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We investigate the effects of parameter uncertainties on the dynamical response of an aeroelastic model representing an oscillating airfoil in pitch and plunge with linear aerodynamics and cubic structural nonlinearities. An approach based on the stochastic normal form is proposed to determine the effects due to the variations in the flow speed and the structural stiffness terms on the stability of the aeroelastic system near the Hopf bifurcation point. This approach allows us to study analytically the bifurcation scenario and to predict the amplitude and frequency of the limit cycle oscillation (LCO). The results show that the amplitude of LCO corresponding to the supercritical Hopf bifurcation increases with the intensity of the noise perturbing the pitch and plunge cubic terms, but there is almost no effect on the LCO frequency. Uncertainties in the flow speed produce a shift in the bifurcation point, and unstable subcritical behavior may occur for values of parameters for which the corresponding deterministic model is stable. The stochastic normal form confirms and extends previously known numerical results regarding the effect of parameter variations, and offers an effective way to perform sensitivity analysis of the system's response.

In this paper we present an expert system to perform steady-state response predictions. We consider an aeroelastic model simulating a two degree-of-freedom airfoil oscillating in pitch and plunge with a freeplay nonlinearity in the pitch degree-of-freedom. In the proposed data-driven methodology, a freeplay is first confirmed, and then the locations of the switching points are determined. A state-space formulation is constructed to model the piece-wise linear system. The parameters of the system are estimated using the Kalman filter and the expectation maximization algorithm. The attractive feature of the present approach is its ability to accurately predict the steady-state behavior of the nonlinear aeroelastic system with freeplay, using only a limited amount of transient input data. To demonstrate the effectiveness of the proposed methodology, we present applications to freeplay aeroelastic data arising from wind tunnel experiments and numerical simulations.

Aging aircraft and combat aircraft that carry heavy external stores potentially face problems arising from nonlinearities in structure. An expert data mining system is proposed that is capable of predicting the asymptotic behavior of an aeroelastic system with structural nonlinearities represented by polynomial restoring forces or freeplay models. The input is represented only by a limited set of transient data. The output provides a long-term nonlinear aeroelastic response, and the prediction is made when certain rule-based reasoning conditions are satisfied. An attractive feature of this new approach is that no information about the system parameters is needed. In the prediction module, we propose two methods, based on nonlinear time series models and the unscented Kalman filter. To our knowledge, these approaches have not been reported so far for predicting the long-term nonlinear aeroelastic responses. Compared with the classical extended Kalman filter, the unscented filter does not require differentiability and can be applied to nonlinear aeroelastic models with freeplay and hysteresis. The performances of the expert data mining system are demonstrated for simulated data and wind-tunnel experimental aeroelastic data resulting from a two-degree-of-freedom airfoil oscillating in pitch and plunge.

A new approach is proposed, based on the use of artificial neural networks, for predicting nonlinear aeroelastic oscillations. Our objective is to reconstruct the asymptotic state of the nonlinear behavior of an aeroelastic model when only a limited segment of the transient data is known. An original neural network architecture is proposed and is used to predict the nonlinear motions of an aeroelastic system modeling a self-excited two-degree-of-freedom airfoil oscillating in pitch and plunge. When a segment of the transient state of the given signal is used for training, the neural network is capable of correctly predicting the corresponding limit-cycle oscillations, damped oscillations, or unstable divergent oscillations. The network training set consists of numerically generated data or data obtained from a wind-tunnel experiment. A neural network used in conjunction with a wavelet decomposition is presented, which proves to be capable of extracting the values of the damping coefficients and frequencies from the predicted signal. Neural networks, thus, prove to be useful tools in nonlinear aeroelastic analysis.

This paper investigates the dynamic response of a two-dimensional aeroelastic system with structural non-linearity represented by hysteresis. The formulations of the point transformation method developed in Part 1 of this study for the aeroelastic system with a freeplay model is extended for a hysteresis model. These formulations can be applied not only to predict the amplitude and frequency of limit cycle oscillations, but also to detect complex aeroelastic responses such as periodic motion with harmonics, period doubling, chaotic motion and the coexistence of stable limit cycles. It is shown that the point transformation technique is the most suitable to analyze the aeroelastic response of systems containing piecewise continuous restoring forces.

A point transformation technique is developed to investigate the non-linear behavior of a two-dimensional aeroelastic system with freeplay models. Two formulations of the point transformation method are presented, which can be applied to accurately predict the frequency and amplitude of limit cycle oscillations. Moreover, it is demonstrated that the developed formulations are capable of detecting complex aeroelastic responses such as periodic motions with harmonics, period doubling, chaotic motions and the coexistence of stable limit cycles. Applications of the point transformation method to several test examples are presented. It is concluded that the formulations developed in this paper are efficient and effective.

