# Neuro-Fuzzy Network-Based Reduced-Order Modeling of Transonic Aileron Buzz

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## Abstract

**:**

_{1}213 airfoil is investigated at different flow conditions, while the aileron deflection and the hinge moment are considered in particular. The aileron is integrated in the wing section without a gap and is modeled as rigid. The dynamic equations of the rigid aileron rotation are coupled with the URANS-based flow model. For ROM training purposes, the aileron is excited via a forced motion and the respective aerodynamic and aeroelastic response is computed using a computational fluid dynamics (CFD) solver. A comparison with the high-fidelity reference CFD solutions shows that the essential characteristics of the nonlinear buzz phenomenon are captured by the selected ROM method.

## 1. Introduction

_{1}213 airfoil with an integrated control surface at transonic buzz condition. Therefore, a high-fidelity model of the aeroelastic system is defined by coupling the CFD aerodynamic model with the dynamics of the rigid aileron.

## 2. Reduced-Order Model Approach

- Model Initialization: As a first step, a global linear model (${\psi}_{1}=1,N=1$) is calculated by estimating the NFM weights (${w}_{i0},{\mathbf{w}}_{i}$) by means of a local linear least-squares algorithm [25]. Therefore, the available CFD training data set is applied.
- LLM error estimation: Within the next step, the worst performing local linear model is located by means of a locally-defined loss function, which is evaluated for all available models ($i=1,..,N$). The LLM yielding the highest prediction error is selected for the subsequent splitting procedure [12]. If only the global model is available, it is automatically chosen for further refinement.
- LLM refinement: The LLM which shows the lowest performance is divided into two models using an axis-orthogonal split [25]. For each resulting model, the centers (${c}_{i}$), widths ($\sigma $) and weights (${w}_{i0},{\mathbf{w}}_{i}$) must be re-computed. The centers are defined as the centers of the corresponding jth hyperrectangle, while the widths are determined by defining the input space extension of the LLM scaled by a factor ${k}_{\sigma}$ [12]. According to Nelles [25], ${k}_{\sigma}$ is chosen as $1/3$. The linear weights are determined by the application of the local weighted least-squares optimization.
- Error evaluation: In order to evaluate the best splitting configuration, a loss function is applied. Therefore, in contrast to the weight estimation procedure in training step one, the available validation data set is applied. Based on the error evaluation, the partition-setup yielding the lowest error is chosen for the last training step.
- Termination: The aforementioned training steps are repeated until the relative change of the error as calculated in the previous step becomes smaller than a user-defined value. As an alternative, the splitting process can be terminated by defining a maximum number of LLMs [12].

## 3. Structural and Numerical Setup

#### 3.1. Structural Model

_{1}213 airfoil with an integrated aileron. The aeroelastic model in the present work represents the coupling between the structural degree of freedom of the aileron, defined by the aileron deflection angle $\delta $ and the aerodynamic response, given by the hinge moment ${M}_{H}$. The structural model is represented by a fixed two-dimensional wing section with a reference length of ${c}_{ref}$ = 1 m. The aileron is hinged by the three-quarter chord location (${x}_{H}$ = 75%) and its degree of freedom is modeled as rigid. Therefore, structural elastic and dissipative contributions are neglected in the present study. The motion of the aileron is described by the aileron deflection angle $\delta \left(t\right)$ about the hinge point, whose dynamics are expressed by the following one-degree of freedom equation:

_{1}213 airfoil with the implemented positive and negative aileron deflection is visualized.

#### 3.2. CFD-Solver

_{1}213 test case is implemented by means of a dynamic mesh deformation, which is defined by a user defined function.

## 4. Application

#### 4.1. Training Data Generation

_{1}213 airfoil appears between a minimum and maximum aileron deflection amplitude of ${\delta}_{min}$ = $-{12}^{\circ}$ and ${\delta}_{max}$ = $+{9}^{\circ}$, respectively. Therefore, the selected training signal covers the amplitude range of interest. Further, according to Steger and Bailey [4], aileron buzz of the NACA65

_{1}213 airfoil is characterized by varying freestream conditions. Therefore, in the present study three different flow conditions are considered, defined by a freestream Mach number of $M{a}_{\infty}$ = [0.8, 0.82, 0.83], a Reynolds number of $Re=20.7\times {10}^{6}$ and an angle of attack of $\alpha =-{1}^{\circ}$. For the first and third condition the initial aileron deflection is constrained to ${\delta}_{start}={0}^{\circ}$, whereas the deflection of the second condition is defined by ${\delta}_{start}={4}^{\circ}$. Further, the corresponding reduced frequencies of the selected buzz conditions are summarized in Table 1:

