LIMITATIONS OF THE THEORY

It is of basic importance to remember that all we discussed above is based on the linearized theory. Since real physical phenomena are not linear, the question always arises how good the linearized theory is as an approximation to the real case, and to what order of magnitude of the variables concerned is the linearized theory valid.

Unfortunately, so little is known about the nonlinear case that the questions so raised cannot be answered. At present, it can only be said that experimental evidences show that the linearized theory of flutter represents fairly closely the real situation in the neighborhood of the critical flutter speed, provided that the amplitude of motion remains neither too small nor too large.

Phenomena that disagree with the linearized theory are often attributed to the nonlinearity of the system. For example, it is often observed that it is possible to exceed the critical flutter speed without encountering flutter, whereas a sufficiently large disturbance may at once initiate flutter with great violence. This is often regarded as a consequence of the nonlinear characteristics of the structural damping, particularly of dry friction. Kiissner7,42 also points out that at very low amplitudes the laws of potential flow do not hold, because of the effect of viscosity of the air. If the amplitude is of the order of the thickness of the boundary layer, the aerodynamic forces induced by the oscillations are probably smaller than would be expected from the potential theory. It seems plausible to assume that a disturbance of certain minimum value is required to initiate flutter.

On the other hand, violent flutter motions cannot be treated by the linearized theory. Thus it is impossible, within the scope of the linearized theory, to trace the divergent flutter motion. Flutter has been observed whose amplitude does not increase indefinitely at super critical speeds. On the contrary, definite maximum amplitudes are often recorded. The prediction of the violence of the fluttering motion can be made only if the nonlinear characteristics of the structures, as well as those of the aerodynamic forces, are allowed.

For control surfaces, the natural frequencies often depend on the amplitude of oscillation, because of dry friction. This may cause some peculiar behavior in control-surface flutter.

Even within the framework of a linearized theory, flutter analyses are not generally made to their full logical extent. Additional assumptions are introduced to simplify the calculation. Examples of these are: (1) A three-dimensional body is replaced by a system of simple beams. (2) The elastic model is replaced by a mechanical substitutional system having only a finite number of degrees of freedom. (3) The “strip” assumption is used to simplify the aerodynamic expressions. (4) The compressibility effect of the air is sometimes neglected. (5) The aerodynamic coefficients are computed for flat-plate airfoils at zero mean angle of attack.

. Of these additional simplifying assumptions, the first and the second have been discussed in § 7.2. The effects of the remaining assumptions vary with the wing planform, Mach number, and flutter mode. They are subjects of current research.

The Effect of Finite Span. The effects of finite span on flutter are complex. The trailing vortices shed from the wing as a result of the spanwise variation of circulation, induce a downwash distribution that is not always negligible. This induced downwash field, however, depends not only on the geometric aspect ratio but also on the mode of deforma­tion. Its theoretical prediction is naturally complicated. Owing to the complexity, flutter analysis based on the finite-span theory is rarely made.

A few examples, incorporating three-dimensional wing theory, seem to show that an increase of the critical flutter speed of the order of 10 to 15 per cent above that computed by the strip theory might be expected as a result of the finite-span effect. The effect is more pronounced for wings of small aspect ratio and low reduced frequency, but tends to be negligible for high-frequency oscillations.

The Effect of the Compressibility of the Fluid. For a flow of sufficiently high Mach number, say M > 0.5, the aerodynamic coefficients differ considerably from those for an incompressible fluid. As was shown in § 6.10, however, the formal flutter analysis can be made in the same way for all Mach numbers.

In a subsonic flow, for Mach numbers below the critical Mach number Garrick7110 concluded that, for an ordinary wing of normal density and low ratio of bending-to-torsion frequency, the compressibility correction to the flutter speed is of the order of a few per cent. There are experi­mental indications that this is true through the transonic speed range, provided that the wing is not stalled.

The character of a supersonic flow differs entirely from that of a sub­sonic flow, and the’ effect of the compressibility of air must be taken into account. Large effect of compressibility at supersonic speeds is expected, not only because of the larger effect of finite span on the transient lift distribution, but also because the flutter modes of a supersonic wing can be very different from those of subsonic wings. For example, a delta wing may exhibit a flapping motion at the wing tip, which cannot be described adequately by bending and pitching about an elastic axis. Whereas under static loading condition it may be a good approximation to assume that the streamwise cross sections remain rigid, such an as­sumption is in general not very good in the flutter analysis of a delta wing. In other words, the location of the nodal line becomes very important in the flutter problem. The use of normal modes of free vibration of the wing is very helpful in such cases. The flutter mode usually approaches one of the higher modes of free vibration.

In the transonic-speed range, the effect of shock waves on the aero – elastic properties of a wing are not yet entirely clarified. The legitimacy of treating transonic-flow problems by means of linearized aerodynamic equations is often questioned, although it has been shown that the familiar singularity (of infinite lift) at M = 1, which occurs in a steady flow, disappears if the flow is unsteady; thus there exists no fundamental contradiction within the linear theory itself (see references 14.26-14.31).

Airfoil Angle of Attack, Camber, and Thickness. Since in flutter calculation only the deviation from the wing’s steady-state configuration need be considered, it is implied by the linearized theory that the actual

angle of attack, camber, and thickness of the airfoil section have no effect. However1, experiments do show the effect of finite angle of attack, camber, and thickness of the wing. The effect of angle of attack can often be detected at angles considerably below the static stalling angle. This is particularly evident for thin wings at transonic speeds. Finite angle of attack usually results in a reduction of the critical flutter speed. As the angle of attack approaches the static stalling angle, very severe drop in critical flutter speed occurs, accompanied by other important changes in the fluttering motion. This is the stall flutter to be discussed in Chapter 9.

When great accuracy is desired, the linearized strip theory alone can hardly be trusted. Experimental model investigations, flight tests, etc., must be performed in conjunction with the theoretical analysis.