According to an analysis of the data on several airplanes in which wing flutter has been observed, and on others that showed no tendency to flutter, Kiissner found in 1929 that, for airplane wings with mass- unbalanced ailerons, of the type of construction prevalent at that time, wing-aileron flutter occurs when the reduced frequency is lower than the following critical value:
^cr = 2t/ = 0.9 ±0.12 (1)
where U = the mean speed of flow, feet per second; со = the fundamental frequency of the wing in torsional oscillation in still air, radians per second; and c — the chord length of the vibrating portion of the wing, feet.
For safety against flutter, the reduced frequency should be higher than k„. In other words, the design speed of the airplane must be lower than
This is Kiissner’s well-known formula. The frequency со may be determined by ground vibration experiments in still air or computed by a
theoretical analysis. It increases with increasing stiffness of the structure. Therefore, the critical speed can be raised by increasing the wing stiffness. From this point of view we see that when the lower limit of Uct is prescribed (e. g., the maximum speed of flight) a minimum value of w, and hence a minimum value of the torsional rigidity is also prescribed.
Thus a stiffness criterion can be specified for the purpose of flutter prevention. If we return to the parameters used at the beginning of the last section, Kiissner’s formula can be expressed in the form
> const №
If the inequality sign is satisfied, flutter will not occur.
The stiffness criterion, regarded as a specification of the wing stiffness with regard to flutter prevention, is convenient for a designer.
Such stiffness criteria arise also in the consideration of steady-state instabilities (cf. Chapters 3 and 4). Most types of aeroelastic instabilities can be avoided by sufficiently high structural stiffness. For the safety against each type of instability, inequalities of the following form are to be satisfied.
Ко — torsional stiffness of the wing =
Kh — flexural stiffness of the wing
maximum bending moment x semispan
linear deflection of tip
s = semispan
c — mean chord of the wing
The constants in the above equations depend on many parameters, such as the type of structural construction, the locations of the elastic axis and the center of mass of the wing, the moment of inertia of the sections, the amount of mass balancing. The final form of the stiffness criteria can be obtained for a particular type of structure only after consideration has been given to all the possible instabilities.
As an example, let us quote the case of flutter of a thin-walled circular cylindrical shell of uniform thickness in a supersonic flow along the axis of the cylinder. Such a shell is used often in large liquid-fueled rockets, particularly at the interstage area. Experiments in a wind tunnel revealed that such a shell may have several types of oscillations: the small amplitude random oscillations, the sinusoidal flutter oscillations, and the sinusoidal flutter motion whose amplitude varies periodically over the circumference of the cylinder and moves as a traveling wave. Tested at a Mach number of 2.49, the amplitude of the last two types of oscillations would suddenly rise at the critical condition
in which q = ipU2, is the dynamic pressure of the main flow, R is the radius of the middle surface of the circular cylinder, h is the thickness of the shell wall, E is the Young’s modulus of the shell material, and /3 = V M2 — 1 is a function of the Mach number. In a limited range of experiments this critical condition is independent of the internal pressure as long as it is positive. The critical condition (6) may be regarded as a stiffness criterion. Details of the experiments are given in the author’s paper, Ref.