Nonlinear Aeroelastic System Research Articles

О динамической устойчивости нелинейной аэроупругой системы

О динамической устойчивости нелинейной аэроупругой системы

Sliding Mode Control for a Nonlinear Aeroelastic System through Backstepping

This paper considers an aeroelastic two-dimensional wing-flap system with uncertainty and external disturbance. Nonlinear stiffness in pitch is expressed as the second-order polynomial form. In the uncontrolled aeroelastic model, limit cycle oscillation (LCO) appears at postflutter speed. By utilizing leading-edge and trailing-edge control surfaces as the control input, a sliding mode control scheme using varying boundary layers based on backstepping design is developed to suppress aeroelastic vibration. The proposed control strategy combines the advantages of both sliding mode control with varying boundary layers and backstepping technique. In contrast to the conventional sliding mode control design, the proposed method not only avoids chattering but also obtains stronger robustness. The results of numerical simulation clearly demonstrate the effectiveness of the proposed control strategy for aeroelastic vibration suppression under saturated controller input.

Bifurcations and stability analysis for nonlinear oscillations of an airfoil

Analytic and numerical analyses of the nonlinear response of a three-degree-of-freedom nonlinear aeroelastic system are performed. Stability parametric regions and multiple bifurcations for the system are investigated. The damping coefficients of the plunge and pitch are considered as perturbation parameters. With aid of center manifold and normal form theories, the stability regions of the initial equilibrium point and the explicit expressions of the critical bifurcation curves are achieved, which can result in static bifurcation and Hopf bifurcation. Nonlinear flutter happens with a phenomenon of limit cycle oscillations (LCOs) when the damping coefficients are less than or greater than the linear critical values. Under certain conditions, quasi-periodic motions on 2T torus may occur. These complex dynamical behaviours are discussed in detail in this paper. This is useful to characterize the effects of different parameters on structural nonlinearities of the system and ensure that aerodynamic flutter of the airfoil does not take place. Finally, the numerical solutions simulated by four-order Runge–Kutta method are illustrated to tally with the analytic results.

Calculation of Aeroelastic Limit Cycles due to Localized Nonlinearity and Static Preload

The presence of a static preload can significantly alter the limit cycle oscillation response of nonlinear aeroelastic systems. This paper reports a numerical study of two distinct types of preload, mechanical (i.e., independent of flow velocity) and aerodynamic (i.e., dependent on flow velocity), and their effects on limit cycle behavior. Simulations are carried out on models of a wing-with-store and an all-moving surface, respectively, with linear potential flow aerodynamics and a localized structural nonlinearity. Novel, computationally efficient methods based on dual-input describing functions are proposed and employed for calculating limit cycles, thereby generalizing earlier work that used single-input describing functions for the no-preload situation. Results are presented for a smooth nonlinearity (cubic hardening) and a nonsmooth one (classical freeplay), along with selected time marching responses. Finally, the feasibility of including nonlinear aerodynamics in the present framework is discussed.

Controlling Limit Cycle Oscillation Amplitudes in Nonlinear Aeroelastic Systems

The paper focuses on the design of nonlinear state feedback controllers to minimize the amplitude of limit cycle oscillations exhibited by nonlinear aeroelastic systems. Nonlinear normal modes are computed for the closed-loop system to represent the flutter mode dynamics using a single mode. The effectiveness of nonlinear normal modes as a tool to capture the limit cycle oscillation growth is demonstrated. The harmonic balance method is used to estimate the amplitude and frequency of the limit cycle oscillations exhibited by the flutter mode. Analytical estimates of sensitivities of limit cycle amplitude with respect to the introduced control parameters are derived and shown to match closely to the sensitivities computed numerically using the finite difference method on a time marching simulation of the complete aeroelastic system. A multi-objective optimization problem that minimizes the estimate of limit cycle oscillation amplitude and an approximate measure of control cost is solved using the analytical sensitivities. Numerical simulation results are used to verify that minimizing the estimate of the limit cycle amplitude of the flutter nonlinear normal mode corresponds closely to minimizing the simulated limit cycle amplitude of the complete aeroelastic system.

