1 Flutter Computation Examples – Simple Cases Flutter Computation Examples A binary aeroelastic system has the following expression Find the stiffness value k that gives a critical flutter speed of V=250 [m/s] and compute the corresponding flutter frequency. Assuming the following solution of the generalized coordinates: with

2 Flutter Computation Examples – Simple Cases Flutter Computation Examples Substituting the expression of the generalized coordinates into the aeroelastic equation, we get For solutions other than the trivial one, we need to impose the determinant of the coefficients to be zero: or which becomes

3 Flutter Computation Examples – Simple Cases Flutter Computation Examples Remembering that and substituting these expressions in the previous equations, two expressions result, one for the real part and the other for the imaginary part: one obtains real part imaginary part

4 Flutter Computation Examples – Simple Cases Flutter Computation Examples Flutter Condition: gives V = 250 [m/s] and real part imaginary part from to from to

5 Flutter Computation Examples – Simple Cases Flutter Computation Examples From the imaginary part, excluding the value ω = 0, we obtain: from which and Which can be substituted into the expression of the real part, resulting in: from to

6 Flutter Computation Examples – Simple Cases Flutter Computation Examples Assuming now V=250 [m/s], the approximate solutions come from the following quadratic equation: In order to compute the divergence speed, we should recall that at the divergence speed the determinant of the stiffness matrix goes to zero. This results in: giving or, consequently

7 Flutter Computation Examples – Simple Cases Flutter Computation Examples Considering k=k 1 the flutter frequency, computed by the equation becomes f=11.33 [Hz] and the divergence speed equal to Considering k=k 2 the flutter frequency, computed by the equation becomes f=3.14 [Hz] and the divergence speed equal to Note that in the case k=k 2 we obtain a divergence speed lower than the flutter speed, therefore in this case the flutter has no physical meaning (the static aeroelastic phenomenon comes before the dynamical one)

8 Flutter Computation Examples – Simple Cases Flutter Computation Examples Another interesting consideration comes when we consider zero speed, which is the case of the GVT (Ground Vibration Test). In such case the equation of motion, neglecting the structural damping (conservative case) becomes: And therefore the frequencies at zero speed are: It may be worth to observe the behaviour of the two frequencies (mode- shapes) as the speed increases.