# Effects of Mass Attachments on Flutter Characteristics of Thin-Walled Panels

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## Abstract

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## 1. Introduction

## 2. Geometry of a Thin-Walled Panel with Mass Attachments

## 3. Dynamic Equations Based on the Assumed Mode Method

## 4. Aerodynamic Force in Modal Coordinate System

## 5. Flutter Equation

## 6. Numerical Example

^{3}, and the air density ratio is taken as ${\rho}_{r}=1.0$.

#### 6.1. The Case of Only One Mass Attachment

#### 6.2. The Case of Two or More Mass Attachments

#### 6.3. Flutter Characteristics with Dampers

#### 6.4. Effects of Aspect Ratio on Flutter Characteristics of the Panel

## 7. Conclusions

- The proposed assumed mode method for panels with mass attachments can well capture the change in the natural frequencies of the panel structure. When there are mass attachments on the panel, the natural frequencies of the panel usually show a downward trend, but it may change dramatically due to different positions of mass attachments;
- The changes in the flutter characteristics of the panel are closely related to the changes in the mass distribution of the panel caused by mass attachments. The flutter velocity of the panel can be improved by mass attachments located at the center of the panel. However, the flutter velocity of the panel drops sharply if the mass attachments are located in a nearby area slightly away from the central point. In addition, generally speaking, the flutter frequency of the panel with mass attachments is lower than that of the panel without a mass attachment. Furthermore, the larger the masses of attachments, the lower the flutter frequencies;
- The flutter characteristics of the panel with mass attachments can be effectively improved by presetting dampers at appropriate locations. Both the present 5P method and the 9P method can effectively improve the flutter velocity and frequency of the panel, but the effect of the nine-point method is due to the five-point method. Additionally, the effects of the 9P method are better than that of the 5P method;
- The study of the flutter characteristics of panels with mass attachments based on the assumed mode method can more realistically simulate the situation encountered during flights. Additionally, it can provide technical reserves for the structural design of panels and the improvement of flight safety.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**A square panel with simply supported boundaries on four sides: (

**a**) with no mass attachment; (

**b**) with several mass attachments.

**Figure 3.**Flutter velocities and flutter frequencies change with the number of truncated modes r ($M{a}_{\infty}=2.0$): (

**a**) flutter velocities, (

**b**) flutter frequencies.

**Figure 4.**Results of flutter calculation for the panel with no mass attachment (NP) ($M{a}_{\infty}=2.0$): (

**a**) $U-g$, (

**b**) $U-f$.

**Figure 5.**Two different positions of a mass attachment: (

**a**) the center point of the panel (CP); (

**b**) the intersection point located at 1/4 span length and 1/4 chord length (QP).

**Figure 8.**Graphs of the change in the first four natural frequencies of the panel with the position $(x,y)$ of the mass attachment: (

