**4. Flutter prediction**

In this section, two flutter-prediction method, namely Zona51of Nastran from MSC Software Corporation [29], and Local piston theory, which is performed by home-made software are employed to obtain the flutter speeds.

#### **4.1 Zona51**

ZONA51, written by MSC Software Corporation, is a supersonic lifting surface theory that accounts for the interference among multiple lifting surfaces. It is similar to the Doublet-Lattice method (DLM) in that both are acceleration potential methods that need not consider flow characteristics in any wake. An outline of the development of the acceleration-potential approach for ZONA51 and its outgrowth from the harmonic gradient method (HGM) are described. ZONA51 is a linearized aerodynamic small disturbance theory that assumes all interfering lifting surfaces lie nearly parallel to the airflow, which is uniform and either steady or vibrating harmonically. As in the DLM, the linearized supersonic theory does neglect any thickness effects of the lifting surfaces.

For aeroelastic analysis, the unsteady aerodynamic forces are obtained using Doublet Lattice for supersonic flight. The rudder section was subdivided into a lattice of 20 chordwise 20 spanwise space vortex panels, yielding a total of 400 vortex panels. **Figure 14** describes aerodynamic trapezoidal panels of the rudder in **Figure 15**.

Through the flutter analysis by Nastran's ZONA51, the V-g and V-f curves of Case 1 are shown in **Figures 16** and **17**, when M = 1.35 and Density Ratio = 0.479.

From **Figure 16**, g of the first bending mode changes from the negative value to the positive at the speed of 380 m/s, and **Figure 17** presents frequencies of the second torsion mode and the first bending mode try to go toward the same value at the speed of 380 m/s, that is, 1.35 M. At this point, flutter occurs.

Through the flutter analysis, the V-g and V-f curves of Case 2 are shown in **Figures 18** and **19**, when M = 2.4 and Density Ratio = 0.327.

**Figure 15.** *DLM grid.*

**Figure 16.** *V-g curve.*

From **Figure 18**, g of the 2st-order bending mode changes from the negative value to the positive at the speed of 550 m/s, and **Figure 19** presents frequencies of the second bending mode and the first torsion mode try to go toward the same value at the speed of 550 m/s, e.g. 2.4 M. At this point, flutter occurs.

As can be seen from the preceding **Figures 16**–**19**, two cases present the same bending-torsion coupling modes that lead to flutter failure in terms of the same flutter mechanisms. However, aeroelastic flutter speeds have somewhat obvious differences, though the first two frequencies are slightly similar.

#### **4.2 Comparison of calculated methods and tests**

Due to the different aerodynamic expressions, Zona51, and Local piston theory, the flutter results are indicated in **Table 5**.

*Verification and Validation of Supersonic Flutter of Rudder Model for Experiment DOI: http://dx.doi.org/10.5772/intechopen.98384*
