[77] By unsteady lift is meant the lift on a wing or airfoil on which the angle of attack is changing as contrasted to the lift under steady conditions, which exist in steady flight or in most static wind-tunnel tests. The primary application of unsteady lift theory is to wing flutter, inasmuch as on airplanes, even in maneuvers, the angle of attack changes slowly enough that the lift is nearly the same as in steady flight. Other applications, however, are frequently encountered, such as in calculating the response of an airplane to gusts and in analyzing the flight of birds and ornithopters.
When I attended MIT, unsteady lift theory was considered a subject for specialists with a knowledge of advanced mathematics and was not taught in the regular courses. At Langley, a similar situation existed. A few specialists in the Physical Research Division had worked in this field and understood the analyses that had been made, but the average engineer in the wind tunnels or Flight Research Division had no knowledge of the subject.
In my work at the Flight Research Division, I soon encountered problems involving unsteady lift theory. For example, the all-movable tail on the XP-42 airplane could not be flown without some kind of flutter check. Spring tabs were known to be subject to flutter. In gust research, the acceleration on an airplane encountering a gust was used to calculate the gust magnitude producing the response. A correction factor was used based on unsteady lift theory of a two-dimensional airfoil, which I considered to have questionable validity for a finite-span wing. As a result of these problems, from time to time I looked up references on unsteady lift theory and became aware of the studies that had been made in this field.
At Langley, the recognized expert in the field was Dr. Theodore Theodorsen, head of the Physical Research Division, who had written a notable report entitled General Theory of Aerodynamic Instability and the Mechanism of Flutter (ref. 9.1). This report, published in 1935, for the first time put the subject of flutter analysis on a rigorous mathematical basis and gave practical solutions to flutter problems. Incidents of flutter had caused airplane crashes since the earliest days of aviation. For example, Lincoln Beachy, a daredevil stuntflyer who in 1913 flew a strongly braced biplane called the Lil' Looper in air shows around the country, was in a high-speed dive in a show at San Diego when the wings and ailerons fluttered, which caused the airplane to disintegrate and crash. During WW II, Anthony Fokker designed the Fokker D-VIII, one of the first monoplanes with a thick unbraced cantilever wing. The airplane had flown successfully, but a change was made that added weight and stiffness to the rear spar. This change evidently reduced [78] the flutter speed, which resulted in the death of several pilots. Since flutter usually occurs at high values of airspeed and involves the instability of a high-frequency mode of oscillation, the wing extracts tremendous amounts of energy from the air stream that results in a practically explosive increase in amplitude and disintegration of the structure.
Theodorsen's report made a very rigorous analysis of flutter theory. From the standpoint of a beginner, however, this was a poor report to study. Theodorsen outlined the mathematics in very elegant form, with many steps omitted, and gave no references to previous work. In the library, I found that only about four reports had been written on the subject of unsteady lift prior to Theodorsen's report and that someone who completely understood these reports would know about as much on this subject as any expert in the field. (I have in mind the reports by Birnbaum (ref. 9.2), Wagner (ref 9.3), Glauert (ref. 9.4), and Walker (ref 9.5)). Unfortunately, I did not have the time or mathematical background to study these reports in detail, but I did find that the analysis by Glauert, the British aerodynamicist, published in 1929 was much easier to understand than the report by Theodorsen. Why, one may ask, was the work by Theodorsen so much more widely acclaimed than that of Glauert? Theodorsen's report did, of course, include the analysis of flutter as well as the theory of unsteady lift. Also, it arrived at the much desired closed-form solution. That is, the results were obtained in terms of Bessel functions. (As shown later by 1. E. Garrick, they could also be put in terms of Hankel functions.) Glauert, on the other hand, evaluated the corresponding functions by numerical integration. From the engineer's standpoint, Glauert's results are just as useful, but from the standpoint of the mathematicians who formed the main body of researchers in the field of unsteady lift, the closed-form solution was much more desirable. I did not fully agree with this reasoning, because Bessel functions themselves had been originally evaluated by numerical methods.
The report by Walker, which had been called the greatest doctoral thesis ever written in the field of aeronautics, was fascinating to read and, through use of flow visualization, gave a clearer physical picture than any of the mathematical reports. A point of interest is that Wagner and Walker studied the response of the lift to a step change in angle of attack, whereas Theodorsen and Glauert considered sinusoidal variations in angle of attack. Mathematicians knew that the two approaches should give equivalent results for the linear systems considered, provided the analyses were correct. Later Garrick, who worked for Theodorsen, put out a report verifying that the two sets of results were in agreement. At that time, neither I nor most other engineers knew that the results were equivalent, a fact that further confused the complicated field of unsteady lift.
