Nonlinear Aeroelastic System Research Articles

Adaptive Fuzzy Sliding Mode Controller Design Using a High-Gain Observer for a Nonlinear Aeroelastic System under Unsteady Aerodynamics

Adaptive Fuzzy Sliding Mode Controller Design Using a High-Gain Observer for a Nonlinear Aeroelastic System under Unsteady Aerodynamics

Data-Driven Bayesian Inference for Stochastic Model Identification of Nonlinear Aeroelastic Systems

The objective of this work is to propose a data-driven Bayesian inference framework to efficiently identify parameters and select models of nonlinear aeroelastic systems. The framework consists of the use of Bayesian theory together with advanced kriging surrogate models to effectively represent the limit cycle oscillation response of nonlinear aeroelastic systems. Three types of sampling methods, namely, Markov chain Monte Carlo, transitional Markov chain Monte Carlo, and the sequential Monte Carlo sampler, are implemented into Bayesian model updating. The framework has been demonstrated using a nonlinear wing flutter test rig. It is modeled by a two-degree-of-freedom aeroelastic system and solved by the harmonic balance methods. The experimental data of the flutter wing is obtained using control-based continuation techniques. The proposed methodology provided up to a 20% improvement in accuracy compared to conventional deterministic methods and significantly increased computational efficiency in the updating and uncertainty quantification processes. Transitional Markov chain Monte Carlo was identified as the optimal choice of sampling method for stochastic model identification. In selecting alternative nonlinear models, multimodal solutions were identified that provided a closer representation of the physical behavior of the complex aeroelastic system than a single solution.

Research on sparse identification method for aeroelastic dynamic response prediction

Nonlinear aeroelastic system has the characteristics of complex structure, difficult modeling and difficult calculation of dynamic response. For the analysis of nonlinear aeroelastic systems, model identification is a very attractive method. However, the models identified by traditional methods are often relatively complex and limited in scope of use, so it is necessary to develop an interpretable equivalent reduced model. In this paper, sparse regression method and sequential threshold least squares technique are used to establish sparse identification method for complex aeroelastic systems. This method has the ability to identify reduced models containing only required nonlinear terms through measurement data. Then, the sparse identification method is used to identify the binary wing with dead zone nonlinearity and cubic stiffness nonlinearity. The obtained model can provide rapid and accurate prediction of the response of the system according to the sensor measurement, and can also be used as an explicit surrogate model for aeroelastic optimization design, thus verifying the superiority of the proposed method.

A modified frequency–time domain method for nonlinear aeroelastic systems with initial conditions

The frequency–time domain method is a very powerful tool for analyzing nonlinear aeroelastic systems; however, it is limited to obtaining steady-state solutions. To investigate nonlinear dynamic behaviors with initial conditions, a modified frequency–time domain method is proposed. Initial conditions, represented as pseudo forces, are introduced into the nonlinear feedback loops and are subsequently transformed from the time domain into the frequency domain. Hence, the proposed method allows nonlinear responses with initial conditions. Owing to the characteristics and limitations of this method, certain skills are presented to simulate both accurately and efficiently. Numerical results are provided for, in addition to the Van Der Pol equation, a two-degree-of-freedom airfoil section, and a three-degree-of-freedom aeroelastic typical section with control surface freeplay. Consequently, the feasibility of introducing initial conditions is validated. The comparison with the Runge–Kutta algorithm demonstrates the accuracy and efficiency of the proposed method, as an alternative to time-marching approaches. Furthermore, several key parameters, such as the sampling frequency and ending time of convolution integral, are investigated to explore the efficiency. The sensitivity of parameters and calculation methods of convolution integral is observed, which contributes to the strategies for adequate accuracy.

Non-normality and transient growth in stall flutter instability.