Forecasting the post-bifurcation dynamics of large-dimensional slow-oscillatory systems using critical slowing down and center space reduction

This paper presents a method for forecasting Hopf bifurcations and the post-bifurcation dynamics in complex large-dimensional nonlinear systems. This method is based on the phenomenon of critical slowing down and the projection of the system dynamics on the center space. The approach does not require a model of the system because forecasting is based only on time-series measurements of the system response to perturbations (only) in the pre-bifurcation regime. Techniques are introduced to increase forecasting accuracy and to enable the use of data from recoveries after large perturbations. In contrast to other existing methods, the proposed technique solves the problem of insufficient data due to low-frequency system oscillations. The proposed approach is applied to a nonlinear aeroelastic system which is exposed to gust loads. Numerical simulations show that the method accurately predicts the bifurcation point, the bifurcation type, and the post-bifurcation dynamics in both supercritical and subcritical cases despite the fact that it uses only pre-bifurcation regime data and it does not use a model of the system.

Concurrent Active Piezoelectric Control and Energy Harvesting of Highly Flexible Multifunctional Wings

Both active actuation and energy harvesting of highly flexible wings using piezoelectric transduction are studied in this paper. The piezoelectric effect is included in a strain-based geometrically nonlinear beam formulation. The resulting structural dynamic equations for multifunctional beams are then coupled with a finite-state unsteady aerodynamic formulation, allowing for piezoelectric energy harvesting and actuation with the nonlinear aeroelastic system. With the development, it is possible to provide an integral aeroelastic and electromechanical solution of concurrent active piezoelectric control and energy harvesting for wing vibrations, with the consideration of the geometrical nonlinear effects of slender multifunctional wings. In this paper, linear quadratic regulator and linear quadratic Gaussian controllers are developed for the active control of wing vibrations. The controllers demonstrate effective gust alleviation capabilities. Furthermore, concurrent active vibration control and energy harvesting can also be realized for the multifunctional wings with embedded piezoelectric materials. From this numerical study, the impact of the piezoelectric actuator and energy harvester placement on wing performance is benchmarked.

Multi-dimensional Fokker-Planck equation analysis using the modified finite element method

The Fokker-Planck equation (FPE) is a frequently used tool for the solution of cross probability density function (PDF) of a dynamic system response excited by a vector of random processes. FEM represents a very effective solution possibility, particularly when transition processes are investigated or a more detailed solution is needed. Actual papers deal with single degree of freedom (SDOF) systems only. So the respective FPE includes two independent space variables only. Stepping over this limit into MDOF systems a number of specific problems related to a true multi-dimensionality must be overcome. Unlike earlier studies, multi-dimensional simplex elements in any arbitrary dimension should be deployed and rectangular (multi-brick) elements abandoned. Simple closed formulae of integration in multi-dimension domain have been derived. Another specific problem represents the generation of multi-dimensional finite element mesh. Assembling of system global matrices should be subjected to newly composed algorithms due to multi-dimensionality. The system matrices are quite full and no advantages following from their sparse character can be profited from, as is commonly used in conventional FEM applications in 2D/3D problems. After verification of partial algorithms, an illustrative example dealing with a 2DOF non-linear aeroelastic system in combination with random and deterministic excitations is discussed.

Identification of nonlinear aeroelastic system using fuzzy wavelet neural network

This paper presents a novel fuzzy wavelet neural network structure to identify the nonlinear uncertain aeroelastic system. The identified aeroelastic system considers stiffness, damping non-linearity, dead zones and uncertainties. The proposed fuzzy wavelet neural network (FWNN) is developed from the interval type-2 fuzzy logic system which has the advantage to model uncertainties. Additionally, taking the rapidity and accuracy of the identification into account, the system is characterized by a set of fuzzy IF–THEN rules, and the consequent parts of which is designed to be single hidden layer wavelet neural network. And then, the sliding mode algorithm based on Lyapunov stability theory is introduced to obtain parameter update rules. Furthermore, numerical simulation for a structurally nonlinear prototypical two-dimensional wing section is investigated to verify the effectiveness of the proposed method.

