THE EFFECT OF STRUCTURAL PARAMETERS OF THE WING ON THE CRITICAL SPEED OF THE TORSION-FLEXURE FLUTTER
Under the assumptions of § 6.3 and using the method of § 6.4, Grossman proved a number of relations expressing the effects of structural parameters of the wing on the critical torsion-flexure flutter speed :f
1. A simultaneous change of the flexural and torsional rigidities by a factor n changes both the critical flutter speed and the flutter frequency by a factor Vn, and leaves the critical reduced frequency unchanged.
For, other things remaining equal, a multiplication of El and GJ
by a factor и’changes the values of the coefficients au, a22, etc., by factors listed below, as can be easily verified according to Eqs. 4, 6, 9 of § 6.4:
Coefficients changed by a factor n: au, a22, Cx, D1; E2, M.
Coefficients changed by a factor n2: Ev N.
Coefficients unaffected: hX2, b22i d^, d±2, d2x, d22, clx, c12 = c2X, c22,
Equations 7 and 10 of § 6.4 then show the conclusion of the theorem at once. The constancy of the reduced frequency follows by definition.
The effect of the individual changes of the flexural and torsional rigidities cannot be stated in such general terms. Examples of flutter analysis generally show that, when the torsional rigidity alone is increased, the flutter speed is also increased, but, when the flexural rigidity alone is varied, the change in critical flutter speed is small. The flutter speed reaches a minimum when the flexural rigidity becomes so high that the frequency of (uncoupled) flexural oscillation is equal to that of the (uncoupled) torsional oscillation. Further increase of flexural rigidity increases the flutter speed.
An important consequence of the above result concerns the accuracy required in determining the rigidity constants: It is permissible to admit considerable error in the flexural rigidity of a wing without causing serious error in the calculated critical flutter speed.
2. A similar investigation gives the following: A change in all the geometric dimensions of a wing by a factor n without a change in the elastic constants (E and G) has no effect on the magnitude of the critical speed, but changes the flutter frequency by a factor 1/я. The critical reduced frequency remains unchanged.
As for the effect of variation of individual geometric parameters, the results of sample calculations can be stated most concisely in the following form, which, however, cannot be proved without introducing further assumptions in addition to those stated in § 6.3:
3. When the characteristic geometric dimensions of a wing are varied individually, with the mass density and elasticity distributions remaining unchanged (while the absolute values of the mass density and the torsional stiffness may vary), the “apparent” reduced frequency for torsion-flexure flutter
remains approximately unchanged. In the formula above, c represents
the chord length at a reference section, and coa the fundamental frequency of the torsional oscillation of the wing.*
Equation 1 is an approximate empirical rule whose validity must be questioned when unconventional wing designs are considered.
4. The effects of the relative positions of the elastic, inertia, and aerodynamic axes are so important that each particular case should be computed separately. Generally speaking, the closer the inertia and elastic axes are to the line of aerodynamic centers, the higher is the critical flutter speed.