# A Parametric Study on the Aeroelasticity of Flared Hinge Folding Wingtips

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## Abstract

**:**

## 1. Introduction

## 2. Modelling

#### 2.1. Structural Dynamics Solver

^{TM}. The lifting surface is discretized into Euler–Bernoulli beam elements as shown in Figure 3. The elastic potential energy $\left(U\right)$ of a beam element, including the effects of the geometric warping, can be expressed as [11]:

#### 2.2. Modelling the Flared Hinge

^{®}. It should be noted that the stiffness matrices of these nonstandard beam elements contain bending–torsion coupling terms that do not exist in the matrices of the other standard Euler–Bernoulli beam elements. The magnitude and sign of these coupling terms depend on the flare angle and the dimensions of the element. The hinge region (shown in Figure 4) is modelled as a swept standard beam element whose sweep angle is equal to the flare angle of the hinge.

#### 2.3. Aerodynamic Solver

#### 2.4. Aeroelastic System

#### 2.5. Validation

## 3. Parametric Study: Linear Aeroelasticity

#### 3.1. Folding the Baseline Wing

#### 3.1.1. Flare Angle

#### 3.1.2. Quasi-Steady Response to Discrete Gusts

#### 3.1.3. Hinge Stiffness

^{8}Nm whilst keeping its torsional rigidity as the baseline wing. Then, its torsional rigidity is varied from 160 to 10

^{8}Nm whilst keeping its bending rigidity as the baseline wing. The outboard segment length is fixed at 1.016 m. The variations of flutter speed, flutter frequency and divergence speed are shown in Figure 9 and Figure 10.

^{2}to 285 m/s at 200 Nm

^{2}, and then it remains constant even when the bending rigidity increases further. On the contrary, flutter speed and frequency are very sensitive to the bending rigidity of the hinge especially for values between 10

^{2}and 10

^{4}Nm

^{2}. In this region, they increase sharply and then drop suddenly regardless of the flare and fold angles. This sudden change in the flutter behavior is associated with a change in the flutter mode from the 2nd bending to 1st torsion. For example, consider the 0° fold angle and 15° flare angle case. At a bending rigidity of 10

^{2}Nm

^{2}, the flutter speed is 121 m/s, and it rises to 189 m/s at 310 Nm

^{2}. Then, it drops sharply to reach 138 m/s at 10

^{4}Nm

^{2}and remains almost constant for higher values of bending rigidity. Similarly, the flutter frequency is 81 rad/s at 10

^{2}Nm

^{2}, and it drops sharply to reach 62 rad/s at 310 Nm

^{2}. Then, it increases again to reach 69 rad/s at 10

^{4}Nm

^{2}and remains almost constant for higher values of bending rigidity. It should be noted that the curve of the 0° fold angle and 15° flare angle case coincides with the 0° fold angle and 0° flare angle case. It can be deduced from Figure 9 that increasing the fold angle alone causes the flutter speed to peak at higher values of bending rigidity. On the other hand, increasing the flare angle alone tends to increase the peak values without affecting the values of bending rigidity at which the peaks occur.

^{4}Nm

^{2}regardless of the flare angle. For 80° fold angle and 0° flare angle, the flutter speed is not sensitive to torsional rigidity, whilst for a 25° flare angle, the flutter speed reduces as the torsional rigidity increases until 490 Nm

^{2}where a further increase in the torsional rigidity does not affect the flutter speed. The flutter frequency tends to increase with the torsional rigidity until 10

^{4}Nm

^{2}, after which it remains constant. The only exception is the 80° fold angle at 0° flare angle where the flutter frequency is insensitive to the torsional rigidity of the hinge. It also worth noting that at 250 Nm

^{2}torsional rigidity, the flutter speed for the 25° flare angle and 80° fold angle case becomes higher than that for the 0° flare angle and 80° fold angle case. The opposite is true for the flutter frequency. Regardless of the torsional rigidity, the flutter mode does not change and remains the first torsion mode. It can be deduced from this subsection that the torsional rigidity of the hinge affects static and dynamic aeroelastic boundaries, whereas its bending rigidity mainly affects the dynamic aeroelastic boundaries. In addition, the variation in torsional rigidity does not change the flutter mode (first torsion), whereas the variation in bending rigidity results in a change in the flutter mode (from second bending to first torsion).

