# BIBLIOGRAPHY

FLUTTER PHENOMENON

5.1 THE PHENOMENON OF FLUTTER

A type of oscillation of airplane wings and control surfaces has been observed since the early days of flight. To describe the physical phenomenon, let us consider a cantilever wing, without sweepback and without aileron, mounted in a wind tunnel at a small angle of attack and with a rigid support at the root. When there is no flow in the wind tunnel, and the model is disturbed, say, by a poke with a rod, oscillation sets in, which is damped gradually. When the speed of flow in the wind tunnel gradually increases, the rate of damping of the oscillation of the disturbed airfoil first increases. With further increase of the speed of flow, however, a point is reached at which the damping rapidly decreases. At the critical flutter speed, an oscillation can just maintain itself with steady amplitude. At speeds of flow somewhat above the critical, a small accidental disturbance of the airfoil can serve as a trigger to initiate an oscillation of great violence. In such circumstances the airfoil suffers from oscillatory instability and is said to flutter.*

Experiments on wing flutter show that the oscillation is self-sustained;

i. e., no external oscillator or forcing agency is required. The motion can maintain itself or grow for a range of wind speed which is more or less wide according to the design of the wing and the conditions of the test. For a simple cantilever wing, flutter occurs at any wind speed above the critical. In other instances, for example, in flutter involving aileron motion, there may be one or more ranges of speed for which flutter occurs, and these are bounded at both ends by critical speeds at which an oscillation of constant amplitude can just maintain itself.

The oscillatory motion of a fluttering cantilever wing has both flexural and torsional components. A rigid airfoil so constrained as to have only the flexural degree of freedom does not flutter. A rigid airfoil with only the torsional degree of freedom can flutter only if the angle of attack is at or near the stalling angle (“stall flutter,” Chapter 9), or for some special mass distributions and elastic-axis locations. In ordinary circumstances, oscillations of a control surface (aileron, flap, etc.), in a single degree of

* In the following text, the terms flutter speed and flutter frequency refer to the critical flutter speed and the frequency at the critical condition.

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freedom, are also damped at all speeds unless a flow separation is involved. Let us restrict the term ‘‘’flutter” to the oscillatory instability in a potential flow, in which neither separation nor strong shocks are involved[13] Then, in general, the coupling of several degrees of freedom is an essential feature for flutter. The steady oscillation that occurs at the critical speed is harmonic. Experiments on cantilever wings show that the flexural movements at all points across the span are approximately in phase with one another, and likewise the torsional movements are all approximately in phase, f but the flexure is considerably out of phase from the torsional movement. It will be seen later that mainly it is this phase difference that is responsible for the occurrence of flutter.

The importance of phase shift between motions in various degrees of freedom suggests at once the importance of the number of degrees of freedom on flutter. An airplane wing, as an elastic body, has infinitely many degrees of freedom. But owing to its particular construction, its elastic deformation in any chordwise section can usually be described with sufficient accuracy by two quantities: the deflection at a reference point, and the angle of rotation about that point, i. e., the flexural and the torsional deformations, respectively. Similarly, for a control surface, such as a flap or an aileron, its freedom to turn about the hinge line is so much more important than its elastic deformation, that ordinarily it is possible to describe the deflection of a control surface simply by the angle of rotation about its hinge line. In general, then, it is sufficient to consider three variables in wing flutter: the flexure, the torsion, and the control-surface rotation. A flutter mode consisting of all three elements is called a ternary flutter. In special cases, however, two of the variables predominate, and the corresponding flutter modes are called binary flutter modes. Similar consideration applies to airplane tail surfaces. In fact, most airplanes can be replaced by a substitutional system of simple beams, so that the elastic deformation can be described by the deflection and torsion of the elastic axes of the substitutional beams, in addition to the rotation of control surfaces about their hinge lines.

These degrees of freedom, together with the freedom of the airplane to move as a rigid body, offer a large number of possible combinations of binary, ternary, and higher modes of flutter. Since it is not clear which of these modes correspond to the actual critical speeds, it is necessary

either to resort to experiments or, in a theoretical approach, to analyze all cases. This is why a successful flutter analysis depends so much on the analyst’s experience. He must be able to choose, among all possible modes, those that are likely to be critical for a given structure.

Since flutter analysis is a rather extensive subject, we shall divide our discussions into three chapters. In the present chapter only some general considerations based on dimensional arguments will be given. The role of the elastic stiffness and the mass balancing in flutter prevention are explained on the empirical basis. The origin of flutter from the aerodynamic point of view is then considered in § 5.4. It will be shown that flutter occurs because the speed of flow affects the amplitude ratios and phase shifts between motion in various degrees of freedom in such a way that energy can be absorbed by the airfoil from the airstream passing by. Some remarks on the experimental approach to the flutter problem are given in § 5.6, and the dynamic similarity rules are discussed in § 5.7. A brief historical review of the earlier developments is included in § 5.8. The details of the dynamical process, however, are left for the next two chapters.

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