My own efforts in the field of flutter and unsteady lift were primarily concerned with two projects, a device to reduce the mass-balance weight required to prevent flutter of spring tabs and an attempted clarification of the role of unsteady lift in explaining snaking oscillations. The following sections will discuss these projects.
Spring-Tab Flutter
The spring-tab problem arose because, as shown by Theodorsen, a reliable way to avoid control surface flutter was to mass balance the surface about its hinge line. If this rule was applied to a spring-tab system, both the tab and control surface would have to be mass balanced. Then additional mass-balance weight would have to be used on the control surface to balance the tab mass-balance weight. This weight penalty seems bad enough, but some British reports had shown that the tab mass balance had to be located very close to the tab hinge line to be effective, which means that this balance weight had to be considerably greater than the weight of the tab itself. The reason for this conclusion is shown in figure 9.1. When....
....the control surface is moved with the control stick held fixed, the tab ordinarily moves in the same direction as the control surface, but through a greater deflection. This movement results from the kinematics of the tab linkage, which are determined by the desire to have the control follow a quick stick deflection with very little overshoot. For prevention of flutter, an angular acceleration of the control surface should cause the tab to jump ahead of the control surface. A little thought will show that the tab must jump ahead of the position where it would be under static conditions. Referring to figure 9.1, it may be seen that a balance weight located at the intersection of the tab and airfoil centerlines would do no good because it would not move. One located at the tab hinge line would be ineffective because it would exert no moment about the tab hinge. The best location for the mass-balance weight is half way between these two points. The result of the heavy tab balance weight and the added weight of the main control surface balance to balance the tab balance weight was an undesirably large weight penalty. I devised a linkage arrangement in which the tab balance weight could be located ahead of the main surface hinge line, where it did not add to the weight behind the hinge line and yet would cause the tab to jump ahead of its static position in a sudden control movement. A drawing of the linkage is shown in figure 9.2. A memorandum was written for the Engineer in Charge, Dr. H. J. E. Reid, proposing this device. As a result, a discussion was held with Dr. Theodorsen, who was familiar with the problems of tab flutter, but who did not appear to know of the recent British reports on the subject. He felt that the device would prevent low-frequency flutter, but might be ineffective in preventing flutter of some....
....high-frequency modes involving flexibility of the structure or the linkage. He proposed building an installation of the system in an actual control surface to allow measurement of the high-frequency modes.
With the endorsement of Dr. Theodorsen, therefore, a vertical tail was obtained from an Air Force B-34 bomber (similar to the Lockheed Hudson) and the system was installed (figs. 9.3 and 9.4). This tail surface was suitable for installation in a wind tunnel for testing as part of an ongoing series of flutter tests.
When Theodorsen was invited to the hanger to see the installation, he hauled back with his fist and hit the tail a solid blow, which caused it to shudder and shake. His comment was, "It will flooter!" Later vibration tests were made, which showed that there were resonant modes at 14 and 25 hertz in which the balance weight moved oppositely from its motion at low frequencies. Most of the flexibility was in the thin-skinned structure of the tail, whereas the linkages themselves were quite rigid. As a result of these tests, it was concluded that the device might be subject to high-frequency flutter, and no further studies were made.
I have heard recently that the elevator on the Curtiss C-46 airplane did have a device similar to the one I tested to move the tab balance weight ahead of the control surface hinge line, though most actual spring-tab installations on service airplanes did not have balance weights on the tab, and the main provision to avoid flutter was to make the tab as light as possible. These installations were tested in flight at gradually increasing speeds to verify that they did not flutter within the desired flight envelope. It therefore appears....
.....that the British analyses requiring the use the balance weights for absolute prevention of flutter were unduly conservative. Also, as mentioned previously, the need for springtab controls rapidly disappeared with the introduction of hydraulic powered controls.
Effect of Unsteady Lift on Snaking Oscillations
A peculiarity of the two-dimensional unsteady lift theory of Theodorsen and other investigators is that the curve showing phase angle of the lift as a function of frequency in an oscillation of angle of attack initially has an infinite negative slope. This result means that at very low frequencies of oscillation, the force in phase with the velocity of the motion is much larger than the force in phase with the displacement. If the surface were used as a vertical tail, for example, this theory would indicate that in an oscillation of low frequency, the tail itself might tend to make the oscillation unstable.
The prevalence of snaking oscillations on many airplanes, as mentioned previously, made it tempting to look for some basic explanation of the oscillation based on aerodynamic theory. Reports by Smilg (ref. 9.6), a flutter expert at Wright-Patterson Field, and by Pinsker (ref. 9.7), a well-known aerodynamicist at the British RAE, made it clear that serious thought was being given to this explanation. Personally, I was well satisfied with the explanation involving control system friction that has been discussed previously, and I felt that it was incorrect to apply [82] two-dimensional theory (that is, a theory that applies to a wing of infinite span) to the case of a low aspect ratio vertical tail. Proving this belief, however, was a difficult problem.