#### 3.2. Adding Folding Wingtip

#### 3.2.1. Wingtip Size

#### 3.2.2. Wingtip Mass

#### 3.2.3. Wingtip Sweep

## 4. Nonlinear Aeroelasticity

#### 4.1. Nonlinear Hinge

#### 4.2. Limit Cycle Oscillations

^{TM}using the ODE23 solver. The ODE23 is a single-step solver based on an explicit Runge–Kutta formulation [24,25]. As in Section 3.1, the baseline Goland wing is split into an inboard and an outboard segment. The length of the outboard segment is set to 1.016 m. The hinge properties are ${a}_{1}=987$ Nm

^{2}and $\gamma =500$ (hardening). To determine the influence of hinge nonlinearity on the aeroelastic response of the wing, three cases are studied here:

- Case 1: 0° flare angle and 0° fold angle;
- Case 2: 0° flare angle and 20° fold angle; and,
- Case 3: 45° flare angle and 20° fold angle.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

Acronyms | |

DLM | Doublet Lattice Method |

DOFs | Degrees Of Freedom |

FE | Finite Element |

FHFWT | Flared Hinge Folding Wingtips |

LCO | Limit Cycle Oscillation |

RBM | Root Bending Moment |

RSHF | Root Shear Force |

SAH | Semi Aeroelastic Hinge |

Nomenclature | |

${C}_{w}$ | warping constant |

$U$ | strain energy |

$T$ | kinetic energy |

${F}_{g}$ | flight profile alleviation factor |

$N$ | shape function |

${U}_{ds}$ | design gust velocity |

${U}_{ref}$ | reference gust velocity |

$\widehat{a}$ | nondimensional position of elastic axis relative to half chord |

${e}_{m}$ | distance between SC and center of gravity |

${m}^{\prime}$ | mass per unit length |

${w}_{g}$ | gust profile |

${\Gamma}_{FWT}$ | fold angle |

${\Lambda}_{hinge}$ | hinge-line angle (or flare angle) |

$\widehat{A}$ | global structural inertia matrix |

$\widehat{B}$ | global aerodynamic damping matrix |

$\widehat{C}$ | global aerodynamic stiffness matrix |

$\widehat{D}$ | global structural damping matrix |

$\widehat{E}$ | global structural stiffness matrix |

X | generalized coordinates vector |

$E$ | Young’s modulus |

$EI$ | spanwise bending rigidity |

GJ | torsional rigidity |

H | gust gradient |

$J$ | torsion constant |

V | true airspeed |

$b$ | half chord |

$dL$ | lift per unit span |

$dM$ | pitch moment per unit span |

$k$ | reduced frequency |

$l$ | length of a beam element |

$t$ | time |

$w$ | plunge displacement |

W | work |

$y$ | spanwise position along the beam |

$\Delta {\theta}_{FWT}$ | change in geometric incidence |

$\theta $ | pitch/twist rotation |

$\varphi $ | bending rotation |

$\rho $ | air density |

$\omega $ | frequency |

Superscripts | |

b | bending |

t | torsion |

T | transpose |

. | first derivatives with respect to time |

.. | second derivatives with respect to time |

Subscripts | |

e | ends of the beam element |

o | amplitude |

## Appendix A

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**Figure 1.**Swept wing equipped with flared hinge folding wingtip “From [1]; reprinted by permission of the American Institute of Aeronautics and Astronautics, Inc.”.