In this study, a frequency relation for limit cycle oscillations of a two-degree-of-freedom aeroelastic system with structural non-linearities represented by cubic restoring spring forces is derived. The centre manifold theory is applied to reduce the original system of nine-dimensional first order ordinary differential equations to a governing system in two dimensions near the bifurcation point. The principle of normal form is used to simplify the non-linear terms of the lower dimensional system. Using the frequency relation and the amplitude–frequency relationships derived from a previous study, limit cycle oscillations (LCOs) for self-excited systems can be predicted analytically. The mathematical technique proposed here has been applied to investigate LCO near a Hopf-bifurcation for an aeroelastic system with cubic restoring forces. Not only that an excellent agreement is obtained compared to the numerical results from solving the original system of eight non-linear differential equations by Runge–Kutta time integration scheme, but we also demonstrate that the use of a mathematical approach leads to a better understanding of non-linear aeroelasticity.

As an effect of continual advances in high performance computing, numerical simulations are widely used to study complex aeroelastic systems. Uncertainty quantification (UQ) methods are an important part of the verification and validation process of the computational simulation results, because small variations in the system parameters can have a large influence on the aeroelastic response. In this paper, we use an approach based on the stochastic normal form to analyze the impact of parameters uncertainty on the Hopf bifurcation and on the amplitude and frequency of the limit cycle oscillations (LCO) of a nonlinear airfoil structure. We start from the following deterministic model describing a two-degree-of-freedom airfoil oscillating in pitch and plunge

Statistical techniques are developed to analyze aeroelastic transient data. By applying a higher order spectral method and the Hinich test for Gaussianity, the presence of the nonlinearity can be identied with condence. The work presented here is motivated by the recent interests in the study of nonlinear aeroelastic behavior using data analysis approach. Statistical procedures which have been frequently employed in dealing with self-exciting autoregressive models are extended to analyze aeroelastic data. The methods are capable of identifying aeroelastic response associated with a freeplay nonlinearity. Moreover, we show that the freeplay parameters can be estimated by a kernel-based non-parametric method. The applications of the statistical analysis tools using numerically simulated aeroelastic data are presented.

An improved neural network-based method for predicting nonlinear oscillations in the aeroelastic response is presented. An articial neural network is trained using the limited available information of a short transient data set, and the asymptotic state of the signal is reconstructed by a multi-step (or recursive) prediction process. An enhanced two-layer feedforward neural network with features that control the propagation of the prediction errors is designed. Methods for consistently choosing the number of network inputs and of neurons in the hidden layer for a given application are reported. The proposed predictor has been applied to wind-tunnel experimental data that model an oscillating airfoil with polynomial restoring forces, as well as to signals generated numerically by solving a dieren tial system that models a self-excited two-degree-of-freedom airfoil oscillating in pitch and plunge.

Aging aircraft and combat aircraft that carry heavy external stores potentially face problems arising from nonlinearities in structure. An expert data mining system is proposed that is capable of predicting the asymptotic behavior of an aeroelastic system with structural nonlinearities represented by polynomial restoring forces or freeplay models. The input is represented only by a limited set of transient data. The output provides a long-term nonlinear aeroelastic response, and the prediction is made when certain rule-based reasoning conditions are satisfied. An attractive feature of this new approach is that no information about the system parameters is needed. In the prediction module, we propose two methods, based on nonlinear time series models and the unscented Kalman filter. To our knowledge, these approaches have not been reported so far for predicting the long-term nonlinear aeroelastic responses. Compared with the classical extended Kalman filter, the unscented filter does not require differentiability and can be applied to nonlinear aeroelastic models with freeplay and hysteresis. The performances of the expert data mining system are demonstrated for simulated data and wind-tunnel experimental aeroelastic data resulting from a two-degree-of-freedom airfoil oscillating in pitch and plunge.

In this paper, the stochastic collocation method (SCM) is applied to investigate the nonlinear behavior of an aeroelastic system with uncertainties in the system parameter and the initial condition. Numerical case studies for problems with uncertainties are carried out. In particular, the performance of the SCM is compared with solutions based on other computational techniques such as Monte Carlo simulation, Wiener chaos expansion and wavelet chaos expansion. From the computational results, we conclude that the SCM is an effective tool to study a nonlinear aeroelastic system with random parameters.