Effectiveness of a nonlinear energy sink in the control of an aeroelastic system

The effectiveness of the nonlinear energy sink in controlling the limit cycle oscillations of a nonlinear aeroelastic system is assessed. The system consists of a rigid airfoil elastically mounted on linear and nonlinear springs. The coupled equations of the airfoil and sink are derived using Lagrange’s equations. The nonlinear quasi-steady aerodynamics are used to model the aerodynamic loads. Parameters including the mass and placement of the passive controller are varied in order to test its efficiency in suppressing undesirable aeroelastic behavior under varying conditions. The nonlinear normal form governing the responses of the airfoil and the energy sink is derived. The contribution of the aerodynamic, structural and sink nonlinearities to the type of instability is quantified. The results show that the nonlinear energy sink has a limited impact on the system’s response in terms of effectively delaying the onset of flutter, changing the type of instability or reducing the amplitude of the limit cycle oscillations.

Improving Performance for Nonlinear Aeroelastic Systems via Sliding Mode Controller

Nonlinear aeroelastic system for controlling the problem of flutter and limit cycle oscillatory (LCO) is proposed. The flight dynamic model describes the plunge and pitch motion of the aircraft section wing with leading and trailing edge control surfaces. Aerodynamic and structural nonlinearities interaction leads to instability phenomena such as flutter and LCO on the wing. Sliding mode control law is designed for the aeroelastic model, and its robustness toward suppressing the LCO and avoiding chatter phenomenon are shown. The results of the simulation show the improvement of the performance of the aeroelastic system. The proposed control law effectively removes the LCO problem and regulates the system states pitch and plunge to origin even under the gust disturbance and limited control surface deflection.

Modeling of Highly Flexible Multifunctional Wings for Energy Harvesting

In this paper, modeling of energy harvesting from transient vibrations of slender wings using piezoelectric transduction is implemented in a strain-based geometrically nonlinear beam formulation. The resulting structural dynamic equations for multifunctional beams are then coupled with a finite-state unsteady aerodynamic formulation, allowing for both energy harvesting and piezoelectric actuation with the nonlinear aeroelastic system. With the development, it is possible to provide an accurate, integral aeroelastic and electromechanical solution of both energy harvesting from and active control for wing vibrations, considering the geometrical nonlinear effects of slender wings. The current paper focuses on modeling the energy harvesting subsystem and exploring its impact on the multifunctional system. Vibrations of a slender multifunctional wing excited by both aeroelastic instability and external wind gusts will be considered as the sources of energy harvesting. All simulations will be completed in the time domain to accurately capture the nonlinear behaviors of the slender multifunctional wing. Based on the time-domain analysis, results of this effort illustrate that the piezoelectric energy harvesting from transient vibrations may provide adequate energy to support onboard sensor operations. In addition, results indicate that a well-tuned piezoelectric energy harvesting system may control the wing vibration using the shunt damping effect.

Uncertainty Quantification of Subcritical Nonlinear Aeroelastic System Using Integrated Interpolation Method and Polynomial Chaos Expansion

The present paper is about analyzing and quantifying parametric uncertainties in a nonlinear aeroelastic system. The response surface of a subcritical system has discontinuities which makes it difficult to analyze using spectral approaches. A quadrature based spectral parametric uncertainty tool called nonintrusive spectral projection (NISP) can capture discontinuities in the response surface only by infinite number of points. As the number of random variable increases, the order of expansion has to increase drastically in order to capture the response accurately. Hence an alternate approach using integrated interpolation scheme which was earlier applied for a thermoacoustic problem is being applied to a classical flutter problem. It is found to be accurate and computationally cheaper than traditional Monte Carlo simulations (MCS) in capturing the response surface. This paper intends to apply the interpolation scheme to the nonlinear aeroelastic model and compares it with quadrature based polynomial chaos expansion (PCE) approach and MCS.