The non-normal nature and transient growth in amplitude and energy of a pitch-plunge aeroelastic system undergoing dynamic stall are explored in this paper through numerical and supporting experimental studies. Wind tunnel experiments, carried out for a canonical pitch-plunge aeroelastic system in a subsonic wind tunnel, show that the system undergoes stall flutter instability via a sub-critical Hopf bifurcation. The aeroelastic responses indicate a transient growth in amplitude and energy-possibly triggering the sub-criticality, which is critical from the purview of structural safety. The system also shows transient energy growth followed by decaying oscillation for certain initial conditions, whereas sustained limit cycle oscillations are encountered for other initial conditions at flow speeds lower than the critical speed. The triggering behavior observed in the wind tunnel experiments is understood better by resorting to study the numerical model of the nonlinear aeroelastic system. To that end, a modified semi-empirical Leishman-Beddoes dynamic stall model is adopted to represent the nonlinear aerodynamic loads of the pitch-plunge aeroelastic system. The underlying linear operator and its pseudospectral analysis indicate that the aeroelastic system is non-normal, causing amplification in amplitude and energy for a short period.

Adaptive Sampling for Interpolation of Reduced-Order Aeroelastic Systems

A new strategy for the interpolation of parametric reduced-order models of dynamic aeroelastic systems is introduced. Its aim is to accelerate the numerical exploration of geometrically nonlinear aeroelastic systems over large design spaces or multiple flight conditions. The parametric reduced-order models are obtained from high-dimensional models by Krylov-subspace projection. They are subsequently interpolated to acquire realizations inexpensively anywhere in the parameter space, where having the state space as opposed to interpolating an output metric permits the use of linear analysis tools. The interpolation scheme is heavily conditioned by the available training points; thus, a novel methodology based on adaptive sampling and a combinatorial use of the available true-systems knowledge is presented, whereby we reuse all known data of the true models as different combinations of training and testing data to build statistical surrogates of the interpolation error across the parameter space and refine the sampling in those regions that need it. This minimizes the number of costly to evaluate function calls and ensures that parameter space regions are sampled according to the underlying system dynamics. The initial implementation of this adaptive sampling strategy is demonstrated on a very flexible wing with a complex stability envelope.

Experimental investigation on the synchronization characteristics of a pitch-plunge aeroelastic system exhibiting stall flutter.

This study focuses on characterizing the bifurcation scenario and the underlying synchrony behavior in a nonlinear aeroelastic system under deterministic as well as stochastic inflow conditions. Wind tunnel experiments are carried out for a canonical pitch-plunge aeroelastic system subjected to dynamic stall conditions. The system is observed to undergo a subcritical Hopf bifurcation, giving way to large-amplitude limit cycle oscillations (LCOs) in the stall flutter regime under the deterministic flow conditions. At this condition, we observe intermittent phase synchronization between pitch and plunge modes near the fold point, whereas synchronization via phase trapping is observed near the Hopf point. Repeating the experiments under stochastic inflow conditions, we observe two different aeroelastic responses: low amplitude noise-induced random oscillations (NIROs) and high-amplitude random LCOs (RLCOs) during stall flutter. The present study shows asynchrony between pitch and plunge modes in the NIRO regime. At the onset of RLCOs, asynchrony persists even though the relative phase distribution changes. With further increase in the flow velocity, we observe intermittent phase synchronization in the flutter regime. To the best of the authors' knowledge, this is the first study reporting the experimental evidence of phase synchronization between pitch and plunge modes of an aeroelastic system, which is of great interest to the nonlinear dynamics community. Furthermore, given the ubiquitous presence of stall behavior and stochasticity in a variety of engineering systems, such as wind turbine blades, helicopter blades, and unmanned aerial vehicles, the present findings will be directly beneficial for the efficient design of futuristic aeroelastic systems.