#### 4.2. Nonlinear System Identification

#### 4.2.1. Aerodynamic System Identification

#### 4.2.2. Aeroelastic System Identification

## 5. Computational Effort

## 6. Conclusions

_{1}213 airfoil test case. The structural model is represented by a fixed two-dimensional airfoil geometry with an integrated, rigid aileron. A one-degree of freedom aeroelastic model is implemented in the solver, which allows for an appropriate aeroelastic coupling for the representation of non-classical aileron buzz.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Nomenclature

$\alpha $ | angle of attack, deg |

$\delta $ | aileron deflection angle, deg |

${\Sigma}_{i}$ | basis function widths of LLM |

$\sigma $ | standard deviation |

$\tau $ | nondimensional time |

${\psi}_{i}$ | fuzzy membership function |

$\omega $ | angular frequency, 1/s |

${c}_{i}$ | centers of LLM |

${C}_{{M}_{H}}$ | hinge moment coefficient |

${c}_{ref}$ | reference chord length, m |

F | nonlinear function mapping |

${g}_{i}$ | linear weights of the MLP neural network |

${\mathbf{G}}_{i}$ | nonlinear weights of the MLP neural network |

${I}_{H}$ | moment of inertia, kgm^{2} |

k | discrete time step |

k | spring constant |

${k}_{red}$ | $2\pi f\xb7{c}_{ref}/{U}_{\infty}$, reduced frequency |

${k}_{\sigma}$ | space extension factor of LLM |

m | dynamic delay order of NFM-MLP input vector |

M | number of neurons of MLP neural network |

$M{a}_{\infty}$ | freestream Mach number |

${M}_{H}$ | hinge moment, N/m |

n | dynamic delay order of NFM-MLP output |

N | number of LLM |

${N}_{MC}$ | number of Monte Carlo iterations |

${N}_{s}$ | number of training samples |

q | input of NFM |

${Q}_{i}$ | fit factor |

$Re$ | Reynolds number |

t | time, s |

${t}_{H}$ | start time of deflection, s |

${U}_{\infty}$ | freestream velocity, m/s |

$\mathbf{v}$ | input vector of MLP |

${w}_{i}$ | weights of LLM |

$\mathbf{x}$ | input vector of NFM |

x | position of points on aileron surface |

${x}_{H}$ | position of hinge point, % |

$\widehat{y}$ | scalar output of the NFM |

$\overline{y}$ | mean deviation |

$\stackrel{\u2018}{y}$ | mean ROM output |

$\tilde{y}$ | scalar output of NFM |

${y}_{c}$ | camber line |

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**Figure 1.**Illustration of the combined recurrent local linear neuro-fuzzy model and multilayer perceptron neural network [12].

**Figure 3.**Mach number contour plots showing the buzz cycle of the NACA65

_{1}213 airfoil ($M{a}_{\infty}$ = $0.82$, $Re$ = $20.7\times {10}^{6}$, $\alpha $ = $-{1}^{\circ}$, $\delta $ = $\pm {12}^{\circ}$). ${T}_{Buzz}$ refers to the buzz period.

**Figure 4.**Comparison of numerical and experimental results in terms of varying aileron angle and shock position with time (NACA65

_{1}213 airfoil, $M{a}_{\infty}=0.8$, $Re=20.7\times {10}^{6}$, $\alpha =-{1}^{\circ}$, $\delta =\pm {9}^{\circ}$).

**Figure 5.**Time series of the training signal (APRBS) for the excitation of the ailerons structural degree of freedom.

**Figure 6.**Grid convergence study with regard to the hinge moment coefficient ${C}_{{M}_{H}}$ (NACA65

_{1}213 airfoil, $M{a}_{\infty}=0.8$, $Re=20.7\times {10}^{6}$, $\alpha ={0}^{\circ}$, $\delta ={0}^{\circ}$).

**Figure 7.**Hinge moment coefficient response caused by the APRBS excitation (NACA65

_{1}213 airfoil, $M{a}_{\infty}$ = 0.82, $Re$ = $20.7\times {10}^{6}$, $\alpha $ = $-{1}^{\circ}$, $\delta $ = $\pm {12}^{\circ}$). Besides the CFD reference solution, the simulation result of the NFM-MLP ROM is shown.