Quantification of Limit Cycle Oscillations in Nonlinear Aeroelastic Systems with Stochastic Parameters

This paper presents an analytical feature for limit cycle oscillation (LCO) in the nonlinear aeroelastic system of an airfoil, with major emphasis on its applications in LCO quantification. The nonlinear stiffness is modeled as the product of the pth power of vibration displacement and the qth power of velocity, with its coefficient as a stochastic parameter. One interesting finding is that the LCO amplitude is directly proportional to the 1/(1-p-q)th power of the coefficient, whereas the frequency is independent of the coefficient. Based on this feature, the statistics and distribution functions of the LCO amplitude are obtained semianalytically, which are validated by Monte Carlo simulations. In addition, we discuss the possible influences of the nonlinear stiffness on flutter suppression of the airfoil subjected to Gaussian white noises. Surprisingly, increasing the nonlinear stiffness alone does not necessarily reduce the vibration amplitude as expected. Instead, it may sometimes induce disastrous subcritical LCOs with much higher vibration amplitudes.

Iterative Learning Control of a Nonlinear Aeroelastic System despite Gust Load

The development of a control strategy appropriate for the suppression of aeroelastic vibration of a two-dimensional nonlinear wing section based on iterative learning control (ILC) theory is described. Structural stiffness in pitch degree of freedom is represented by nonlinear polynomials. The uncontrolled aeroelastic model exhibits limit cycle oscillations beyond a critical value of the free-stream velocity. Using a single trailing-edge control surface as the control input, a ILC law under alignment condition is developed to ensure convergence of state tracking error. A novel Barrier Lyapunov Function (BLF) is incorporated in the proposed Barrier Composite Energy Function (BCEF) approach. Numerical simulation results clearly demonstrate the effectiveness of the control strategy toward suppressing aeroelastic vibration in the presence of parameter uncertainties and triangular, sinusoidal, and graded gust loads.

A REDUCED ORDER MODEL BASED ON BLOCK ARNOLDI METHOD FOR AEROELASTIC SYSTEM

A reduced-order model (ROM) based on block Arnoldi algorithm to quickly predict flutter boundary of aeroelastic system is investigated. First, a mass–damper–spring dynamic system is tested, which shows that the low dimension system produced by the block Arnoldi method can keep a good dynamic property with the original system in low and high frequencies. Then a two-degree of freedom transonic nonlinear aerofoil aeroelastic system is used to validate the suitability of the block Arnoldi method in flutter prediction analysis. In the aerofoil case, the ROM based on a linearized model is obtained through a high-fidelity nonlinear computational fluid dynamics (CFD) calculation. The order of the reduced model is only 8 while it still has nearly the same accuracy as the full 9600-order model. Compared with the proper orthogonal decomposition (POD) method, the results show that, without snapshots the block Arnoldi/ROM has a unique superiority by maintaining the system stability aspect. The flutter boundary of the aeroelastic system predicted by the block Arnoldi/ROM agrees well with the CFD and reference results. The Arnoldi/ROM provides an efficient and convenient tool to quick analyze the system stability of nonlinear transonic aeroelastic systems.

Bayesian model selection for nonlinear aeroelastic systems using wind-tunnel data

