# Investigation of the Free-Fall Dynamic Behavior of a Rectangular Wing with Variable Center of Mass Location and Variable Moment of Inertia

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## Abstract

**:**

## 1. Introduction

## 2. Research Methods and Validations

#### 2.1. Quasi-Steady Analytical Model

#### 2.2. CFD Numerical Method

#### 2.3. Experiment Method of Freely Falling Wing

#### 2.4. Validation and Discussion

## 3. Effect of MOI

#### 3.1. Quasi-Steady Analytical Model

#### 3.2. Experiment Results

## 4. Effect of COM Position on Freely Falling Wing

#### 4.1. Simulation of the Analytical Model

#### 4.2. Experiment Results

## 5. Results and Discussion

#### 5.1. Quasi-Steady Analytical Model Results

- 1.
- Effect of MOI: In the case of wings with different MOIs, there exists a limit cycle of tumbling motion that all wings will eventually converge to after releasing, reaching a stable state shown in Figure 11. The phase from the initial state to the final stable tumbling limit cycle is defined as the transition phase. As the MOI of the wing increases, the trajectory of the free-falling wing becomes steeper, and the transition phase becomes longer, meaning that the wing will take more time to converge to the limit cycle.
- 2.
- Effect of COM: When the COM moves forward, the transition phase from the initial state to the tumbling motion limit cycle becomes longer, and the phase trajectory during the transition phase becomes more complex. A new limit cycle emerges when the COM is positioned 40 mm ahead of the wing’s geometric center, as can be seen in Figure 13f, corresponding to the quasi-periodic fluttering motion of the wing. At this point, the limit cycle of the fluttering motion is unstable, and the phase trajectory eventually diverges from this limit cycle, converging towards the limit cycle of the tumbling motion; Figure 15 shows this process well. As the COM continues to move forward, the limit cycle corresponding to the tumbling motion disappears, and the limit cycle of the fluttering motion becomes a stable limit cycle, resulting in the wing freely falling with periodic fluttering motion.

#### 5.2. Experimental Results

- 1.
- Effect of MOI: By symmetrically altering the position of the clump weights block, the MOI of the wing is changed, and tumbling motion is observed under different conditions of MOI, as shown in Figure 12. As the MOI of the wing increases, the trajectory of the freely falling wing becomes steeper, and the transition phase from the initial descent to stable tumbling becomes longer. This observation aligns with the conclusions drawn from the analytical model.
- 2.
- Effect of COM: Experimental results show that maintaining the actual wing trajectory in the $XZ$ plane is challenging because of asymmetrical disturbances. Typically, the wing descends along a spiral path, as shown in Figure 16. As the COM moves forward, the transition phase from the initial state to tumbling motion increases. Further forward movement of the COM results in the fluttering motion of the wing. However, no tumbling motion is observed in the experiment after shifting the COM forward by 40 mm, as shown in Figure 16d.

#### 5.3. Comparison of the Analytical Model with Experimental Results

## 6. Conclusions

- 1.
- A quasi-steady analytical model was developed based on the Andersen–Pesavento–Wang model. The analytical model was derived in the two-dimensional plane and can reflect the dynamic behavior well in the longitudinal plane during the three-dimensional falling of a real wing.
- 2.
- After deployment, the wing undergoes a transition phase before eventually entering a stable motion, which can be characterized as either tumbling or fluttering, as revealed by the quasi-steady analysis model and experiment method employed in this study. Through CFD analysis, it is observed that, during the transition phase, the shedding frequency of vortices gradually increases and stabilizes, resulting in periodic aerodynamic oscillations due to the cyclic generation and separation of vortices. These periodic oscillations eventually lead to the emergence of a stable periodic motion. Both of these stable motions exhibit relatively low translational velocities. In the case of rolling motion, the pitch rate of the wing remains approximately 15 rad/s. In the case of fluttering motion, the pitch rate oscillates between positive and negative values. Therefore, both of these stable motion patterns should be avoided in the air-drop launched UAVs.
- 3.
- For a wing with elliptical airfoil, the higher the MOI about the ${Y}^{\prime}$-axis, the steeper the trajectory, the lower the angular rate of tumbling, and the shorter time for it staying in the air. For most aircraft, the pitch tumbling is not beneficial. Without changing the shape of the wing and the position of the COM, and only changing the MOI of the wing about the ${Y}^{\prime}$-axis, the initial drop attitude cannot prevent the pitch tumbling of the wing during the falling process.
- 4.
- The position of the COM has a crucial influence on the handling performance and stability of the UAVs. The traditional flight dynamics theory suggests that the forward shift of the COM position can make the vehicle obtain better static stability. For the freely falling wing, the forward shift of the COM will delay the appearance of tumbling, i.e., the transition phase becomes longer. When the forward shift of the COM exceeds a certain value, a new relatively stable motion of falling appears, which is expressed as fluttering in the analytical model, while the wing shows a heaving/ pitching composite motion along the spiral line in the real three-dimensional falling experiment. In the future, for air-drop launch UAVs with relaxed longitudinal static stability, the possibility of its tumbling needs to be considered.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

