Natural Frequencies Correlation
Attention has been focused on the lower frequency modes because they are the most significant for the dynamic behavior of the structure, especially for the flutter phenomenon. The wind tunnel tests have shown that the main flutter instability involves the 1st and 3rd modes, respectively, out-of-plane symmetric bending and symmetric torsional of the wing.
Table 7 compares the frequencies of the first three modes of the mass-backward configuration since they are the ones used for the optimization process. The flutter results have been checked using different sets of modes, reaching a total of 40 modes to compute the response and looking at the possible variations. The result remained unaffected.
Starting from the stick model results, a very limited error can be noted (wheb dealing with modal frequencies, a threshold error of 1–2% is usually considered). Nevertheless, its prediction for flutter behavior is very bad, as described below in this section, demonstrating that a good correlation level on modal frequencies is not sufficient to also have the same level for aeroelastic ones. The transition from the stick model to the hybrid one has not produced better results concerning the the frequency errors but has paved the way for a better description in terms of mode shapes. For this reason the updating process has been applied to the hybrid model, thus setting up an optimization problem aiming at the minimization of the frequency errors of the first three low frequency modes, i.e., the first bending and two torsional ones working mainly on the stiffness of the wing spar and the wing–fuselage connection.
During the first step, updating only three model parameters has updated, i.e., the three wing spar thicknesses describing the Omega section, keeping them fixed spanwise, so to correct the wing stiffness distribution in a global fashion. In the objective function the frequencies to be reached have been set equal to Hz, Hz, and Hz.
As already anticipated, the reason why the frequencies
and
do not coincide with GVT’s results is due to the fact that the numerical model is perfectly symmetric showing two torsional modes (i.e., symmetric and anti-symmetric) with the same modal participation of left and right half wings. On the contrary, the actual physical model shows a slightly asymmetric behavior able to split any torsional mode into two separate modes with different participation on left and right side half wings (see
Figure 12). This partially justifies the high error on the second torsional mode. In terms of the final design variables, the solution obtained shows a thickening of the vertical webs and side flanges of the spar, which intends to increase the torsional inertia and so the related frequencies, and a lightening of the central web that tries to decrease the bending frequency.
During the second step, a more extended model tuning was applied to improve the correlation level. The set of design variables were defined looking at the sensitivity of modal results showing that:
Decreasing the Young modulus of the wing root connection elements allows us to decrease the first bending frequency.
Decreasing the Young modulus or the thickness of the top flange of the spar mainly decreases the bending frequency.
Changing the webs and flanges thicknesses of the spar impact on the torsional frequencies.
At the same time, further modification to the finite element model was applied, such as:
Transverse shear flexibility set in order to have an infinitely rigid in transverse direction plate and trying to increase the torsional stiffness of the wing.
Classical and not burdensome slender body aerodynamics formulation of the pod was introduced during the flutter analysis, to better reproduce the flutter behavior of the updated model.
The second updating step allowed us to improve the correlation level that will be now discussed in detail. Looking at the modal frequencies reported in
Table 7 and the Cross MAC matrices ([
24]) of
Figure 13, it is possible to draw the following remarks:
With the last optimization run, the error committed on the modal frequencies falls below an adequate threshold, thus achieving an excellent level.
Starting from Hybrid model, the updating process has managed to reduce the error at every step, as can be seen in
Table 7. The algorithm has impacted the target frequencies of 0.72% for the 1st symmetric bending, 6.28% for the 1st anti-symmetric torsional, and 6.4% for the 1st symmetric torsional, during the first optimization loop; the values were 5.05%, 1.92%, and 2.19% during the second loop.
The 1st first bending mode has been captured very well, both in terms of frequency and shape. This is very important because the flutter mechanism involves the exactly 1st bending and the 2nd torsional (mode # 3).
Due to the presence of two split torsional modes, as identified during the GVT campaign and reported
Figure 12b,c, due to the fact that the actual wind tunnel model is not perfectly symmetric, the correlation results are still poor.
Despite this low correlation related to the torsion modes, the frequency error reduction as well the model improvement allowed us to capture the global aeroelastic behavior and it will be now discussed.
Concerning the flutter instability, commonly it is studied in terms of
and
plot typically obtained using the well known p-k method [
32], where the aeroelastic damping
g is the quantity tracked to identify flutter onset; the instability is associated with negative damping, reflected in a positive value of
g.
In such a case, these values have been extracted through post-processing activities. No dedicated Operational Modal Analyses were carried out.
Figure 14 and
Figure 15 depict the trend of the three modes involved in the flutter mechanism and compare the experimental results with the numerical ones. The preliminary wind tunnel tests had the scope to identify the flutter; no active flutter suppression controllers were available at that time. Thus, the experimental velocities were limited by the flutter point (
Table 3). The error in predicting the flutter velocity is summarized in
Table 8. During these tests, the presence of unsteady aerodynamic forces practically made the left/right torsion difference no more significant and almost disappeared during the identification.
Looking at
Table 7 and
Table 8 it is possible to say that both step 1 and step 2 updated models have produced significant results in terms of error reduction with respect to the experimental flutter test results. In the first instance, the hybrid model alone did not improve the error in flutter velocity prediction but did prepare the ground for the optimization process. In fact, after the second updating loop, the final model was the best compromise between the errors in the flutter velocity and frequency prediction and the aeroelastic behavior represented in the
plot.
Of special interest to us is the correlation obtained after the updating process in the case of the model with the mass of the safety flutter device in the forward position, which has not been directly involved in the updating. Indeed, since in reality the only difference between the two configurations is represented by the position of two concentrated masses on the tips, it is reasonable to expect some progress in the correlation level, demonstrating the global meaning of the model changes made.
Concerning the modal parameters, the modes participating in flutter refined from time to time their correlation, in particular the 1st bending mode (
Table 9 and
Figure 16). For the two torsional modes, the improvements were limited by the non-symmetry of the real model, as explained previously.
Looking at the flutter behavior reported in
Figure 17 and
Figure 18, the
and
plots have confirmed what was expected: the updating process applied to the FEM model with the safety mass in the backward positions produced an updated model able to capture the global aeroelastic behavior also in the case of the safety masses in the forward position.