*2.3.2 Damping analysis*

At altitude 9.65 km, the Mach number of air velocity was varied from 0.29 to 0.59 (M = 0.29; 0.35; 0.41; 0.47; 0.53; 0.59). From Eq. (4), the aerodynamic damping coefficient was calculated and presented in **Figure 5a**. The zero damping

meshed in 9257 nodes and 1350 elements (**Figure 3a**). The fluid domain in CFD problem was meshed in 67,949 nodes and 279,535 elements (**Figure 3b**).

*Computational grids. (a) AGARD 445.6 wing grids. (b) Fluid computational grids.*

**Material property E11 E22 E33 G υ ρ** Value 3.151 0.416 0.416 0.439 0.310 397.5 Unit GPa GPa GPa GPa N/m2 kg/m<sup>3</sup>

and from 0.47 M to 0.73 M at an altitude of 14 km.

*Mechanical properties for the weakened AGARD 445.6 wing.*

**2.3 Results**

**58**

**Figure 3.**

**Figure 2.**

*Aerodynamics*

**Table 1.**

*Semispan AGARD 445.6 wing model.*

*2.3.1 Modal analysis*

The free-stream air velocity was from 0.29 to 0.59 M at an altitude of 9.65 km

The mode shapes are obtained from the finite element analysis of the modeled wing. The deflection contours between the modal analysis and experiment [13] were compared as shown in **Figure 4**. The natural frequencies between the developed solution, experiment [13], and other researches were also compared as shown in **Table 2**. It could be concluded that the obtained results were in good agreement

coefficients were interpolated at Mach number 0.46. While the experimental zero damping was 0.499 in [13], the difference between simulation results and results of [13] was about 8%.

At altitude 14 km, the Mach numbers of air velocity were varied from 0.47 to

0.73 (M = 0.47; 0.53; 0.59; 0.65; 0.67; 0.73). From Eq. (4), the aerodynamic damping coefficient was calculated and presented in **Figure 5b**. The zero damping coefficients were interpolated at Mach number 0.58. While the experimental zero damping was 0.678 in [13], the difference within simulation results was about 14%.

*Research on Aeroelasticity Phenomenon in Aeronautical Engineering*

*DOI: http://dx.doi.org/10.5772/intechopen.91748*

*Vibration of wing at altitude H = 9.65 km. (a) M = 0.29 – ζ = 0.0071. (b) M = 0.47 – ζ = 0.000125.*

**Figure 6.**

**61**

*(c) M = 0.53 – ζ = 0.006.*


### **Table 2.**

*Natural frequencies of modal analysis.*

**Figure 5.** *Aerodynamic damping coefficients. (a) At altitude 9.65 km. (b) At altitude 14 km.*

## *Research on Aeroelasticity Phenomenon in Aeronautical Engineering DOI: http://dx.doi.org/10.5772/intechopen.91748*

coefficients were interpolated at Mach number 0.46. While the experimental zero damping was 0.499 in [13], the difference between simulation results and results of

Modal analysis 9.96 41.5 52.95 99.99 Kolonay [4] 9.63 37.12 50.50 89.94 Erkut and Ali [6] 10.85 44.57 56.88 109.10 Yates [13] 9.6 38.1 50.7 98.5

**Mode 1 Mode 2 Mode 3 Mode 4**

[13] was about 8%.

*Aerodynamics*

*Natural frequencies of modal analysis.*

**Table 2.**

**Figure 5.**

**60**

*Aerodynamic damping coefficients. (a) At altitude 9.65 km. (b) At altitude 14 km.*

At altitude 14 km, the Mach numbers of air velocity were varied from 0.47 to 0.73 (M = 0.47; 0.53; 0.59; 0.65; 0.67; 0.73). From Eq. (4), the aerodynamic damping coefficient was calculated and presented in **Figure 5b**. The zero damping coefficients were interpolated at Mach number 0.58. While the experimental zero damping was 0.678 in [13], the difference within simulation results was about 14%.

#### **Figure 6.**

*Vibration of wing at altitude H = 9.65 km. (a) M = 0.29 – ζ = 0.0071. (b) M = 0.47 – ζ = 0.000125. (c) M = 0.53 – ζ = 0.006.*

#### *Aerodynamics*

In both the two considered altitudes (9.65 and 14 km), the numerical results agreed well with the experimental results of [13] with a relative error about 14%. This difference would be from the computation such as the quality of mesh and order of model in CFD and CSD.

would create the damage of the wing such as loss of control for the flap, aileron,

*Research on Aeroelasticity Phenomenon in Aeronautical Engineering*

*DOI: http://dx.doi.org/10.5772/intechopen.91748*

two of the most important problems in the dynamic aeroelastic analysis.

simulation settings were appropriate for solving the transonic flow.

Dynamic aeroelastic analysis was a problem related to fluid-structure interaction over a period of time. Therefore, the quality of aerodynamic grid and the time step strongly influenced the results of aeroelastic analysis. These parameters were also

In order to evaluate the quality of aerodynamic grid, the coefficient of pressure of AGARD 445.6 wing was first estimated at 26% semispan and at 75.5% semispan and then was compared with Ref. [6] as shown in **Figure 7**. The presented results were in good agreement with the results in Ref. [6]. It could be concluded that these

To evaluate the time step size, three different time sizes were examined such as 0.001, 0.002, and 0.005 s. As it could be seen in **Figure 8**, the displacement of the wing tip was reduced with the reduction of time step size up to 0.002 s until the aeroelastic simulation did not change [6]. Therefore, the value 0.002 s of time step size in the numerical solution was chosen for both aerodynamic and structural

The limit of flutter was identified by using damping estimations for a large test point at each Mach number. At M = 0.499, the oscillation of the displacement of the wing tip was harmonic (**Figure 9**), and it was considered as a flutter point. At this limit of flutter, the air speed was calculated as 174.26 m/s, and the density of air

values: 172.46 m/s for flutter speed and 0.428 kg/m<sup>3</sup> for density of air (**Table 3**).

. These values were very close to the experimental

fracture of wing, etc.

*2.3.3 Simulation analysis*

analysis.

**Figure 8.**

**Figure 9.**

**63**

was calculated as 0.432 kg/m3

*Wing-tip oscillation depends on the selected time step.*

*Wing-tip oscillation of flutter point at M = 0.499.*

For more details of the stability of the wing structure, the vibrated value of the wing tip position at three Mach numbers were plotted as shown in **Figure 5**.

At a Mach number smaller than the flutter value, the damping coefficient was positive, and the wing was stable (**Figure 6a**).

At a Mach number near the flutter value, the damping coefficient was zero, and the vibration was harmonic oscillation (**Figure 6b**).

At a Mach number greater than the flutter value, the damping coefficient was negative, and the vibration was divergent (**Figure 6c**). This divergent vibration

would create the damage of the wing such as loss of control for the flap, aileron, fracture of wing, etc.