**Figure 2.**The AlbatrossONE flying demonstrator equipped with SAH “Reprinted from [10]”.

**Figure 10.**Variation of aeroelastic parameters with torsional rigidity of the hinge for a tip length of 1.016 m.

DOFs | Case 1 | Case 2 | Case 3 | Case 4 | Case 5 | Case 6 |
---|---|---|---|---|---|---|

w_{1} | 1 | 0 | 0 | 0 | 0 | 0 |

φ_{1} | 0 | 1 | 0 | 0 | 0 | 0 |

θ_{1} | 0 | 0 | 1 | 0 | 0 | 0 |

w_{2} | 0 | 0 | 0 | 1 | 0 | 0 |

φ_{2} | 0 | 0 | 0 | 0 | 1 | 0 |

θ_{2} | 0 | 0 | 0 | 0 | 0 | 1 |

**Table 2.**Properties of representative wings used for validation “Reprinted from [17]”.

Specifications | HALE Wing | Goland Wing | Representative Wing |
---|---|---|---|

Half span (m) | 16 | 6.096 | 3 |

$2b$ (m) | 1 | 1.8288 | 1 |

${m}^{\prime}$ (kg/m) | 0.75 | 35.71 | 6 |

Moment of inertia per unit length (kgm) | 0.1 | 8.64 | 0.75 |

Spanwise elastic axis (from LE) | 50% | 33% | 35% |

Center of gravity (from LE) | 50% | 43% | 45% |

$EI$ (Nm^{2}) | 2 × 10^{4} | 9.77 × 10^{6} | 6 × 10^{5} |

$GJ$ (Nm^{2}) | 1 × 10^{4} | 0.987 × 10^{6} | 6 × 10^{4} |

$\rho $ (kg/m^{3}) | 0.0889 | 1.225 | 1.225 |

Wing | Method | ||||||
---|---|---|---|---|---|---|---|

Present Work (First 8 Modes) | Ref. [17] | Ref. [18] | Ref. [19] | Ref. [20] | Ref. [21] | Ref. [22] | |

HALE Wing | |||||||

Flutter Speed (m/s) | 32.61 | 33.43 | 32.21 | − | − | 32.51 | − |

Flutter Freq. (rad/s) | 22.27 | 21.38 | 22.61 | − | − | 22.37 | − |

Divergence Speed (m/s) | 37.34 | 37.18 | 37.29 | − | − | 37.15 | − |

Goland Wing | |||||||

Flutter Speed (m/s) | 136.99 | 137.11 | − | 135.60 | 136.22 | 137.16 | 133 |

Flutter Freq. (rad/s) | 69.97 | 69.90 | − | 70.20 | 70.06 | 70.70 | 72.70 |

Divergence Speed (m/s) | 252.46 | 252.80 | − | − | 250.82 | − | − |

Representative Wing | |||||||

Flutter Speed (m/s) | 79.07 | 78.33 | − | − | − | − | 77 |

Flutter Freq. (rad/s) | 149.60 | 148.94 | − | − | − | − | 149.60 |

Divergence Speed (m/s) | 207.34 | 206.70 | − | − | − | − | − |

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**MDPI and ACS Style**

Ajaj, R.M.; Saavedra Flores, E.I.; Amoozgar, M.; Cooper, J.E.
A Parametric Study on the Aeroelasticity of Flared Hinge Folding Wingtips. *Aerospace* **2021**, *8*, 221.
https://doi.org/10.3390/aerospace8080221

**AMA Style**

Ajaj RM, Saavedra Flores EI, Amoozgar M, Cooper JE.
A Parametric Study on the Aeroelasticity of Flared Hinge Folding Wingtips. *Aerospace*. 2021; 8(8):221.
https://doi.org/10.3390/aerospace8080221

**Chicago/Turabian Style**

Ajaj, Rafic M., Erick I. Saavedra Flores, Mohammadreza Amoozgar, and Jonathan E. Cooper.
2021. "A Parametric Study on the Aeroelasticity of Flared Hinge Folding Wingtips" *Aerospace* 8, no. 8: 221.
https://doi.org/10.3390/aerospace8080221