# The Effect of Torsional and Bending Stiffness on the Aerodynamic Performance of Flapping Wing

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## Abstract

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## 1. Introduction

## 2. Theoretical Methods

#### 2.1. Aerodynamic Model

#### 2.2. Structural Model

#### 2.3. Aeroelastic Coupling

#### 2.4. Performance Parameters

#### 2.5. Solver Validation

#### 2.5.1. Aerodynamic Force

#### 2.5.2. Geometric Nonlinearity

#### 2.5.3. Fluid–Structure Coupling

## 3. Wing Model

#### 3.1. Computational Model

#### 3.2. Structural and Kinematic Parameters

## 4. Results and Discussion

#### 4.1. Variation of Parameters with Initial Geometric Twist Angle

#### 4.2. Variation of Parameters during the Period

#### 4.3. Pressure Coefficient at Different Portions during the Period

## 5. Conclusions

- By comparing with experiments and calculations, the method proposed in this paper has relatively high accuracy in calculating three-dimensional inviscid unsteady aerodynamic forces and three-dimensional slender geometric nonlinear beams, and is suitable for solving the problem in this article.
- For a passively twisted wing, an increase in torsional stiffness leads to an increase in the lift and thrust coefficients and a concomitant decrease in propulsive efficiency. This illustrates the need to make trade-offs during design.
- Under the same torsional stiffness, as the initial geometric twist angle of the wing increases, the lift coefficient increases, and the thrust coefficient and propulsion efficiency first increase and then decrease. For a certain wing torsional stiffness, there is a suitable initial geometric twist angle that maximizes propulsion efficiency. An inappropriate initial geometric twist angle will lead to an excessive angle of attack at a certain time during the period, thereby reducing propulsion efficiency.
- For a passively twisted wing, the wing is in a bending–torsion coupling state. Compared with a spanwise rigid wing, a wing with some spanwise flexibility can increase the thrust coefficient of the outer wing, making the thrust distribution along the wing span more reasonable, thereby increasing the overall thrust coefficient and propulsion efficiency.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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EI = Rigid | EI = EI1 | EI = EI2 | |
---|---|---|---|

GJ = GJ1 | Case 1 | Case 2 | Case 3 |

GJ = GJ2 | Case 4 | Case 5 | Case 6 |

GJ = GJ3 | Case 7 | Case 8 | Case 9 |

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**MDPI and ACS Style**

Qi, M.; Zhu, W.; Li, S.
The Effect of Torsional and Bending Stiffness on the Aerodynamic Performance of Flapping Wing. *Aerospace* **2023**, *10*, 1035.
https://doi.org/10.3390/aerospace10121035

**AMA Style**

Qi M, Zhu W, Li S.
The Effect of Torsional and Bending Stiffness on the Aerodynamic Performance of Flapping Wing. *Aerospace*. 2023; 10(12):1035.
https://doi.org/10.3390/aerospace10121035

**Chicago/Turabian Style**

Qi, Ming, Wenguo Zhu, and Shu Li.
2023. "The Effect of Torsional and Bending Stiffness on the Aerodynamic Performance of Flapping Wing" *Aerospace* 10, no. 12: 1035.
https://doi.org/10.3390/aerospace10121035