rate of turn is a
little slippery for most people. Emphasize degrees
of heading change per second.

The objective is to have the student understand
what happens (to the radius and rate of turn) when
you increase and decrease the bank angle (with a
constant airspeed).

And with a constant bank angle, what happens (to
the radius and rate of turn) when you increase and
decrease the airspeed?

If the student can answer the two above questions,
they have a strong understanding of the material,
which can be very difficult to understand without
any practical experience, for most people.

I find it helps to look at extreme examples.

For example, look at a J-3 Piper Cub at 60 mph in a
30 degree banked turn. That's a very safe airspeed
for that very slow aircraft, and it can fly a circle inside
the infield, with only 30 degrees of bank! Now, look at
an SR-71 going mach 3 at 80,000 feet. When it
puts on 30 degrees of bank, it's going to take a couple
of time zones for it to fly a complete circle. So as
airspeed goes up (with a constant bank angle) hopefully
the student will realize that the radius goes up (a lot)
and the rate of turn goes down. This is an unfortunate
problem with a really fast airplane, but the rest of us
that don't fly mach 3 have trouble generating much
sympathy!

Now to talk about bank angle. Still working the above
scenario, I might ask the student if he has ever been
to an airshow, and seen an F-16 fly a very tight circle
at very high speed and very high G, with the afterburner
on.

This leads to a discussion of the effect of bank angle.

As the aircraft goes faster and faster, more bank is
required to get the same radius and rate of turn - the
9G turn in an F-16 is a great extreme example of that.

The speedy F-16 has to pull 9 G's at 80 degrees of
bank to fly the same radius and rate of turn, as a piper
cub at 30 degrees of bank!

This is a rather long-winded explanation, and you might
wonder about the value of such a lengthy diatribe on
such a seemlingly simple question.

But the subject matter is actually, "How does an airplane
turn?" which is a fundamental and important concept
that every pilot needs to understand, at least on an
intuitive level, even if he can't write out any of the
equations (and virtually none can).

Even pilots with many licences and ratings and thousands
of hours can struggle with deceptively simple questions
on this topic, because they lack an understanding of
the basic physics.

For example, an airline pilot simply cannot understand
how I can use extremely high angles of bank in the circuit
without stalling, or pulling much G. I routinely use a 90
degree banked sideslip on short final, for example. It is
a very gentle maneuver. Infinite G is not required, and
no singularity in the space-time continuum results. We
need not even reach for a La Place transform to deal
with the resulting discontinuity.

It is terribly sad how many pilots simply don't understand
that angle of bank doesn't stall the wing - pulling G does.

Again, this is deceptively simple but understood by very
few people. This is easily proved. Visit any FTU in Canada
and they will erroneously tell you that "for safety" you
should not exceed X degrees of bank, where X is a random
number depending upon what FTU you visit. Not only
is this completely wrong, it is also a horrible lesson to
teach students, whom in a vain attempt to keep their
low-time instructors happy, use inside rudder to increase
their rates of turn with low angles of bank - "for safety",
you understand. This kills people, and that's ok, I guess,
because everyone does it except me.

[Chuck Ellsworth]

Now that was the type of teaching that everyone who fly's needs Colonel.

Logical and explained in a context everyone can picture.

I seldom read this training forum when these questions designed to look like picking fly shit out of pepper come up....

....good post..thanks.

[pdw]

Again, this is deceptively simple but understood by very
few people. This is easily proved. Visit any FTU in Canada
and they will erroneously tell you that "for safety" you
should not exceed X degrees of bank, where X is a random
number depending upon what FTU you visit. Not only
is this completely wrong, it is also a horrible lesson to
teach students, whom in a vain attempt to keep their
low-time instructors happy, use inside rudder to increase
their rates of turn with low angles of bank - "for safety",
you understand. This kills people, and that's ok, I guess,
because everyone does it except me.

Not OK. Our FTU did not allow the rudder errors ... very sticky on them. It's also true, ... "low-time instructors" have to be more on the ball to correct rudder error.

Limiting the bank angle was for safety during the education ... to learn to understand the "Connection" first. In that sense limiting the bank angle was for safety, excepting when breaking the rules (with "inside rudder" error) to "increase their rates of turn".

"Inside rudder" sure was tempting, ... but i remember that it forced you to consider turning earlier and then also came with a warning to keep the circuit pattern wider.

[Bede]

When I teach new pilots, I try to explain it using tensors so it's applicable in all inertial reference frames and spherical coordinates as well.

[Skyhunter]

I like visuals/practical descriptions, even guys like me can get it then. Maybe try this.

Get a small ball on the end of a string (or other weight). Hold on to string with ball hanging a foot or two below. This is level flight, ball not turning, string vertical (lift vector = string angle, in this case straight up).

Start spinning ball until string is close to horizontal, note how fast ball is moving, note the force increase on fingers by string (g loading), note string angle (lift vector/bank angle, note radius and rate). Slow spinning down till string is about 30 degrees from perpendicular, note the decreased ball speed, decreased radius, decreases force on fingers.

Have student hold string and repeat so they can feel difference as well as see.... now take a few minutes to relate the demonstration to an airplane.

No math, no formula, but demonstrates the relationships. Take it or leave it, tell me its stupid if you want, but think it works just fine.

Oh,, and Col, I am now one of those airline pilots... but still do high bank in the traffic pattern with a 172. Funny haven't hurt myself yet.

[Colonel Sanders]

I am now one of those airline pilots

I won't mention it to anyone - we can try to
keep it as our secret