. THE EXTRACTION OF ENERGY FROM AIRSTREAM

Since flutter is an oscillation induced by the aerodynamic forces without any external source of energy other than the airstream, it is possible only if the oscillating body, the mean position of which is assumed stationary, can extract energy from the airstream.* Hence, the possibility of flutter can be discussed by considering the energy relation.

An oscillation will be called aerodynamically unstable if the oscillating body gains energy from the airstream in completing a cycle. If the oscil­lating body has neither external excitation nor internal friction, then the aerodynamic instability can be identified with flutter. Internal friction dissipates energy, external excitation imposes a source of energy exchange; both modify the kinematic relations (the amplitude ratio and phase relationship between various degrees of freedom) of the oscillation of an aeroelastic system. Hence when there is external excitation or internal

* The inertia force and the elastic force are both conservative and do not contribute any net gain or loss of energy in each complete cycle of motion.

friction, the aerodynamic instability alone cannot be directly identified with flutter.

Consider an airfoil performing a vertical translatory oscillation with a constant amplitude h0. Let the vertical displacement be described by the expression

h = hQeiad (1)

We shall define h as positive downward. The speed of downward motion is therefore

h = icoh0eiwt (2)

where a dot indicates a differentiation with respect to time. If h were a constant, the downward motion will induce a lift force L0 on the airfoil:

Подпись:Lo==2 pU8-d^U

where pU2 is the dynamic pressure and S is the wing area. The lift is defined as positive upward in the usual sense. When the airfoil is oscil­lating, the true instantaneous lift acting on the airfoil differs from L0 both in magnitude and in phase. Let us call L0 the quasi-steady lift and write the true instantaneous lift as

L = (4)

Then r represents the ratio of the absolute value of the instantaneous lift to that of the quasi-steady lift, and if the phase angle by which the actual lift leads the quasi-steady value. The quantities r and if depend on the reduced frequency k, the Mach number M, and the Reynolds number R. For a nonviscous incompressible fluid, r and if are functions of к alone. The ratio L/L0 — гегу> can be plotted as vector with length r and angle ip. Such a vector diagram, for a flow of an incompressible fluid (M = 0), with к as a parameter, is given in Fig. 5.1. The theoretical derivation of this diagram will be given in Chapter 13.

When the airfoil moves through a distance dh, the work done by the lift is, in real variables,

dW = — Ldh= —Lhdt (5)

It must be recognized that, when L and h are expressed in the complex forms 1 and 4, the physical quantities are represented only by the real parts of the complex representations. For example, the physical dis­placement h represented by Eq. 1 is h0 cos cot. The proper form of the Work, in the complex representation, is therefore

Подпись: (5 a)dW=~ Rl [L] ■ Rl [h] dt

. THE EXTRACTION OF ENERGY FROM AIRSTREAM

Integrating through a cycle of oscillation, we obtain the total work done by the air on the airfoil:

Подпись: 2Ґ da. (6)

Hence, the gain of energy W by the airfoil from the airstream is propor­tional to (— cos y>). If — Я-/2 < if < Я-/2, W is negative; i. e., the oscil­lating airfoil will lose energy to the airstream. The oscillation is therefore stable. If we refer back to Fig. 5.1, it is seen that the condition is satisfied. Hence, in a nonviscous incompressible fluid, the vertical translation oscillation is aerodynamically stable.

This example shows the importance of the phase angle between the aerodynamic force and the oscillatory motion. Although purely trans­lational flutter is impossible, it is conceivable that, when several degrees of freedom are involved, a certain combination of the phase relations will render the energy input to the airfoil positive.

Thus the fundamental cause of flutter is quite clear. The airfoil, by adjusting its phase shift, extracts energy from the airstream. In fact, the airfoil can be regarded as a flutter engine.5-2 The fact that the phase shift and amplitude ratio of the flexural and torsional wing motions which follow an imposed disturbance depend largely on the speed of flow over the wing is of fundamental importance. It is this dependence that makes flutter occur at certain critical speed of flow.

