**2. Two-way FSI method**

### **2.1 FSI method**

The generalized equations of aeroelasticity motion:

$$[M]\{\ddot{q}(t)\} + [C]\{\dot{q}(t)\} + [K]\{q(t)\} = \{F(t)\} \tag{1}$$

Following [8], the first four modes of vibration were sufficient to accurately model the wing structure response. So, in order to determine the time step of the unsteady problem, a modal method was applied to estimate first the natural frequency of the first four modes and then calculate the time step using the following

*Research on Aeroelasticity Phenomenon in Aeronautical Engineering*

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

The flutter velocity was estimated from the vibration of the wing tip position, the most dangerous position of the wing [5, 8, 9, 13]. From the variation of this position, the damping coefficient except the influence of structural damping was measured. It means that there was an effect of aerodynamic damping coefficient on the vibration of the wing structure. The aerodynamic damping coefficient was

The damping coefficient increased/decreased when the air velocity decreased/ increased. The type of vibration was distinguished from the value of damping

• ζ > 0: The vibration was convergent. The wing structure was stable.

• ζ < 0: The vibration was divergent. The wing structure was unstable.

• ζ = 0: The vibration was harmonic oscillation. The wing was in critical state.

AGARD 445.6 wing with aspect ratio of 1.65, taper ratio of 0.66, and 45<sup>o</sup> sweep angle at quarter chord line was studied as seen in **Figure 2**. The cross section of the wing was NACA65A004 airfoil in the stream-wise direction. This NACA65A004 airfoil was a symmetric airfoil with a maximum thickness of 4% of the local chord. The dimensions of the wing were root chord of 0.558 m, tip chord of 0.368 m, and semispan of 0.762 m. The wing model used in aeroelastic experiments [13] was constructed by laminated mahogany which was modeled as an orthotropic material with different material properties in different directions. The properties of the laminated mahogany are given in **Table 1**. The modal analysis was performed using mechanical APDLs solver to evaluate the accuracy of the constructed model.

The AGARD 445.6 wing was modeled at a zero attack angle and at altitudes of 9.65 and 14 km, the same conditions of experimental study in [13]. The wing was

ð3Þ

ð4Þ

formula:

where: Δt: time step f: natural frequency

calculated as follows:

Xi: ith peak of vibration. n: number of periods.

ζ: aerodynamic damping coefficient.

The air velocity was in the flutter velocity.

where:

coefficient:

**2.2 Wing model**

**57**

$$\left\{\mathbf{w}\left(\mathbf{x},\boldsymbol{\mathcal{y}},\boldsymbol{z},t\right)\right\} = \sum\_{i=1}^{N} q\_i\left(t\right) \left\{\phi\_i\left(\mathbf{x},\boldsymbol{\mathcal{y}},\boldsymbol{z},t\right)\right\} \tag{2}$$

where:

w: structural displacement at any time instant and position.

q: generalized displacement vector.

[M]: generalized mass matrice.

[C]: damping matrice.

[K]: stiffness matrices.

ϕ: normal modes of the structure.

N: total number of modes of the structure.

F: generalized force vector, which is responsible for linking the unsteady aerodynamics and inertial loads with the structural dynamics.

Eq. (1) shows that there are distinct terms representing the structure, aerodynamic, and dynamic disciplines. This equation was solved numerically by integrated CFD-CSD tool on the ANSYS software. This method was also called the two-way fluid-solid interaction (FSI). Aerodynamic loads were first calculated by CFD solver. Then, these loads were used to calculate the structural response of the wing structure through the fluid-solid interface. By using CSD solver, the structural deflection was estimated, and the mesh in each time step was deformed. The simulation of the two-way FSI is presented in **Figure 1** [10].

**Figure 1.** *Two-way FSI algorithm.*

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

Following [8], the first four modes of vibration were sufficient to accurately model the wing structure response. So, in order to determine the time step of the unsteady problem, a modal method was applied to estimate first the natural frequency of the first four modes and then calculate the time step using the following formula:

$$
\Delta t = \frac{1}{20\,f} \tag{3}
$$

where: Δt: time step f: natural frequency

**2. Two-way FSI method**

The generalized equations of aeroelasticity motion:

w: structural displacement at any time instant and position.

