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.”
He has the honour 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, even 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 that they have the mysterious property of existing only at moderate angles of bank. The FAA merely asserts that overbanking tendencies must be controlled with aileron, but does not say how much; so this is not really a debatable proposition 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 centre 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 angle increases 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 aeroplane 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 centre 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% farther than the inner in the same period of time, and so must be going 1.4% 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 ten feet out along the outer wing is moving two 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 point, such as to make the object rotate. The pivot point in this case is the rolling axis of the aeroplane – roughly, a line running from nose to tail down the middle of its fuselage – and the effect of the couple is to make the aeroplane 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% more lift than the inner. Now, the total lift is 5,000 pounds, because this is a 2G 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 * 10 * 2).
I assume that by now your eyes have glazed over, 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. How large is it, in practical terms? To figure that out, we need to know how those rolling moments compare to ones generated by various amounts of aileron deflection.
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% 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-foot-pound 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 mid- span points of the two wings was 1.27 percent, and that this was “insignificant” in comparison to the full rolling moment available from the ailerons. Indeed it is, just as running a car into a brick wall at 20 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 aeroplane. The question is merely whether some opposite aileron is needed to keep a steep bank angle from getting steeper, and the answer is yes.
To summarise, then, for those who go straight to the end to know how the story turns out:
- There is an overbanking tendency. The FAA is right.
- The overbanking tendency does not diminish at large bank angles – quite the contrary. Zweng was wrong.
- 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.
- Don’t believe everything you read.