By calculating the energy input from the airstream the stability of more complicated motions can be determined. The bending-torsion case, in an incompressible fluid, has been calculated by J. H. Greidanus5 43 whose result, with a slight addition, is reproduced in Fig. 5.2. The airfoil is assumed rigid. The vertical translation, called bending, is denoted by h and is positive downward. The rotation about the 1/4-chord point, called torsion, is denoted by a and is positive nose up. The fluid is assumed nonviscous and incompressible, and the linearized two-dimensional aerodynamic theory is used. Let Ё be the mean work per unit time done by the aerodynamic force per unit span in a harmonic oscillation of

Подпись: Fig. 5.1. Vector diagram of lift in vertical translation oscillation. A vector drawn from the origin to an appropriate point on the curve gives a complex number (real part, abscissa; imaginary part, ordinate) that is the value of LjL0 given by the linearized theory. Curve given by von Karman and Sears, Ref. 13.21. (Courtesy of the Institute of the Aeronautical Sciences.)
Подпись: frequency со (radians per second). The expressions given by Greidanus are

Ё = {:77pc4co3a02Cs

CE =Mk)P – Шк) sin ф + ffk) cos ф]І ~ і

where

I = dimensionless ratio /г0/(«ос) h = h0eiu>t

Here ф represents the phase lag of the torsion behind the bending. The functions /1; /2, /3 are functions of the reduced frequency к derived from the theory of oscillating airfoils which will be presented in Chapter 13. All critical oscillations are given by the equation

CE = 0 (9)

The solutions of this equation are plotted in Fig. 5.2. Inside each loop, CE is positive and the oscillation is unstable.

. THE EXTRACTION OF ENERGY FROM AIRSTREAM

Fig. 5.2. Energy coefficient in bending-lorsion oscillations according to Greidanus, Ref. 5.43. (Courtesy of the Institute of the Aeronautical Sciences.)

For U -»• 00, к 0, the entire half-strip | > 0, 0 < < я – becomes

unstable, к -» 0 also when со -»• 0 if V remain finite, then Ё tends to zero as fast as со3.

For U -> 0, к -> oo, only one limiting point reaches the critical con­dition of neutral stability. This is

І = і, Ф = tt

which implies that the wing is oscillating about the 3/4-chord point, because the downward displacement at a point located at a distance xc behind the 1/4-chord point is

z = h0eM + ос0хсе1<-ы~ф) (10)

which vanishes when x = f and ф = ъ. When £ = 1/2, the 3/4-chord point is stationary. Thus, in the absence of structural damping, a wing reaches the critical flutter condition at zero airspeed if the oscillation node is

located at the 3/4-chord point. In practice, structural damping always exists and flutter in this case does not occur; but Biot and Arnold5-42 have shown that, if the nodal line of oscillation of a wing is located close to the 3/4-chord line, flutter at low airspeed is likely to occur.

For intermediate values of U, к is finite. The loops of instability become smaller as к increases.

From Eq. 10, the location x where the amplitude of |z| becomes a minimum can be calculated. The result is

X = — I COS ф (11)

It coincides with the 3/4-chord point if £ cos <f> = —1/2, a relation repre­sented by a dotted curve in Fig. 5.2, which appears to pass right through the dangerous area. This indicates again that flutter is liable to occur if the node of oscillation is located near the 3/4-chord point.

It is particularly interesting to consider the possibility of one-degree-of- freedom flutter. A purely translation motion corresponds to a0 = 0 or £ = oo, which by Fig. 5.2 is stable for all к > 0, confirming a result obtained previously. A purely rotational motion exists if there is a node. From Eq. 10 we see that z vanishes at all time t if and only if

h0eiml + «0схет~ф) = 0 (12)

Eq. 12 is satisfied by pitching about the 1/4-chord point, corresponding to x = h0 = 0, which, for к > 0, is seen from Fig. 5.2 to be stable for all values of ф between 0 and tt. For nonvanishing x, Eq. 12 is satisfied if

(a) ф = п, x = I (13)

(b) ф = 0, x = — I

Case a is the oscillation about the 3/4-chord point discussed above. Case b leads to flutter, according to Fig. 5.2, only if к < 0.0435, and for axis of rotation located forward of the 1/4-chord point, yet not too far forward of the airfoil leading edge. This torsional flutter was first found by Glauert in 1929. It is discussed in detail by Smilg.5-50 Recently, several types of single-degree-of-freedom flutter involving control surfaces at both subsonic and supersonic speeds have been found,5.45-5.49 ац requiring the fulfillment of certain special conditions on the rotational-axis locations, the reduced frequency, and the mass moment of inertia. Pure – bending flutter is possible for a cantilever swept wing if it is heavy enough relative to the surrounding air and has a sufficiently large sweep angle.5-46

The preceding analysis is purely kinematical. It does not tell how the phase shift ф and the reduced frequency к will vary with the flow speed U for a given structure. The latter information must be obtained from a consideration of the balance of the inertia and elastic forces witH the
aerodynamic forces. Thus a dynamic analysis is necessary to determine which point on a figure such as Fig. 5.2 corresponds to a given structure at a given airspeed U. Such a dynamic analysis is the subject matter of the next two chapters.

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