Vibration control and trajectory tracking for nonlinear aeroelastic system based on adaptive iterative learning control

Vibration control and trajectory tracking for nonlinear aeroelastic system of 2D airfoil of wind turbine subjected to flap-wise vibration and independent pitch motion, are investigated based on hydraulic pitch control and adaptive iterative learning (AIL) control. The aeroelastic system is simplified by an external slubber, the host material of which is a composite material with shape memory alloy (SMA) wires embedded in. The independent pitch motion is driven by two-way-rack cylinder and hydraulic servo valve. The control of divergent unstable motions is implemented by tracking the vibrations of the preset displacements. The control law of AIL algorithm combines iterative algorithm and proportional-derivative (PD) control. The control parameters of PD controller are adjusted by proportional-derivative control based on single neuron (PDC/SN) algorithm. Based on the convergence analysis of the iterative algorithm, the tracking results (including position tracking and speed tracking) of the iterative process are shown, and the convergence of the absolute values of the relative errors during tracking is verified. The experiment based on process control and a refined OPC Technology verifies the feasibility of the engineering application of the proposed algorithm in nonlinear aeroelasticity. The consistency between the real-time dynamic tracking process shown on the experimental panel and the theoretical tracking process illustrated by simulation verifies the effectiveness of the control algorithm for engineering application. The research implication for the design of current turbine manufacturers is that it provides a design idea of blade buffer based on smart materials, and an implementation idea for the engineering application of adaptive AIL algorithm.

Nonlinear aeroelastic analysis of a damped elastica-aerofoil system

This work formulates a comprehensive model of a nonlinear aeroelastic system developed for the analysis of complex aeroelastic phenomena related to structural and aerodynamic nonlinearities. The system is formulated as a two-dimensional cantilevered elastica with a rigid airfoil section firmly attached at its tip undergoing large displacements in the crosswind conditions. The system can demonstrate a wide range of domain specific as well as coupled nonlinear phenomena. The structural model is developed by means of the Rayleigh–Ritz approach, with shape functions discretizing both vertical and horizontal displacements and Lagrangian multipliers enforcing inextensibility. Damping is modeled based on a non-local strain-based mechanism in the Kelvin–Voigt arrangement. The resulting structural model is examined through studying the behavior under a follower load and with a tip-attached tendon under tension to study the shape convergence properties and the alignment of the results with known characteristics in the literature. The ONERA dynamic stall model is used to model the aerodynamics of the problem to accurately capture post-stall behavior at large deformations. The LCO responses of the aeroelastic problem are evaluated through time-marched simulations, and the combined airspeed–damping interactions are studied in this manner.

Nonlinear State Dependent Riccati Equation Controller for 3-DOF Airfoil with Cubic Structural Nonlinearity

In this paper, Limit Cycle Oscillations (LCOs) of a Two-Dimensional (2-D) airfoil with cubic structural nonlinearity in the plunge Degree of Freedom (DOF) are investigated. The aerodynamic loads are modelled using the unsteady Theodorsen’s theory. Wagner function and Jones’ approximation are used to transform the unsteady aerodynamic loads from frequency domain into time domain. The aeroelastic differential equations are solved using the routine ODE45 in MATLAB to get the system response. Both subcritical and supercritical LCOs are observed in the 3-DOF airfoil with structural nonlinearity in the plunge DOF. Such LCOs are undesirable phenomena and should be suppressed within the flight envelope. A nonlinear state feedback controller is designed to minimize the amplitude of LCOs presented in the considered nonlinear aeroelastic system. The State-Dependent Riccati Equation (SDRE) approach in combination with Kalman filter technique is used to design a controller for the 2-D airfoil with trailing edge control surface (flap). The forces and moments produced by this flap’s action are used to stabilize the system and suppress the existence of LCOs. The efficiency of the proposed nonlinear controller by the SDRE and Kalman filter in suppressing the existence of subcritical and supercritical LCOs are verified by the numerical simulation results.

