Nonlinear Aeroelastic Model Research Articles

Dynamically Linearized Time-Domain Approach for Limit-Cycle Oscillation Prediction for an Elastic Panel

A nonlinear aeroelastic model for a flexible panel is extended to consider the Euler equation as the aerodynamic model. A highly efficient method is presented to compute the generalized aerodynamic forces from a CFD simulation, and the results are compared to those results previously obtained using full potential flow aerodynamics and the linear piston theory. The flutter onset condition as well as limit cycle oscillation amplitudes are computed for a range of supersonic Mach numbers. Additionally, the effects of a shock wave and a viscous boundary layer (Reynolds-averaged Navier–Stokes) are considered in the computational fluid dynamics simulation and are implemented in the aeroelastic model via the generalized aerodynamic force obtained by the new approach. Conclusions are made based on the use and application of this more complete aerodynamic theory, including cases with shock impingement and near-transonic Mach numbers.

Nonlinear-Model-Inversion Control for Stall-Flutter Suppression of an Airfoil via Camber Morphing

This paper presents a nonlinear-model-inversion control law to suppress stall flutter of an airfoil with active trailing-edge morphing. First, a nonlinear aeroelastic model is proposed utilizing two nonlinear autoregressive neural networks with exogenous inputs , which are used to predict aerodynamic moments on an airfoil due to large-amplitude oscillation and camber morphing, respectively. Afterward, a nonlinear-model-inversion control system is designed upon the mentioned aeroelastic system to suppress stall flutter via the camber morphing. A fluid–structure–control (FSC) coupling strategy is developed with structure and control systems embedded in the high-fidelity computational-fluid-dynamics environment to validate the control effect. The FSC high-fidelity simulations show that the nonlinear-model-inversion controller can suppress pitching oscillation completely, whereas a linear proportional–derivative controller without time delay only performs a limited suppression rate by 25.6%. The flowfield evolution result infers that the active camber morphing can generate a converse training-edge vortex, which counteracts the leading-edge vortex during stall flutter. From the perspective of an energy hysteresis, active camber morphing works well by converting injected aerodynamic energy from positive to negative.

Corrosion fatigue analysis of NREL’s 15-MW offshore wind turbine with time-varying stress concentration factors

Over the last twenty years, significant development in wind turbine technologies has led to a dramatic increase in the scale of wind turbines with many now beginning to be installed in offshore locations. Consequently, modern multi-megawatt offshore wind turbines are exposed to increased cyclic loading in addition to an increased risk of corrosion attack. The combination of these two factors may result in wind turbine support structures becoming increasingly vulnerable to fatigue corrosion. The objective of this work is to investigate the impact of material thinning in fatigue-prone areas with respect to fatigue loading and ultimately to examine the potential repercussions on the lifespan of wind turbine support structures. To achieve this, a composite model is constructed coupling results from a multi-body structural dynamic model with time-varying Stress Concentration Factors (SCF) obtained from a finite element model (FEM) of NREL’s 15-MW monopile-based offshore wind turbine. The nonlinear aeroelastic multi-body dynamic model of the wind turbine is used to generate stress time histories for a set of environmental conditions based on the operational conditions of the wind turbine. The finite element model of the wind turbine is then used to identify fatigue-vulnerable regions in the wind turbine support structure and calculate SCFs for these specific areas. The integration of SCFs into the fatigue calculations reduced the lifespan of the turbine tower by a factor of 4, demonstrating the importance of precisely modelling such local stress concentrations for effective fatigue analysis. A novelty of this work arises in the ability of the finite element model to update the SCFs of the fatigue-prone areas over time as corrosion-induced wastage alters the substructure’s geometry, thereby inducing a global redistribution of stresses. A fatigue analysis is carried out availing of the SCFs which vary annually in addition to the stress-time histories produced by the multi-body dynamic model. The results illustrate that the phenomenon of corrosion thinning induced an 8.9% reduction in the fatigue life of the wind turbine tower, thus emphasising the significant importance of proactive maintenance strategies to mitigate the impact of corrosion.