CFD | Computational Fluid Dynamics |

COM | Center of Mass |

DOAJ | Directory of open access journals |

LD | Linear dichroism |

MDPI | Multidisciplinary Digital Publishing Institute |

MOI | Moment of Inertia |

TLA | Three Letter Acronym |

UAV | Unmanned Aerial Vehicle |

Symbols | |

a | Semimajor axis of the ellipse (m) |

b | Semiminor axis of the ellipse (m) |

e | Eccentricity of the airfoil |

${F}_{{x}^{\prime}}^{v}$ | Translational drag force in the ${X}^{\prime}$ direction (N) |

${F}_{{z}^{\prime}}^{v}$ | Translational drag force in the ${Z}^{\prime}$ direction (N) |

$\mathsf{\Gamma}$ | Circulation (m${}^{2}$/s) |

g | Gravitational acceleration (m/s${}^{2}$) |

I | Moment of inertia about the ${Y}^{\prime}$-axis (kg·m${}^{2}$) |

J | Added moment of inertia (kg·m${}^{2}$) |

${l}_{\tau}$ | Length of the torque arm (m) |

m | Mass of the wing (kg) |

${m}_{{x}^{\prime}}$ | Added mass coefficient in the ${X}^{\prime}$ direction |

${m}_{{z}^{\prime}}$ | Added mass coefficient in the ${Z}^{\prime}$ direction |

${\rho}_{f}$ | Density of the air (kg/m${}^{3}$) |

${\rho}_{s}$ | Density of the wing (kg/m${}^{3}$) |

${\rho}_{\theta}$ | Volume density of the fluid displaced by the wing (kg/m${}^{4}$) |

${\tau}_{\theta}$ | Torque about the ${Y}^{\prime}$-axis (N m) |

${v}_{{x}^{\prime}}$ | Speed in the ${X}^{\prime}$ direction of the body coordinate system (m/s) |

${v}_{{z}^{\prime}}$ | Speed in the ${Z}^{\prime}$ direction of the body coordinate system (m/s) |

Subscripts | |

C | Centroid |

f | Zone of fluid |

s | Zone of wing |

${X}^{\prime}$ | ${X}^{\prime}$ direction of the body coordinate system |

${Y}^{\prime}$ | ${Y}^{\prime}$-axis vertical to the ${X}^{\prime}O{Z}^{\prime}$ plane |

${Z}^{\prime}$ | ${Z}^{\prime}$ direction of the body coordinate system |

## Appendix A. Method Description

#### Appendix A.1. Range-Kutta Method

#### Appendix A.2. The Ω-Method Vortex Identification Criterion

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**Figure 3.**Two-dimensional CFD calculation mesh (

**a**) Complete domain (

**b**) Partial enlarged view of overset component.

**Figure 4.**The WT901 inertial measurement unit. WT901 has a mass of 20 g and is able to transmit data via WiFi with a transmission frequency of 200 Hz.

**Figure 5.**(

**a**) Wing and clump weights assembly. (

**b**) The inner structure of the wing. (

**c**) Clump weights.

**Figure 6.**(

**a**) Experimental scheme for freely falling wing. (

**b**) Wing and release system. (

**c**) Wing free-fall experimental platform.

**Figure 7.**Comparison of simulation data with experimental data for pitch angle $\theta $, $x$-directional force ${F}_{x}$, and $z$-directional force ${F}_{z}$ simulated by the analytical model and CFD, where (I = 0.00045 kg·m${}^{2}$).

**Figure 8.**The vorticity street around the freely falling wing. Positive vortices are red and negative vortices are blue (

**a**–

**d**), respectively, represent the distribution of vortices on the falling trajectory of the wing with different MOIs in two seconds, where the unit of I is kg·m${}^{2}$.

**Figure 9.**Streamline diagram of the first one second in the transition phase of the wing for (I = 0.00045 kg·m${}^{2}$), with vortex contour with force change curve. (

**a**–

**h**) Vortical contour ($\mathsf{\Omega}$-criterion) and streamline around the wing: positive vortices are in red and negative vortices are in blue. (

**i**) Aerodynamic forces in x and z directions.

**Figure 10.**Simulation of falling trajectories of airfoils with different MOIs using the analytical model, where the unit of I is kg·m${}^{2}$.