In this paper we present a Bayesian framework for parameter estimation and model selection for nonlinear aeroelastic oscillatory systems using their measured response. The framework is motivated by the fact that (1) the oscillations involve the complex nonlinear fluid–structure interaction whose numerical modeling poses both conceptual and computational challenges, (2) several plausible models fit the experimental data with reasonable accuracy, and (3) the model calibration involves estimating the joint probability density function (pdf) of high-dimensional (between 10 and 22) parameter vectors including the variance of model prediction error (modeling error). A four-stage process is followed. First, a candidate model set consisting of parameterized stochastic models is proposed based on the prior understanding of the physics. Second, the joint posterior pdf of the model parameters of each model is estimated using Bayesian inference which involves Markov Chain Monte Carlo (MCMC) sampling complemented by a nonlinear filter for state estimation. Owing to the dense observational data, the state estimation is performed by the extended Kalman filter (EKF). The unnormalized, high-dimensional, joint posterior pdf is sampled through an MCMC method, whereby a parallel version of single-block, adaptive proposal-based Metropolis–Hastings (MH) algorithm is employed to tune the proposal pdf. Third, using Bayes’ theorem each model is weighted according to the evidence provided by the data in its favor. For single-block MCMC sampling, an estimate of the evidence is readily available through the Chib–Jeliazkov method using the posterior samples generated after the burn-in. Fourth, the evidence is used to compute model plausibilities of each model, which are later incorporated in the Bayesian model averaging to make robust predictions. The resulting prediction incorporates uncertainties in the measurements, model parameters, model structure and model selection. A correction term handling the use of improper priors is shown to mimic the effect that diffuse (but proper) priors have on Bayesian model selection. The methodology is applied to wind-tunnel data of a nonlinear aeroelastic oscillator consisting of a rigid wing performing self-sustained small amplitude (<6°) limit cycle oscillation (LCO) in pitch involving complex fluid–structure interactions.

Calculations of the bounds on limit cycle oscillations in nonlinear aeroelastic systems based on equivalent linearization

The nonlinear aeroelastic system of an airfoil with an external store was investigated, with emphasis on the bounds of limit cycle oscillations (LCOs). Based on the equivalent linearization, an approach was proposed to calculate the bounds on LCOs over the full flight envelope. The bounds are determined directly without solving LCOs one by one as the flow speed varies. The presented approach can provide us with the maximal LCO amplitudes and the lower threshold for flow speed beyond which LCOs may arise. Numerical examples show that the obtained bounds are in nice agreement with numerical simulation results. The speed threshold can be predicted to a relative error less than 0.1%, and the maximal LCO amplitude to about 3%. The influences of the system parameters on the speed threshold for speed were investigated efficiently by the proposed approach.

Nonlinear Reduced-Order Modeling for Multiple-Input/Multiple-Output Aerodynamic Systems

The paper presents a novel nonlinear reduced-order modeling approach for multi-input/multi-output aerodynamic systems. The nonlinear reduced-order model for an aerodynamic system includes a finite sum of Wiener-type cascade models. The nonlinear reduced-order model approach starts with fitting a Wiener-type cascade path between the inputs and outputs of the aerodynamic system first. Then, the approach computes the outputs of the path and subtracts them from the measured outputs. The second path is then fitted between the inputs and the output residuals. This process is repeated until the residuals contain only noise. To obtain an optimal path at each stage, a novel nonlinear model, a linear dynamic state-space element followed by a single-layer neural network model, is selected as the Wiener-type cascade model. The Wiener-type cascade model can be optimized by using the Levenberg–Marquadt algorithm. To demonstrate the performance of the proposed nonlinear reduced-order model in modeling the statically nonlinear and dynamically linearized behavior of a nonlinear aerodynamic system, the unsteady transonic compressible flow over a two-degree-of-freedom wing section with the NACA 64A010 airfoil is presented. The numerical results indicate that the proposed nonlinear reduced-order model can accurately identify the outputs of aerodynamic systems subject to a weak excitation. Then, the nonlinear reduced-order model is applied to the transonic flutter analysis of the Isogai wing model. Compared with the direct computational fluid dynamics and linear reduced-order model, the proposed nonlinear reduced-order model is accurate and efficient for transonic flutter prediction of nonlinear aeroelastic systems.

A data-driven quadratic stability condition and its application for stabilizing unknown nonlinear systems

This paper proposes a data-driven stability criterion for quadratic stabilization of unknown nonlinear discrete-time systems. The novelty of this quadratic stability criterion lies in the direct use of the time series of system states, instead of using mathematical models. The data-driven stability criterion is utilized to design a control for stabilizing unknown nonlinear systems using online black-box system identification. The effectiveness and the adaptability of the proposed approach are compared with those of adaptive feedback linearization method with an example of stabilizing a nonlinear aeroelastic system.