Prescribed Performance Adaptive Control for a Nonlinear Aeroelastic System with Input Constraint

A constrained prescribed performance compensation controller is proposed for a nonlinear aeroelastic system in the presence of wind gust, system uncertainties, and input saturation. To deal with the effects of the nonsmooth saturation nonlinearity, an approximate saturation function is introduced into the controller design, which can smoothly approximate the real saturation with arbitrarily prescribed precision. Specifically, by designing the prescribed performance function, a fixed-time control framework is designed to ensure that the closed-loop system has the prescribed tracking performance. The designed control algorithm can not only compensate the adverse effect caused by disturbances and uncertainties but also restrain the excessive amplitude of control input. Finally, the stability analysis shows that all the signals in the closed-loop system are semiglobally uniformly ultimately bounded via the Lyapunov stability analysis method, and simulation results are presented to demonstrate the feasibility and effectiveness of the proposed method.

Stall-induced fatigue damage in nonlinear aeroelastic systems under stochastic inflow: Numerical and experimental analyses

This study focuses on characterizing the fatigue damage accumulated in nonlinear aeroelastic systems subjected to stochastic inflows through both numerical simulations and wind tunnel experiments. In the mathematical model, nonlinearities are assumed to exist either in the structure (via a cubic hardening nonlinearity in the pitch stiffness), or in the flow (via dynamic stall condition), or simultaneously in both the structural and aerodynamic counterparts. The aerodynamic loads in the attached flow and dynamic stall conditions are estimated using Wagner’s formulation and semi-empirical Leishman–Beddoes model, respectively. To augment the findings to in-field flow conditions, the oncoming wind flow is considered to be randomly time-varying in nature. The stochastic input flow fluctuations are modeled using a Karhunen–Loeve Expansion formulation. The response dynamics and the associated fatigue damage of the aeroelastic system, possessing different sources of nonlinearities, are systematically investigated under isolated cases of deterministic and stochastic input flows. Specifically, the pertinent role of stochasticity in the input flow is brought out by presenting the response dynamics and the associated fatigue damage accumulation for different values of noise intensity and time scale of the input flow fluctuation. It is demonstrated that under fluctuating flow conditions, the dynamics intermittently switch between attached flow and the dynamic stall regimes even at low mean flow speeds. The intermittent nature of the response varies as the time scale and intensity of the oncoming flow are varied. The role of torsional stresses as the predominant component dictating the fatigue damage accumulation irrespective of the source of nonlinearity is illustrated. Using the rainflow counting method and Miner’s linear damage accumulation theory, it is shown that the accumulated fatigue damage is substantially higher under stochastic flow conditions as compared to deterministic input flows. Importantly, it is observed that different time scales and intensities of the oncoming flow fluctuation play a pivotal role in dictating the fatigue damage in aeroelastic systems. Finally, fatigue damage is observed to be significantly higher for torsionally dominant oscillations in the dynamical stall regime compared to the oscillations at the attached flow regime. The numerical findings are strengthened by drawing comparisons with the preliminary results obtained from wind tunnel experiments performed on a NACA 0012 airfoil undergoing dynamic stall. To the best of our knowledge, this is the first study that systematically bridges the dichotomy between the stall-induced dynamical signatures in stochastic aeroelastic systems and maps the same to the corresponding structural damage.

Amplitude Control of Stall-Induced Nonlinear Aeroelastic System Based on Iterative Learning Control and Unified Pitch Motion

In this study, vibration control, a behavior which subordinates to stall-induced nonlinear vibration and amplitude control of a wind turbine’s blade section, based on unified pitch motion driven by slider-linkage mechanism, is investigated by using an iterative learning control (ILC) method. The nonlinear dynamical system is a nonlinear aeroelastic system. The aeroelastic system equations consist of three parts: the nonlinear structural equations derived by using Lagrange’s equations, the improved stall-induced nonlinear ONERA (ISNO) aerodynamic equations, and the pitch control equation. The ISNO model is not only suitable for the actual external pitch motion, but also suitable for the solution by using an ILC algorithm due to its fitted nonlinear aerodynamic coefficients. The ILC algorithm used here is an improved iterative learning algorithm (IILC) which considers the large-range, linearized, residual terms, and realizes gain adaptive tuning based on PID controller. On the one hand, it can control the amplitude of an unsteady flutter through trajectory tracking. On the other hand, when the preset value of the amplitude of the ideal trajectory is very small, it can make the system directly tend to convergence and stability of a nonlinear aeroelastic system. To simplify the extremely difficult iterative process, the pitch movement can track the elastic twist displacement in time, thus simplifying the aeroelastic equations and accelerating the IILC iteration process. Therefore, amplitude control for flap-wise/lead-lag displacements is realized by the unified pitch motion and the trajectory tracking controlled by using the IILC algorithm.