**Figure 8.**Responses of the hinge moment coefficient resulting from harmonic aileron pitching motion with ${k}_{red,Ex}$ = [0.5, 0.6, 0.7, 0.8]. The results of the NFM-MLP ROM are compared to the reference CFD solution (NACA65

_{1}213 airfoil, $M{a}_{\infty}$ = 0.82, $Re$ = $20.7\times {10}^{6}$, $\alpha $ = $-{1}^{\circ}$, ${\delta}_{min,max}$ = $-12$/+${9}^{\circ}$).

**Figure 9.**Frequency domain responses of the hinge moment coefficient resulting from harmonic aileron pitching motion with ${k}_{red,Ex}$ = [0.5, 0.6, 0.7, 0.8]. The results of the NFM-MLP ROM are compared to the reference CFD solution (NACA65

_{1}213 airfoil, $M{a}_{\infty}$ = 0.82, $Re$ = $20.7\times {10}^{6}$, $\alpha $ = $-{1}^{\circ}$, ${\delta}_{min,max}$ = $-12$/$+{9}^{\circ}$).

**Figure 10.**Stable aeroelastic response of the hinge moment coefficient (NACA65

_{1}213 airfoil, $M{a}_{\infty}$ = 0.8, $Re$ = ${10}^{6}$, $\alpha $ = ${0}^{\circ}$). The result of the NFM-MLP ROM is compared to the reference CFD solution.

**Figure 11.**Responses of the resulting aileron deflection angle and hinge moment coefficient at a selected buzz condition (NACA65

_{1}213 airfoil, $M{a}_{\infty}$ = 0.82, $Re$ = $20.7\times {10}^{6}$, $\alpha $ = $-{1}^{\circ}$, ${\delta}_{start}=-{4}^{\circ}$). The results of the NFM-MLP ROM are compared to the reference CFD solution.

**Figure 12.**Responses of the resulting aileron deflection angle and hinge moment coefficient at a selected buzz condition (NACA65

_{1}213 airfoil, $M{a}_{\infty}$ = 0.83, $Re$ = $20.7\times {10}^{6}$, $\alpha $ = $-{1}^{\circ}$, ${\delta}_{start}={0}^{\circ}$). The results of the NFM-MLP ROM are compared to the reference CFD solution.

**Figure 13.**Responses of the resulting aileron deflection angle and hinge moment coefficient at a buzz condition with time decay (NACA65

_{1}213 airfoil, $M{a}_{\infty}$ = 0.8, $Re$ = $20.7\times {10}^{6}$, $\alpha $ = $-{1}^{\circ}$, ${\delta}_{start}={0}^{\circ}$). The results of the NFM-MLP ROM are compared to the reference CFD solution.

**Table 1.**Corresponding reduced frequencies ${k}_{\mathrm{red},\mathrm{Buzz}}$ of selected buzz conditions.

${\mathrm{Ma}}_{\infty}$ | 0.8 | 0.82 | 0.83 |

${k}_{\mathrm{red},\mathrm{Buzz}}$ | 0.67 | 0.76 | 0.79 |

${k}_{\mathrm{red},\mathrm{Ex}}$ | 0.5 | 0.6 | 0.7 | 0.8 |

$\mathit{Q}$(${C}_{{M}_{H}}$) | 91.37% | 92.89% | 92.45% | 91.06% |

**Table 3.**Evaluation of fit factors for the aeroelastic investigation ($Re$ = $20.7\times {10}^{6}$, $\alpha $ = $-{1}^{\circ}$).

${\mathrm{Ma}}_{\infty}$ | 0.8 | 0.82 | 0.83 |

${\mathit{Q}}_{\mathit{i}}$($\mathit{\delta}$) | 84.37% | 90.89% | 88.47% |

${\mathit{Q}}_{\mathit{i}}$(${\mathit{C}}_{{\mathit{M}}_{\mathit{H}}}$) | 85.12% | 89.05 % | 88.83% |

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**MDPI and ACS Style**

Zahn, R.; Breitsamter, C.
Neuro-Fuzzy Network-Based Reduced-Order Modeling of Transonic Aileron Buzz. *Aerospace* **2020**, *7*, 162.
https://doi.org/10.3390/aerospace7110162

**AMA Style**

Zahn R, Breitsamter C.
Neuro-Fuzzy Network-Based Reduced-Order Modeling of Transonic Aileron Buzz. *Aerospace*. 2020; 7(11):162.
https://doi.org/10.3390/aerospace7110162

**Chicago/Turabian Style**

Zahn, Rebecca, and Christian Breitsamter.
2020. "Neuro-Fuzzy Network-Based Reduced-Order Modeling of Transonic Aileron Buzz" *Aerospace* 7, no. 11: 162.
https://doi.org/10.3390/aerospace7110162