A remarkable report had been published in 1940 by Robert T. Jones on The Unsteady Lift of a Wing of Finite Aspect Ratio (ref. 9.8). Jones had avoided the complex mathematics of previous investigators by considering the response to a step change in angle of attack, then converting this result to the response to sinusoidal oscillations. To make this conversion in a simple way, Jones fit the variation of lift following a step input with an exponential curve. I was not sure what effect this approximation would have at low frequencies; therefore, I attempted to make an analysis similar to that of Jones, but working from the start with a sinusoidal input.
Today I would have less concern about the accuracy of Jones' results, but at that time I was less familiar with the transformation between step and sinusoidal inputs. I hoped to be able to make an analysis in which the mathematics, at least, was exact, thereby obtaining a direct basis of comparison with Theodorsen's two-dimensional results.
The basis of the analysis in Jones' theory was to convert the vortex system of an infinite wing, which consisted of a bound vortex and a starting vortex, to that of a finite wing by superimposing a pair of horseshoe vortices that may be called "canceling vortices." These vortices cancel the parts of the vortex system beyond the wing tips and add the trailing vortices of the finite wing. I used the same technique, but allowed all these vortices to vary sinusoidally with time and position. The effects of each element of each vortex on the downwash on the wing was obtained by integration. The resulting integrals were evaluated with the help of Keith Harder, a mathematician in the Physical Research Division. All the integrals could be found in tables of integrals except one term of one integral. This term was evaluated by graphical integration. As a result, I did not realize my hope of obtaining a closed-form mathematical solution.
In 1952, the results were put in the form of a report in which my analysis and the work of various investigators in the field of unsteady lift of finite wings were compared, and the effect of these results on lateral oscillations of airplanes was demonstrated. The report was reviewed by an editorial committee consisting mainly of members of the Physical Research Division, including mathematicians who had devoted much of their careers to studying unsteady lift effects. These mathematically inclined individuals were much more concerned with the mathematical development than with the application to airplane oscillations. They concluded that the theory really did not contribute any new results in unsteady lift theory, and therefore they recommended that the report should not be published. I was disappointed that the paper was not published, mainly because I felt that the two-dimensional unsteady lift theory had been frequently applied incorrectly in the past. Also, I felt that my report would make the subject more understandable to the average engineer.
An important point in the presentation of my
results was the method of expressing reduced frequency. In previous
reports on unsteady lift theory, the frequency had been expressed in
nondimensional form by use of the parameter 
c/V where is the frequency of the
oscillation in radians per second, c is the wing chord, and V is the
airspeed. I found that at low frequencies, the parameter b/V, where b is
the wing span, was much more suitable. This parameter would be
meaningless in the two-dimensional case because the span b would be
infinite. In the case of the finite wing, however, the results
showing the phase lag of the lift was nearly independent of aspect
ratio when the reduced frequency based on span was used. This result
shows that at low frequencies, the trailing vortices, which disappear
in the two-dimensional case, remain important and, at low
frequencies, are mainly responsible for the phase...
as functions
b/V, where is the frequency in radians per
second, b is the wingspan, and V is the airspeed.
....variations of the lift. The results indicate that the slope of the phase angle curve remains finite as the frequency approaches zero, though my analysis, because of the use of graphical integration, did not demonstrate this point rigorously.
To illustrate these points, figure 9.5 shows the variation of the amplitude and phase angle of the circulatory lift with frequency based on wing span. Note that the two-dimensional case can not even be shown when the frequency is based on wing span. The damping of oscillations based on two-dimensional and three-dimensional theories are shown in figure 9.6. The damping based on the three-dimensional theory is almost the same as that based on conventional stability theory, called the quasi-static case, which considers the angle-of-attack change at the tail due to yawing velocity, but then neglects unsteady lift considerations altogether. The damping based on two-dimensional theory, on the other hand, is considerably less over the whole range of frequencies shown.
One beneficial result of this project was that I obtained a working knowledge of unsteady lift theory. I prepared a talk on this subject that minimized the mathematical aspects. This talk was presented on several occasions to engineers working in my division. Since....
, where is frequency
in radians per second, b is span of weather vane, and v is velocity.
Frequency varied by changing the inertia.
....this work was completed, the use of high-speed computers has greatly extended the work on unsteady lift, which allows more accurate calculations for wings with various planforms and mode shapes at speeds throughout the Mach number range. This field still remains a subject for specialists, though some joint programs have been conducted in recent years to combine the efforts of control theorists, structural dynamicists, and unsteady lift specialists to allow applications of control theory to predict flutter of a flexible structure and to design automatic flutter damping systems.