**a**) the first order natural frequency ${f}_{1}$; (

**b**) the second order natural frequency ${f}_{2}$; (

**c**) the third order natural frequency ${f}_{3}$; (

**d**) the fourth order natural frequency ${f}_{4}$.

**Figure 9.**The changes in the flutter characteristics of the panel with the location $(x,y)$ of the mass attachment: (

**a**) flutter velocity ${U}_{F}$, (

**b**) flutter frequency ${f}_{F}$.

**Figure 10.**The changes in flutter characteristics caused by the change in attachment mass: (

**a**) flutter velocity ${U}_{F}$, (

**b**) flutter frequency ${f}_{F}$.

**Figure 11.**Two position relations of the panel with two or more mass attachments: (

**a**) one point located at CP and the other at QP (TP); (

**b**) there are 5 mass attachments on the panel (MP).

**Figure 12.**Results of flutter calculation for the panel with two mass attachments (TP) ($M{a}_{\infty}=2.0$): (

**a**) $U-g$, (

**b**) $U-f$.

**Figure 13.**Results of flutter calculation for the panel with five mass attachments (MP) ($M{a}_{\infty}=2.0$): (

**a**) $U-g$, (

**b**) $U-f$.

**Figure 14.**The first 4 modes of the panel with no mass attachment: (

**a**) the 1st mode (64.55 Hz); (

**b**) the 2nd mode (161.4 Hz); (

**c**) the 3rd mode (161.4 Hz); (

**d**) the 4th mode (258.1 Hz).

**Figure 15.**Two different damper configurations to suppress panel flutter with mass attachments: (

**a**) 5P method, (

**b**) 9P method.

**Figure 16.**Two different damper configurations to suppress panel flutter with mass attachments: (

**a**) 5P method, (

**b**) 9P method.

**Figure 17.**Results of flutter calculation for the panel with the 5P method damper configuration ($M{a}_{\infty}=2.0$): (

**a**) $U-g$, (

**b**) $U-f$.

**Figure 18.**Results of flutter calculation for the panel with the 9P method damper configuration ($M{a}_{\infty}=2.0$): (

**a**) $U-g$, (

**b**) $U-f$.

**Figure 19.**Effects of the two damper schemes on flutter characteristics of the panel: (

**a**) ${U}_{F}-g$, (

**b**) ${f}_{F}-c$.

**Figure 20.**Two kinds of panels with different aspect ratio (QP): (

**a**) $a/b=0.5$ (AR1), (

**b**) $a/b=2.0$ (AR2).

**Figure 21.**Results of flutter calculation for the panel with no mass attachment (NP) ($M{a}_{\infty}=2.0$): (

**a**) $U-g$ (AR1), (

**b**) $U-f$ (AR1).

**Figure 22.**Results of flutter calculation for the panel with no mass attachment (QP) ($M{a}_{\infty}=2.0$): (

**a**) $U-g$ (AR1), (

**b**) $U-f$ (AR1).

**Figure 23.**Results of flutter calculation for the panel with the 9P method damper configuration ($M{a}_{\infty}=2.0$): (

**a**) $U-g$ (AR1), (

**b**) $U-f$ (AR1).

**Figure 24.**Results of flutter calculation for the panel with no mass attachment (NP) ($M{a}_{\infty}=2.0$): (

**a**) $U-g$ (AR2), (

**b**) $U-f$ (AR2).

**Figure 25.**Results of flutter calculation for the panel with no mass attachment (QP) ($M{a}_{\infty}=2.0$): (

**a**) $U-g$ (AR2), (

**b**) $U-f$ (AR2).

**Figure 26.**Results of flutter calculation for the panel with the 9P method damper configuration ($M{a}_{\infty}=2.0$): (

**a**) $U-g$ (AR2), (

**b**) $U-f$ (AR2).

Physical Quantity | Value |
---|---|

Young’s modulus (Pa) | $E=7.1\times {10}^{10}$ |

Poisson’s ratio | $v=0.32$ |

Mass density (kg/m3) | ${\rho}_{s}=2768$ |

Modal damping ratio (%) | $\zeta =0.0$ |

Case | U_{F} | f_{F} | s_{F} |
---|---|---|---|

NP | 543.4 | 133.9 | 0.0155 |

CP | 602.2 | 125.4 | 0.0073 |

QP | 536.3 | 117.5 | 0.0083 |

TP | 574.0 | 111.8 | 0.0012 |

MP | 537.1 | 97.92 | 0.0155 |

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**MDPI and ACS Style**

Qi, W.; Wang, M.; Tian, S. Effects of Mass Attachments on Flutter Characteristics of Thin-Walled Panels. *Aerospace* **2022**, *9*, 748.
https://doi.org/10.3390/aerospace9120748

**AMA Style**

Qi W, Wang M, Tian S. Effects of Mass Attachments on Flutter Characteristics of Thin-Walled Panels. *Aerospace*. 2022; 9(12):748.
https://doi.org/10.3390/aerospace9120748

**Chicago/Turabian Style**

Qi, Wuchao, Meng Wang, and Sumei Tian. 2022. "Effects of Mass Attachments on Flutter Characteristics of Thin-Walled Panels" *Aerospace* 9, no. 12: 748.
https://doi.org/10.3390/aerospace9120748