*(November 2011)* It is a bracing feeling to stand up and deny accepted knowledge. So bracing, in fact, that I try to do it as often as possible. I have argued (countless times) that downwind turns are no different from upwind ones, dismissed as a wives' tale the common notion that the horizontal tail of an airplane always provides a downward force, and attributed P factor to several things but not to the supposedly increased angle of attack of the downgoing blade. I have denounced countless physics textbooks for their stupid explanations of wing lift, but defended Bernoulli against the assaults of revisionist Newtonians.

I am always on the lookout for some piece of accepted wisdom to deny, and so I was immediately interested when a friend sent me a copy of an article with the title “Questioning the Overbanking Tendency.” I did not have to read far to discover the author’s answer: “For all practical purposes,” he states, “there is no such thing as an overbanking tendency.”

My fellow antinomian has the sublime satisfaction of contradicting the FAA’s Airplane Flying Handbook, which he quotes: “As the radius of the turn becomes smaller, a significant difference develops between the speed of the inside wing and the speed of the outside wing. ... This creates an overbanking tendency that must be controlled by the use of the ailerons.” He prefers the view of Charles Zweng in his 1946 classic Flight Instructor: “Overbanking tendencies approach their minimum in steep turns. The steeper the turn, the less they are present.”

So here we have two apparently authoritative sources taking diametrically opposite views of the question. Does it matter? Not really. We use our ailerons to create and maintain a bank angle without troubling ourselves about their absolute positions. The overbanking tendency, if it exists, must be quite weak, at least at normal bank angles; no one complains of tired arms after making a two-minute turn. But still, the question is an interesting one. Why is there disagreement about it?

To be sure, my author is hedging his bets. Notice the weasel words, “for all practical purposes,” by which he situates himself in neutral territory somewhere between Zweng and the FAA. Zweng, after all, allows that there are overbanking tendencies, but claims that they have the mysterious property of existing only at moderate bank angles. The FAA merely asserts that overbanking tendencies must be controlled with aileron, but does not say how much; so it’s not really sticking its neck out either. None of our commentators on this important topic seems to be willing to make a definite, quantitative statement.

But I am.

The argument for the existence of the overbanking tendency, as the FAA handbook says, is that since the outer wing in a turn is slightly farther from the center of the circle than the inner wing is, it moves a little faster, and so it has more lift. The argument against it is — I assume — that as the bank steepens the radii of turn of the two wings get closer and closer together, and so the difference in speed and lift must shrink. In a 90-degree bank, after all, the difference is zero; where is the overbanking tendency now? Of course, in a coordinated 90-degree bank the turn radius is zero and the G-loading is infinite, so ordinary arithmetic fails us.

The question is best illustrated with a concrete example. Let’s use a hypothetical Cherokee-like 2,500-pound airplane of 35-foot wingspan. For the sake of simplicity, let’s look at relative speeds at points 20 feet apart along the wing. This is a bit beyond the midpoint of each wing panel, but lift is a function of the square of speed and so whatever speed-related effects arise in turning flight are going to be biased toward the outer part of the wings. We’ll try bank angles of 10, 30, 60 and 80 degrees. Let’s say that our true airspeed is 119 knots — a convenient 200 feet per second — which yields turn radii of 7,043, 2,157, 718 and 220 feet respectively and G-loadings of 1.015, 1.155, 2.0 and 5.76. As the wings tilt increasingly, the 20 feet measured horizontally between our reference points shrinks, becoming 19.7 feet at 10 degrees, 17.3 at 30, 10 at 60 and 3.5 at 80.

Taking the 60-degree bank as an example, since it is the steepest coordinated turn that most pilots are likely to encounter, the outer wing is five feet farther from the center and the inner one five feet closer. If the radius of the turn is 718 feet, their radii are 723 and 713 feet respectively. The ratio of those two numbers, 1.014, is the ratio of the speeds of those points on the wing. In other words, the outer wing travels 1.4 percent farther than the inner in the same period of time, and so must be going 1.4 percent faster.

If we do the same calculation for each bank angle, we find that the ratio increases gradually, but not very rapidly, as the bank angle increases. At 89 degrees, the point 10 feet out along the outer wing is moving 2 percent faster than the corresponding point on the inner.