**Figure 11.**The trajectory obtained from the analytical model and its $({v}_{{x}^{\prime}},{v}_{{z}^{\prime}},\dot{\theta})$ phase diagram. (

**a**) Trajectory visualization for the simulation of freely falling wings with different MOIs, (

**b**–

**e**) Extend the simulation time to 20 s and make the $({v}_{{x}^{\prime}},{v}_{{z}^{\prime}},\dot{\theta})$ phase diagram, where the unit of I is kg·m${}^{2}$.

**Figure 12.**The trajectory of the freely falling wing with different MOIs observed in the experiment, where the unit of I is kg·m${}^{2}$.

**Figure 13.**The falling trajectory and phase diagram as predicted by the analytical model. (

**a**) The falling trajectories corresponding to the six cases in Table 1. (

**b**–

**g**) The plots of $({v}_{{x}^{\prime}},{v}_{{z}^{\prime}},\dot{\theta})$ phase diagrams for the six cases over a simulation time of 20 s. The units of the linear velocities ${V}_{{x}^{\prime}}$ and ${V}_{{z}^{\prime}}$ are m/s, and the unit of angular velocity $\dot{\theta}$ is rad/s.

**Figure 14.**Phase diagram and the projection of the phase trajectory for case 5 in two planes. (

**a**) Phase diagram for case 5. (

**b**) Projection of the phase trajectory in the ${v}_{{z}^{\prime}}$, $\dot{\theta}$ plane. (

**c**) Projection of the phase trajectory in the ${v}_{{x}^{\prime}}$, ${v}_{{z}^{\prime}}$ plane. The units of the linear velocities ${V}_{{x}^{\prime}}$ and ${V}_{{z}^{\prime}}$ are m/s, and the unit of angular velocity $\dot{\theta}$ is rad/s.

**Figure 15.**The simulation results for case 5 as predicted by the quasi-steady model: (

**a**) Local magnification of the drop trajectory when the wing exhibits fluttering behavior. (

**b**) Local magnification of the drop trajectory showing the transformation of the wing from fluttering to tumbling.

**Figure 16.**The free−falling trajectory of the wing after altering the position of the COM in the experiment. (

**a**) Flight trajectory for case 1. (

**b**) Flight trajectory for case 2. (

**c**) Flight trajectory for case 3. (

**d**) Flight trajectory for case 5. (

**e**) Flight trajectory for case 4. (

**f**) Flight trajectory for case 6.

**Table 1.**The relationship between the coordinates of the clump weights, the COM position, and the MOI.

Case | ${\mathit{C}}_{\mathit{xl}}$ (mm) | ${\mathit{C}}_{\mathit{xr}}$ (mm) | ${\mathit{X}}_{\mathit{c}}$ (mm) | ${\mathit{I}}_{\mathit{c}}$ (kg·m${}^{2}$) |
---|---|---|---|---|

1 | −400 | 500 | 4.547 | 0.008 |

2 | −300 | 500 | 10.126 | 0.007 |

3 | −125 | 500 | 19.89 | 0.006 |

4 | 100 | 500 | 32.44 | 0.006 |

5 | 250 | 500 | 40.813 | 0.006 |

6 | 400 | 500 | 49.138 | 0.008 |

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## Share and Cite

**MDPI and ACS Style**

Dou, Y.; Wang, K.; Zhou, Z.; Thomas, P.R.; Shao, Z.; Du, W.
Investigation of the Free-Fall Dynamic Behavior of a Rectangular Wing with Variable Center of Mass Location and Variable Moment of Inertia. *Aerospace* **2023**, *10*, 458.
https://doi.org/10.3390/aerospace10050458

**AMA Style**

Dou Y, Wang K, Zhou Z, Thomas PR, Shao Z, Du W.
Investigation of the Free-Fall Dynamic Behavior of a Rectangular Wing with Variable Center of Mass Location and Variable Moment of Inertia. *Aerospace*. 2023; 10(5):458.
https://doi.org/10.3390/aerospace10050458

**Chicago/Turabian Style**

Dou, Yilin, Kelei Wang, Zhou Zhou, Peter R. Thomas, Zhuang Shao, and Wanshan Du.
2023. "Investigation of the Free-Fall Dynamic Behavior of a Rectangular Wing with Variable Center of Mass Location and Variable Moment of Inertia" *Aerospace* 10, no. 5: 458.
https://doi.org/10.3390/aerospace10050458