# Tonal-Noise Assessment of Quadrotor-Type UAV Using Source-Mode Expansions

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## Abstract

**:**

## 1. Introduction

- -
- thickness noise associated with fluid-displacement effects around moving blades;
- -
- loading noise generated by the steady and unsteady blade forces, essentially the lift;
- -
- flow noise due to flow inhomogeneities around the blades.

## 2. Basic Aerodynamic Features of Quadrotors

## 3. Prediction of Tonal Loading Noise

#### 3.1. General Formulation

- -
- the angle of the dipoles is the same value $\gamma \left(r\right)$ as for the actual fluctuating lift;
- -
- the angular frequency $\omega $ of the stationary dipoles is $mB\phantom{\rule{0.166667em}{0ex}}\mathrm{\Omega}$;
- -
- the dipole strength at the azimuth $\alpha $ along the circle is defined as ${F}_{s}\phantom{\rule{0.166667em}{0ex}}{\mathrm{e}}^{\mathrm{i}\phantom{\rule{0.166667em}{0ex}}n\alpha}$.

**Figure 2.**Source and observer coordinates in the individual reference frame of a rotor. Similar coordinates are defined for other rotors, with both possible directions of rotation.

#### 3.2. Far-Field Formulation

#### 3.3. Steady-Loading Noise and Forward-Flight Induced Unsteady-Loading Noise

#### 3.4. Generic Potential-Interaction Noise Model

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Generic quadrotor architecture with main geometrical parameters (

**a**) and addressed test cases of hover (struts) and of forward flight (no strut) (

**b**).

**Figure 3.**Idealized instantaneous velocity triangle for a blade cross-section in forward flight with negative pitch (red) and main definitions.

**Figure 4.**Weighting factors on the steady-loading noise (

**a**) and on the low-order blade-loading harmonics of unsteady-loading noise $s=1$ and $s=-1$ ((

**b**) and (

**c**), respectively). Values of $X=Msin\theta $ by steps of 0.1 from 0.2 to 0.9. Value $X=0.18$ featured in red added as representative of the drone D1 ($B=2$). Value $X=0.7$ highlighted in blue as representative of the quadrotor reported by Yoon et al. [19] ($B=3$). Vertical dashed lines standing for the first three BPF orders $n=B,\phantom{\rule{0.166667em}{0ex}}n=2B,\phantom{\rule{0.166667em}{0ex}}n=3B$.

**Figure 5.**(

**a**) Isocontours of the sectional thrust coefficient on a front rotor of the quadrotor D1 in forward flight, from [18]. Typical values given on the plot. (

**b**) Modulus of the associated Fourier coefficients for ten circular cuts.

**Figure 6.**Two periods of the unwrapped angular profiles of sectional thrust for the 10 circular cuts in the map of Figure 5a (numbered on the plot). Inner and outer profiles highlighted on the left and right parts by black and colored markers, respectively, for clarity.

**Figure 7.**Three-dimensional directivity patterns of forward-flight noise, at $10\phantom{\rule{0.166667em}{0ex}}{R}_{T}$ from the quadrotor center. Single rotor (gray-blue) and pair of front rotors (dark red). Flight in the direction ${\mathbf{e}}_{X}$. Synchronized rotors with zero phasing (top) and half-blade passage phasing (bottom). Configuration D1, scales in arbitrary decibels.

**Figure 8.**Computed instantaneous pressure maps of the steady-loading noise (

**a**–

**d**) and of the total noise (

**e**,

**f**), in the plane of the front rotors of configuration D1, at the BPF. (

**a**,

**b**) Single-rotor signature; (

**c**,

**e**) zero phasing; (

**d**,

**f**) half-blade passage phasing. Rotations indicated by arrows, flight direction along ${\mathbf{e}}_{X}$, tip-radius circles drawn in white.

**Figure 9.**Azimuthal profiles of the upwash on the blades (normal to the chord) according to the potential-interaction model [25]. Configurations D1 (

**a**) and D2 (

**b**).

**Figure 11.**Three-dimensional directivity patterns of potential-interaction noise at $10\phantom{\rule{0.166667em}{0ex}}{R}_{T}$ from the quadrotor center. Single rotor (gray-blue) and pair of front rotors (dark red). Flight in the direction ${\mathbf{e}}_{X}$. Half-blade passage phasing synchronization. Configurations D1 (upper plots) and D2 (lower plots), same scales in arbitrary decibels.

**Table 1.**Main parameters of the quadrotor configurations used as test cases. Drone D1 used in both Section 3.3 and Section 3.4, drone D2 used only in Section 3.4.

L (m) | D (m) | ${\mathit{R}}_{\mathit{H}}$ (m) | ${\mathit{R}}_{\mathit{T}}$ (m) | d (m) | a (m) | $\mathbf{\Omega}$ (rpm) | |
---|---|---|---|---|---|---|---|

drone D1 | 0.229 | 0.323 | 0.019 | 0.152 | 0.076 | 0.014 | 4058 |

drone D2 | 0.177 | 0.250 | 0.01 | 0.083 | 0.025 | 0.012 | 4058 |

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**MDPI and ACS Style**

Roger, M.; Moreau, S. Tonal-Noise Assessment of Quadrotor-Type UAV Using Source-Mode Expansions. *Acoustics* **2020**, *2*, 674-690.
https://doi.org/10.3390/acoustics2030036

**AMA Style**

Roger M, Moreau S. Tonal-Noise Assessment of Quadrotor-Type UAV Using Source-Mode Expansions. *Acoustics*. 2020; 2(3):674-690.
https://doi.org/10.3390/acoustics2030036

**Chicago/Turabian Style**

Roger, Michel, and Stéphane Moreau. 2020. "Tonal-Noise Assessment of Quadrotor-Type UAV Using Source-Mode Expansions" *Acoustics* 2, no. 3: 674-690.
https://doi.org/10.3390/acoustics2030036