# Verification of a Body Freedom Flutter Numerical Simulation Method Based on Main Influence Parameters

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## Abstract

**:**

## 1. Introduction

## 2. The Rigid–Elastic Coupled Dynamic Modeling Method: Taking a Two-Dimensional Airfoil as an Example

_{θ}is the distance from the fuselage centroid to the elastic axis, and the centroid in front is positive; Xα is the distance from the wing centroid to the rigid axis, and the centroid in front is positive. Note that the mass of the fuselage and wing are M and m, respectively. The pitching moment of the inertia of the fuselage wing are I

_{θ}and I

_{α}. K

_{h}and K

_{α}are the bending stiffness and torsional stiffness of fuselage wing connection, respectively.

## 3. CFD and Fluid–Solid Coupling Calculation Method

## 4. CFD and Fluid–Solid Coupling Calculation Method

#### 4.1. Results and Discussion of the Two-Dimensional Model

#### 4.1.1. Model Parameters

#### 4.1.2. Numerical Solution by the CFD Method

^{+}≈ 1 and the calculated height of the first layer of the grid Y

_{min}≈ 0.00001. The inflow static temperature was 300 K, the density was 1.225 kg/m

^{3}, the angle of attack was 0° and Δt = 0.001 s. The Mach number and Reynolds number were obtained according to different incoming flow velocity.

#### 4.1.3. Theoretical Solution by the Theodorsen Unsteady Aerodynamic Model

_{h}= 2 N/mm (shown in the left-hand figure), the flutter point was 79 m/s and the flutter circle frequency was 24.73 rad/s, i.e., 3.94 Hz. In this case, body freedom flutter occurred. In the right-hand figure, the results are shown for a higher bending stiffness. In this case, the bending mode frequency was relatively high and bending torsional flutter occurred through with the torsional mode coupling with the bending mode. The flutter velocity was 81 m/s and the flutter circle frequency was 91.76 rad/s, i.e., 14.6 Hz.

#### 4.1.4. Discussion and Validation of the BFF Calculation Method Using a Navier–Stokes Fluid Model

#### 4.2. Results and Discussion of the Three-Dimensional Model

^{6}, 0.9 × 10

^{6}, and 2.5 × 10

^{6}. The lift coefficients C

_{L}were 0.2925, 0.2856 and 0.2856, respectively. The pressure distribution curves of the fuselage and wing are also shown in Figure 8. The results show that increasing the number of grids has no effect on the calculation results when the number of grids is greater than 0.9 × 10

^{6}.

_{j}until the stiffness multiple N at the time that flutter occurred. Therefore, given the flight Mach number and flight altitude, only one steady aerodynamic calculation was performed. The theoretical derivation showed that the stiffness of the calculation model was N times the original stiffness, the flutter dynamic pressure of the calculated model was Q

_{F,m}and the flutter frequency was ω

_{F,m}. The flutter velocity pressure Q

_{F,a}and flutter frequency ω

_{F,a}of the original stiffness aircraft at a given Mach number were, respectively:

_{yy}is the pitching moment of inertia, α is the angle of attack, C

_{MZα}is the static derivative of the pitching moment, ${C}_{Mz\dot{a}}$ is the dynamic derivative of the pitching moment, Q is the dynamic pressure, S is the reference area and c

_{ref}is the reference length. The solution of the equation is:

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 6.**Comparisons of the CFD and Theodorsen unsteady aerodynamic forces (Kh = 2 N/mm; inflow velocity, 79 m/s; forced vibration frequency, 3.9 Hz).

**Figure 8.**(

**a**) Computing grid: surface grid, (

**b**) local space grid, (

**c**) pressure distribution on the fuselage (y = 0.0 m), (

**d**) the pressure distributed at the wing tip (y = 1.0 m).

**Figure 10.**Calculation results of the generalized coordinate time response in the benchmark status (BFF-04).

**Figure 11.**Calculation results of the generalized coordinate time response when the center of gravity moves forward 40 mm (BFF-11).

**Figure 12.**Comparison of time response of BFF-04 and BFF-11 in the first two generalized coordinates.

Parameters | Values | Parameters | Values |
---|---|---|---|

Fuselage mass M | 4 kg | Centroid position of the wing | 20%c |

Wing mass m | 4 kg | Radius of gyration of the fuselage ${R}_{\theta}$ | 0.18 m |

Fuselage pitching moment of inertia ${I}_{\theta}$ | 0.1312 kg·m^{2} | Radius of gyration of the wing ${R}_{\alpha}$ | 0.18 m |

Wing pitching moment of inertia ${I}_{\alpha}^{}$ | 0.1312 kg·m^{2} | Bending stiffness ${K}_{h}$ | 1, 2, 4, 12 (N/mm) |

Distance between the elastic center and the centroid position of the fuselage ${X}_{\theta}$ | 5%c | Torsional stiffness ${K}_{\alpha}$ | 600 (Nm/rad) |

Distance between the elastic center and the centroid position the of wing ${X}_{\alpha}$ | 5%c | Elastic center position | 15%c |

Airfoil chord length c | 0.4 m | Wing segment length | 1.5 m |

Centroid position of fuselage | 20%c |

Model Name | Pitching Mode | Symmetric First Wing Bend | Symmetric Wing Second Bend | Symmetric Wing First Twist | Symmetric Wing Third Bend |
---|---|---|---|---|---|

FEM | 0.0 | 5.19 | 24.55 | 47.16 | 62.18 |

GVT | 0.0 | 5.10 | 23.60 | 44.17 | - |

Error | 0.0% | 1.7% | 4.0% | 6.8% | - |

Calculation or Experimental Status | Flutter Velocity (m/s) | Flutter Frequency (Hz) | Vibration Frequency and Damping under the Experimental Flutter Velocity | ||
---|---|---|---|---|---|

Frequency (Hz) | Damping (%) | ||||

Benchmark status | Experiment | 22.3 | 1.67 | ||

CFD/CSD | 19.21 | 1.31 | 1.47 | 7.7% | |

Center of gravity moved forward 40 mm | Experiment | 24.2 | 2.73 | ||

CFD/CSD | 19.77 | 1.88 | 2.62 | 10.8% |

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**MDPI and ACS Style**

Lei, P.; Guo, H.; LYu, B.; Chen, D.; Yu, L.
Verification of a Body Freedom Flutter Numerical Simulation Method Based on Main Influence Parameters. *Machines* **2021**, *9*, 243.
https://doi.org/10.3390/machines9100243

**AMA Style**

Lei P, Guo H, LYu B, Chen D, Yu L.
Verification of a Body Freedom Flutter Numerical Simulation Method Based on Main Influence Parameters. *Machines*. 2021; 9(10):243.
https://doi.org/10.3390/machines9100243

**Chicago/Turabian Style**

Lei, Pengxuan, Hongtao Guo, Binbin LYu, Dehua Chen, and Li Yu.
2021. "Verification of a Body Freedom Flutter Numerical Simulation Method Based on Main Influence Parameters" *Machines* 9, no. 10: 243.
https://doi.org/10.3390/machines9100243