F: generalized force vector, which is responsible for linking the unsteady aero-

Eq. (1) shows that there are distinct terms representing the structure, aerodynamic, and dynamic disciplines. This equation was solved numerically by integrated CFD-CSD tool on the ANSYS software. This method was also called the two-way fluid-solid interaction (FSI). Aerodynamic loads were first calculated by CFD solver. Then, these loads were used to calculate the structural response of the wing structure through the fluid-solid interface. By using CSD solver, the structural deflection was estimated, and the mesh in each time step was deformed. The

q: generalized displacement vector. [M]: generalized mass matrice.

ϕ: normal modes of the structure.

N: total number of modes of the structure.

dynamics and inertial loads with the structural dynamics.

simulation of the two-way FSI is presented in **Figure 1** [10].

[C]: damping matrice. [K]: stiffness matrices. ð1Þ

ð2Þ

**2.1 FSI method**

*Aerodynamics*

where:

**Figure 1.**

**56**

*Two-way FSI algorithm.*

The flutter velocity was estimated from the vibration of the wing tip position, the most dangerous position of the wing [5, 8, 9, 13]. From the variation of this position, the damping coefficient except the influence of structural damping was measured. It means that there was an effect of aerodynamic damping coefficient on the vibration of the wing structure. The aerodynamic damping coefficient was calculated as follows:

$$\frac{1}{m}\ln\left(\frac{X\_i}{X\_{i\leftrightarrow n}}\right) = \frac{2\pi\zeta'}{\sqrt{1-\zeta'^2}}\tag{4}$$

where: Xi: ith peak of vibration. n: number of periods. ζ: aerodynamic damping coefficient.

The damping coefficient increased/decreased when the air velocity decreased/ increased. The type of vibration was distinguished from the value of damping coefficient:


#### **2.2 Wing model**

AGARD 445.6 wing with aspect ratio of 1.65, taper ratio of 0.66, and 45<sup>o</sup> sweep angle at quarter chord line was studied as seen in **Figure 2**. The cross section of the wing was NACA65A004 airfoil in the stream-wise direction. This NACA65A004 airfoil was a symmetric airfoil with a maximum thickness of 4% of the local chord. The dimensions of the wing were root chord of 0.558 m, tip chord of 0.368 m, and semispan of 0.762 m. The wing model used in aeroelastic experiments [13] was constructed by laminated mahogany which was modeled as an orthotropic material with different material properties in different directions. The properties of the laminated mahogany are given in **Table 1**. The modal analysis was performed using mechanical APDLs solver to evaluate the accuracy of the constructed model.

The AGARD 445.6 wing was modeled at a zero attack angle and at altitudes of 9.65 and 14 km, the same conditions of experimental study in [13]. The wing was

#### **Figure 2.**

*Semispan AGARD 445.6 wing model.*


#### **Table 1.**

*Mechanical properties for the weakened AGARD 445.6 wing.*

**Figure 3.** *Computational grids. (a) AGARD 445.6 wing grids. (b) Fluid computational grids.*

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**).

The free-stream air velocity was from 0.29 to 0.59 M at an altitude of 9.65 km and from 0.47 M to 0.73 M at an altitude of 14 km.

> with the experimental results in [4, 6, 13] within an error relative of 8%. The frequency of the first mode was around 9.96 Hz. Following Eq. (3), the time step of

*Comparison of the mode shapes. (a) Mode 1. (b) Mode 2. (c) Mode 3. (d) Mode 4.*

*Research on Aeroelasticity Phenomenon in Aeronautical Engineering*

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

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

the unsteady problem was estimated about 0.005 s.

*2.3.2 Damping analysis*

**Figure 4.**

**59**

#### **2.3 Results**

#### *2.3.1 Modal analysis*

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

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

**Figure 4.** *Comparison of the mode shapes. (a) Mode 1. (b) Mode 2. (c) Mode 3. (d) Mode 4.*

with the experimental results in [4, 6, 13] within an error relative of 8%. The frequency of the first mode was around 9.96 Hz. Following Eq. (3), the time step of the unsteady problem was estimated about 0.005 s.