Event-triggered intelligent critic control with input constraints applied to a nonlinear aeroelastic system

In this paper, we establish an event-triggered intelligent control scheme with a single critic network, to cope with the optimal stabilization problem of nonlinear aeroelastic systems. The main contribution lies in the design of a novel triggering condition with input constraints, avoiding the Lipschitz assumption on the inverse hyperbolic tangent function. Based on an improved weight updating criterion that eliminates the requirement of initial admissible control, the control law is obtained approximately by online training of a single critic network. The Lyapunov stability and the Zeno phenomenon of the closed-loop system are analysed. The feasibility of the established algorithm is verified by applying it to an optimal stabilization task of a nonlinear aeroelastic system. The results reveal that the developed approach can handle input-constrained optimal control problems, with performance comparable to the time-based method that updates control inputs at each instant, while reducing the computational and communication's load.

Combined selection of the dynamic model and modeling error in nonlinear aeroelastic systems using Bayesian Inference

We report a Bayesian framework for concurrent selection of physics-based models and (modeling) error models. We investigate the use of colored noise to capture the mismatch between the predictions of calibrated models and observational data that cannot be explained by measurement error alone within the context of Bayesian estimation for stochastic ordinary differential equations. Proposed models are characterized by the average data-fit, a measure of how well a model fits the measurements, and the model complexity measured using the Kullback–Leibler divergence. The use of a more complex error models increases the average data-fit but also increases the complexity of the combined model, possibly over-fitting the data. Bayesian model selection is used to find the optimal physical model as well as the optimal error model. The optimal model is defined using the evidence, where the average data-fit is balanced by the complexity of the model. The effect of colored noise process is illustrated using a nonlinear aeroelastic oscillator representing a rigid NACA0012 airfoil undergoing limit cycle oscillations due to complex fluid–structure interactions. Several quasi-steady and unsteady aerodynamic models are proposed with colored noise or white noise for the model error. The use of colored noise improves the predictive capabilities of simpler models.

Influence of Bistable Plunge Stiffness on Nonlinear Airfoil Flutter

Abstract Aeroelastic systems have nonlinearities that provide a wide variety of complex dynamic behaviors. Nonlinear effects can be avoided in practical applications, as in instability suppression or desired, for instance, in the energy harvesting design. In the technical literature, there are surveys on nonlinear aeroelastic systems and the different manners they manifest. More recently, the bistable spring effect has been studied as an acceptable nonlinear behavior applied to mechanical vibration problems. The application of the bistable spring effect to aeroelastic problems is still not explored thoroughly. This paper contributes to analyzing the nonlinear dynamics of a typical airfoil section mounted on bistable spring support at plunging motion. The equations of motion are based on the typical aeroelastic section model with three degrees-of-freedom. Moreover, a hardening nonlinearity in pitch is also considered. A preliminary analysis of the bistable spring geometry's influence in its restoring force and the elastic potential energy is performed. The response of the system is investigated for a set of geometrical configurations. It is possible to identify post-flutter motion regions, the so-called intrawell and interwell. Results reveal that the transition between intrawell to interwell regions occurs smoothly, depending on the initial conditions. The bistable effect on the aeroelastic system can be advantageous in energy extraction problems due to the jump in oscillation amplitudes. Furthermore, the hardening effect in pitching motion reduces the limit cycle oscillation (LCO) amplitudes and also delays the occurrence of the snap-through.

