# Transient Thrust Analysis of Rigid Rotors in Forward Flight

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Experimental Testing

#### 2.1.1. Description of Experimental Setup and Conventions

#### 2.1.2. Verification of the Rotor-Test Stand

#### 2.1.3. T-MOTOR 18x6.1 Rotor Testing

#### 2.2. Computational Modeling

#### 2.2.1. Overview of the Method

#### 2.2.2. Unsteady Aerodynamic Predictions

#### 2.2.3. Rotor Performance Predictions

#### 2.2.4. Validation of the DDE Method

## 3. Results

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

${C}_{T}$ | Thrust coefficient |

${\overline{C}}_{T}$ | Mean thrust coefficient |

${\widehat{C}}_{T}$ | Normalized thrust coefficient |

c | Chord length |

D | Diameter |

$DDE$ | Distributed doublet element |

J | Propeller advance ratio |

M | Number of chordwise elements |

n | Revolutions per second |

V | Velocity |

${V}_{\infty}$ | Freestream velocity |

${\alpha}_{tpp}$ | Rotor tip-path plane angle of attack |

$\mathsf{\Gamma}$ | Circulation |

$\Delta {x}_{c}$ | Chord length of a DDE element |

$\Delta {x}_{w}$ | Distance traveled by lifting surface in one timestep |

$\widehat{\lambda}$ | Normalized thrust oscillation amplitude |

$\mu $ | Rotor advance ratio |

${\mu}_{\infty}$ | Normalized freestream velocity |

## References

- Brandt, J.B. Small-Scale pRopeller Performance at Low Speeds. Ph.D. Thesis, University of Illinois at Urbana-Champaign, Champaign, IL, USA, 2005. [Google Scholar]
- University of Illinois at Urbana-Champaign, Department of Aerospace Engineering. UIUC Propeller Data Site. 2015. Available online: https://m-selig.ae.illinois.edu/props/propDB.html (accessed on 3 November 2018).
- Kolaei, A.; Barcelos, D.; Bramesfeld, G. Experimental analysis of a small-scale rotor at various inflow angles. Int. J. Aerosp. Eng.
**2018**, 2018, 2560370. [Google Scholar] [CrossRef] [Green Version] - Serrano, D.; Ren, M.; Qureshi, A.J.; Ghaemi, S. Effect of disk angle-of-attack on aerodynamic performance of small propellers. Aerosp. Sci. Technol.
**2019**, 92, 901–914. [Google Scholar] [CrossRef] - Misiorowski, M.; Gandhi, F.; Oberai, A.A. Computational Study on Rotor Interactional Effects for a Quadcopter in Edgewise Flight. AIAA J.
**2019**, 57, 5309–5319. [Google Scholar] [CrossRef] [Green Version] - Carroll, T. A Design Methodology for Rotors of Small Multirotor Vehicles. Master’s Thesis, Ryerson University, Toronto, ON, Canada, 2015. [Google Scholar]
- Katz, J.; Plotkin, A. Low-Speed Aerodynamics; Cambridge Aerospace Series; Cambridge University Press: Cambridge, UK, 2001. [Google Scholar]
- Murua, J.; Palacios, R.; Graham, J.M.R. Assessment of wake-tail interference effects on the dynamics of flexible aircraft. AIAA J.
**2012**, 50, 1575–1585. [Google Scholar] [CrossRef] [Green Version] - Nguyen, A.T.; Kim, J.K.; Han, J.S.; Han, J.H. Extended unsteady vortex-lattice method for insect flapping wings. J. Aircr.
**2016**, 53, 1709–1718. [Google Scholar] [CrossRef] - Simpson, R.J.; Palacios, R.; Murua, J. Induced-drag calculations in the unsteady vortex lattice method. AIAA J.
**2013**, 51, 1775–1779. [Google Scholar] [CrossRef] [Green Version] - Tan, J.F.; Sun, Y.M.; Barakos, G.N. Vortex Approach for Downwash and Outwash of Tandem Rotors in Ground Effect. J. Aircr.
**2018**, 55, 2491–2509. [Google Scholar] [CrossRef] [Green Version] - Tan, J.F.; Wang, H.W. Simulating unsteady aerodynamics of helicopter rotor with panel/viscous vortex particle method. Aerosp. Sci. Technol.
**2013**, 30, 255–268. [Google Scholar] [CrossRef] - Singh, P.; Friedmann, P.P. Application of vortex methods to coaxial rotor wake and load calculations in hover. J. Aircr.
**2018**, 55, 373–381. [Google Scholar] [CrossRef] - Hirato, Y.; Shen, M.; Gopalarathnam, A.; Edwards, J.R. Vortex-sheet representation of leading-edge vortex shedding from finite wings. J. Aircr.
**2019**, 56, 1626–1640. [Google Scholar] [CrossRef] - Saberi, H.; Khoshlahjeh, M.; Ormiston, R.A.; Rutkowski, M.J. Overview of RCAS and application to advanced rotorcraft problems. In Proceedings of the American Helicopter Society 4th Decennial Specialists’ Conference on Aeromechanics, San Francisco, CA, USA, 21–23 January 2004. [Google Scholar]
- Johnson, W. Rotorcraft aerodynamics models for a comprehensive analysis. Annual Forum Proceedings of the American Helicopter Society. Am. Helicopter Soc.
**1998**, 54, 71–94. [Google Scholar] - Russell, C.R.; Sekula, M.K. Comprehensive Analysis Modeling of Small-Scale UAS Rotors. In Proceedings of the AHS International 73rd Annual Forum, Fort Worth, TX, USA, 9–11 May 2017. [Google Scholar]
- Leishman, J.G.; Bhagwat, M.J.; Bagai, A. Free-vortex filament methods for the analysis of helicopter rotor wakes. J. Aircr.
**2002**, 39, 759–775. [Google Scholar] [CrossRef] - Govindarajan, B.; Leishman, J. Curvature corrections to improve the accuracy of free-vortex methods. J. Aircr.
**2016**, 53, 378–386. [Google Scholar] [CrossRef] - Barcelos, D.; Kolaei, A.; Bramesfeld, G. Aerodynamic Interactions of Quadrotor Configurations. J. Aircr.
**2020**, 57, 1074–1090. [Google Scholar] [CrossRef] - Bramesfeld, G.; Maughmer, M. Relaxed-wake vortex-lattice method using distributed vorticity elements. J. Aircr.
**2008**, 45, 560–568. [Google Scholar] [CrossRef] - ATI Industrial Automation. F/T Sensor: Mini45. 2019. Available online: https://www.ati-ia.com/Products/ft/ft_models.aspx?id=Mini45 (accessed on 5 June 2017).
- Krebs, T. A Distributed Doublet-Based Method for Unsteady Aerodynamic Analysis with Relaxed Wakes. Ph.D. Thesis, Ryerson University, Toronto, ON, Canada, 2021. [Google Scholar]
- Drela, M. Flight Vehicle Aerodynamics; MIT Press: Cambridge, MA, USA, 2014. [Google Scholar]
- Johnson, F. A General Panel Method for Analysis and Design of Arbitrary Configurations in Incompressible Flows; NASA CR-3079; Boeing Airplane Company: Chicago, IL, USA, 1980. [Google Scholar]
- Ashby, D.L.; Dudley, M.R.; Iguchi, S.K.; Browne, L.; Katz, J. Potential Flow Theory and Operation Guide for the Panel Code PMARC. 1991. Available online: https://ntrs.nasa.gov/citations/19920023178 (accessed on 12 May 2018).
- Murua, J. Flexible Aircraft Dynamics with a Geometrically-Nonlinear Description of the Unsteady Aerodynamics. Ph.D. Thesis, Imperial College London, London, UK, 2012. [Google Scholar]
- Simpson, R.J.; Palacios, R. Numerical aspects of nonlinear flexible aircraft flight dynamics modeling. In Proceedings of the 54th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, Boston, MA, USA, 8–11 April 2013; p. 1634. [Google Scholar]
- Rusak, Z.; Wasserstrom, E.; Seginer, A. Convergence characteristics of a vortex-lattice method for nonlinear configuration aerodynamics. J. Aircr.
**1985**, 22, 743–749. [Google Scholar] [CrossRef] - Cole, J.A.; Maughmer, M.D.; Bramesfeld, G.; Melville, M.; Kinzel, M. Unsteady Lift Prediction with a Higher-Order Potential Flow Method. Aerospace
**2020**, 7, 60. [Google Scholar] [CrossRef] - Leishman, J. Principles of Helicopter Aerodynamics; Cambridge Aerospace Series; Cambridge University Press: Cambridge, UK, 2002. [Google Scholar]
- Kayran, A. Küssner’s Function in the Sharp Edged Gust Problem-A Correction. J. Aircr.
**2006**, 43, 1596–1599. [Google Scholar] [CrossRef] - Barcelos, D.F.; Kolaei, A.; Bramesfeld, G. Performance prediction of multirotor vehicles using a higher order potential flow method. In Proceedings of the 2018 AIAA Aerospace Sciences Meeting, Kissimmee, FL, USA, 8–12 January 2018; p. 1528. [Google Scholar]