Now, the overbanking tendency, if it exists, is what is called a “couple” — a force applied at a distance from an object’s pivot, such as to make the object rotate. The pivot in this case is the rolling axis of the airplane — roughly, a line running from nose to tail down the middle of its fuselage — and the effect of the couple, if any, is to make the airplane roll into a steeper bank. The total force consists of an excess of lift on the outer wing and a deficit on the inner, and the lever arm, which is the same regardless of the bank angle, is 10 feet for each wing.

To get the size of this couple, or “rolling moment,” at 60 degrees of bank, we use the square of the speed ratio, because lift is proportional to the square of speed. The square of 1.014 is 1.028. The outer wing therefore has 2.8 percent more lift than the inner. Now, the total lift is 5,000 pounds, because this is a 2-G turn, and so it turns out that the inner wing’s lift is 2,465 pounds (5,000 / 2.028), the outer one’s 2,535, and the rolling moment is 700 pound-feet (35 x 10 x 2).

I assume that by now your eyes have glazed over or you’re experiencing uncomfortable flashbacks to high school, but that’s all right. This is all just a back-of-an-envelope exercise to get a sense of the magnitude of the overbanking force, if such there be. It’s not exact, because we’re using a single point on the wing to represent the entire wing, and all these distances and forces really vary along the wing. They’re smaller closer to the fuselage and larger out toward the wingtip.

But we have a rough idea now of the overbanking force in a 60-degree bank, and it seems quite large. The corresponding figures for the other bank angles are 140 pound-feet at 10 degrees, 230 at 30 and 2,270 at 80 degrees.

So, just as common sense and the FAA would suggest, there is an overbanking tendency, and it is not negligible, and it increases with bank angle. How large is it, in practical terms?

At this point the assumptions become more arbitrary and the math more complicated. Let us agree, to paraphrase Mr. Bennett in Pride and Prejudice, that I have delighted you long enough with my calculations, and that you are willing to take my word for the rest. It happens that a plain flap like an aileron produces a change in lift, per degree of deflection, equal to about 45 percent of the change that would be produced by changing the angle of attack of the flapped portion of the wing by a like amount. It turns out that in a 60-degree bank, a total aileron deflection of around two degrees, up plus down, is required to nullify the 700-pound-foot rolling moment. The corresponding figures for 10, 30 and 80 degrees of bank are 0.4, 0.7 and 7 degrees.

More important than the magnitude of the ailerons’ deflections, however, is the fact that they are required at all. If you left the ailerons untouched, the bank angle would increase of its own accord.

The author of the article purporting to debunk the overbanking tendency used as an example an Arrow doing a 50-degree banked turn at 100 knots. He concluded that the difference in lift between the midspan points of the two wings was 1.27 percent, and that this was “insignificant” in comparison with the full rolling moment available from the ailerons. Indeed it is, just as running a car into a brick wall at 10 mph is insignificant compared with doing so at top speed. But there is no logical reason to compare the magnitude of the overbanking tendency to the full rolling capability of the airplane. The question is merely whether some opposite aileron is needed to keep a steep bank angle from getting steeper, and the answer is yes.

All of the foregoing is, I admit, only an intellectual exercise. I find it interesting to reflect on various aspects of how airplanes fly, and I hope you do; but the actual flying is instinctive and tactile and does not require any arithmetic at all. You roll into a bank, you stop the roll, and you adjust the bank angle as necessary. You do not look out at the wings to see precisely where the ailerons are, and even if you did it would do you little good, because they are seldom perfectly rigged and tend to float upward in flight anyway. At any rate, to overcome the overbanking tendency never requires more than a few degrees of aileron deflection.

To summarize, then, for those who go straight to the end to know how the story turns out:

1. There is an overbanking tendency. The FAA is right.

2. The overbanking tendency does not diminish at large bank angles — quite the contrary. Zweng was wrong.

3. The reason for teaching pilots about the overbanking tendency is to get them to think about the dynamics of flight. It is not necessary to think about it while flying; it takes care of itself.

4. Unless you read it here, don’t believe everything you read.

*Send reader mail to: edit@flyingmag.com or P.O. Box 8500, Winter Park, FL 32789.*

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