Fuzzy control of nonlinear aeroelastic system based on neural network identification

This article presents a fuzzy control method for the limit cycle oscillation (LCO) suppression of nonlinear aeroelastic systems based on the neural network identification algorithm. A prototypical 2D wing section with a single control surface at the trailing edge of the main wing, which contains a symmetrical free play nonlinearity in the pitch degree of freedom, is modeled to illustrate the proposed method. A neural network is used to identify the fuzzy control rules from the existing LCO suppression input and output data. A new fuzzy control rate of the nonlinear aeroelastic system is obtained by adjusting the parameters of the fuzzy control surface. Numerical simulations are conducted to verify the effectiveness of the proposed method.

Adaptive fuzzy control of nonlinear aeroelastic system with measurement noise

This paper presents a limit cycle oscillation (LCO) suppression method of nonlinear aeroelastic system based on adaptive neuro-fuzzy control. A prototypical 2D wing section with a single control surface at the trailing edge of the main wing, which contains a symmetrical freeplay nonlinearity in the pitch degree of freedom, is modelled by SIMULINK (Matlab 2016R) to illustrate the proposed method. Proportional integral differential (PID) controller is used to suppression the LCO of nonlinear aeroelastic system. The control law of the PID controller is identified by neural network. A new fuzzy control law of the nonlinear aeroelastic system is obtained by adjusting the parameters of the fuzzy control system. A nonlinear aeroelastic system with measurement noise in the measurement feedback loop is conducted to verify the effectiveness of the proposed method.

Feedback control of a nonlinear aeroelastic system with non-semi-simple eigenvalues at the critical point of Hopf bifurcation

Abstract The feedback control of Hopf bifurcation of nonlinear aeroelastic systems with asymmetric aerodynamic lift force and nonlinear elastic forces of the airfoil is discussed. For the Hopf bifurcation analysis, the eigenvalue problems of the state matrix and its adjoint matrix are defined. The Puiseux expansion is used to discuss the variations of the non-semi-simple eigenvalues, as the control parameter passes through the critical value to avoid the difficulty for computing the derivatives of the non-semi-simple eigenvalues with respect to the control parameter. The method of multiple scales and center-manifold reduction are used to deal with the feedback control design of a nonlinear system with non-semi-simple eigenvalues at the critical point of the Hopf bifurcation. The first order approximate solutions are developed, which include gain vector and input. The presented methods are based on the Jordan form which is the simplest one. Finally, an example of an airfoil model is given to show the feasibility and for verification of the present method.

Investigating amplitude death in a coupled nonlinear aeroelastic system

Coupling nonlinear dynamical systems can lead to a host of phenomena, one of which leads to the complete cessation of their oscillations. This phenomenon is referred to as amplitude death (AD) in the dynamical systems literature. Recently, there is a growing interest to mitigate oscillatory or dynamic instabilities in a variety of engineering systems using AD. Deriving impetus from the same, we investigate the possibility of cessation of oscillatory instabilities in aeroelastic systems using the concept of AD. In specific, the suppression of flutter instability that arises in aeroelastic systems due to highly nonlinear fluid–structure interactions is investigated from the purview of AD. To that end, we consider two identical airfoils that are exposed to input fluid forces along the axial directions. The airfoils, via translational and rotational springs, are allowed to oscillate in plunge and pitch degrees of freedom. To augment the findings to in-field scenarios, we consider the individual cases of the input flow to be either uniform (deterministic) or possess randomly time varying components, respectively. To promote coupled interactions, the airfoils are coupled using a linear torsional spring. The coupled interaction of the airfoils at the flutter regime are then studied by obtaining the pitch and plunge responses. Further, the strength of the coupling and the time delay between the airfoils are systematically varied to investigate its effect on the regime of AD. The ability of AD to suppress flutter instability and thereby serve as a possible flutter suppression mechanism in both deterministic and stochastic input flow cases is then demonstrated. Finally, the phase delay values between the airfoils is computed to heuristically present a relationship between the coupling parameters and the post coupling signatures.