**Figure 1.**Ryerson University’s subsonic wind-tunnel [3].

**Figure 2.**The rotor-test stand configuration and angle of attack convention. (

**a**) The rotor-test stand. (

**b**) Definition of tip-path plane angle of attack [3].

**Figure 3.**Comparison between UIUC and RU wind-tunnel test data for the Master Airscrew 11x7E propeller.

**Figure 4.**T-MOTOR 18x6.1 rotor geometry [3].

**Figure 9.**Comparison of time-averaged thrust forces for various advance ratios and angles of attack.

**Figure 10.**Sectional thrust coefficient $\mathrm{d}{C}_{T}/\mathrm{d}(r/R)$ predictions for ${\alpha}_{tpp}={5}^{\circ}$, $\mu =0.162$. (

**a**) CFD predictions [5]. (

**b**) DDE metond predictions.

**Figure 13.**Rotor and wake visualization of the APC 12x5.5MR (${\alpha}_{tpp}={0}^{\circ}$, ${\mu}_{\infty}=0.25$) as modeled in the DDE method.

**Figure 14.**Normalized thrust oscillation amplitude vs. normalized freestream velocity as predicted with the DDE method.

**Figure 16.**Normalized single blade thrust oscillation amplitude vs. normalized freestream velocity as predicted with the DDE method.

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Krebs, T.; Bramesfeld, G.; Cole, J.
Transient Thrust Analysis of Rigid Rotors in Forward Flight. *Aerospace* **2022**, *9*, 28.
https://doi.org/10.3390/aerospace9010028

**AMA Style**

Krebs T, Bramesfeld G, Cole J.
Transient Thrust Analysis of Rigid Rotors in Forward Flight. *Aerospace*. 2022; 9(1):28.
https://doi.org/10.3390/aerospace9010028

**Chicago/Turabian Style**

Krebs, Travis, Goetz Bramesfeld, and Julia Cole.
2022. "Transient Thrust Analysis of Rigid Rotors in Forward Flight" *Aerospace* 9, no. 1: 28.
https://doi.org/10.3390/aerospace9010028