Reduced-order modelling of floating offshore wind turbine: Aero-hydro-elastic stability analysis

A reduced-order aero-hydro-elastic analysis tool for performing the stability analysis on floating offshore wind turbines is presented. In this work, a high-order non-linear aero-elastic model initially developed for onshore wind turbines is extended by utilizing a super-element body, which models the floating platform, the mooring system and the hydrodynamic contributions. The turbine structure is modelled by a Finite beam Element Method, and the aerodynamic loads are modelled by the Blade Element Momentum method coupled with a Beddoes-Leishman type dynamic stall model in a state-space formulation. The linearization is performed around steady state equilibrium at any given mean wind speeds, rotor speeds and collective blade pitch angles utilizing Coleman transformation to remove the periodic terms. The order reduction is based on two projections to reduce the number of structural states and aerodynamic states separately using the structural modal project matrix and the aerodynamic shape functions. Concurrently, we investigate the effect of unsteady aerodynamic states on the stability analysis of a floating wind turbine. The results show the low-order aero-hydro-elastic model expands the scope of numerical tools available for conducting structural modal analysis and aero-hydro-elastic stability analysis on floating wind turbines.

Correlation of supersonic wind tunnel measurements with a nonlinear aeroelastic theoretical/computational model of a thin plate

Supersonic wind tunnel test data is compared with a nonlinear aeroelastic computational model, considering a freestream flow with Mach nearly 2, coupled cavity, and no-shock impingement. The measurements include Limit Cycle Oscillations (LCO) for the given set of flow and structural parameters. The effect of a static pressure differential, temperature differential, and the type of LCO is also studied: periodic or chaotic. A fully coupled aero-thermal–acoustic–elastic analysis was carried out, where the strong dependency of the dynamic instability upon the in-plane boundary stiffness was discovered. Associated with aerodynamic and acoustic formulations, the nonlinearities from the structure are found to be the strongest factor determining the occurrence and character of LCO. At the same time, the cavity effect was found to be negligible for this experimental configuration. Additionally, a comparison analysis was performed on the aerodynamic model for this flow condition between the Linear Piston Theory and the Full Potential Aerodynamics. The calculation of the critical flutter boundary was also performed in terms of the static pressure differential between the cavity pressure and the panel surface pressure, presenting regions where the panel is in static deformation due to the pressure differential versus LCO response. The comparison with the analytical model showed good agreement with the measured data, achieving a similar panel displacement at three-quarters of the panel length as well as the expected behavior of periodicity or chaos for specific values of temperature and static pressure differential.

Variable Blade Inertia in State-of-the-Art Wind Turbine Structural-Dynamics Models

This paper presents a comparison of two methods to represent variable blade inertia in two codes for aero-servo-elastic simulations of wind turbines: the nonlinear aeroelastic multi-body model HAWC2 and the nonlinear geometrically exact beam model BeamDyn for OpenFAST. The main goal is to enable these tools to simulate the dynamic behavior of a wind turbine with variable blade inertia. However, current state-of-the-art load simulation tools for wind turbines cannot simulate variable blade inertia, so the source code of these tools must be modified. The validity of the modified codes is proven based on a simple beam model. The validation shows very good agreement between the modified codes of HAWC2, BeamDyn and an analytical calculation. The add-on of variable blade inertias is applied to reduce the mechanical loads of a 5-megawatt reference wind turbine with an integrated hydraulic-pneumatic flywheel in its rotor blades.

Examination of occurrence probability of vortex-induced vibration of long-span bridge decks by Fokker–Planck–Kolmogorov equation

Vortex-induced vibration (VIV) is a major concern for long-span bridge decks, which may happen especially for closed-box girders under certain environmental wind conditions. Traditionally, sustained VIV is observed as long as the wind speed falls within the lock-in region in wind tunnel tests. However, structural vibration monitoring of long-span bridges has indicated that frequency of VIV events may exhibit a probabilistic pattern. This paper proposes an analytical framework to evaluate the probability of VIV occurrence. First, a nonlinear aeroelastic model with multi-stability limit cycles is used to simulate the bridge deck VIV response within the VIV-triggering wind conditions. The first structural dynamic equilibrium point is usually unstable; the bridge deck instantaneous oscillation amplitude, which is excited by external environmental loads, must exceed this fixed point to trigger the VIV event. The external environmental excitation applied on bridge can be identified from the deck vibration without VIV occurring, then the VIV occurrence probability can be evaluated by Fokker–Planck–Kolmogorov equation. This study utilizes the field measurement data from the Humen Bridge, on which VIV event occurred the first time after 23-year servicing period, because water-filled barriers were placed along the two edges of bridge deck. Afterwards, VIV occurred frequently for over one month. The bridge deck VIV occurrence probability is evaluated by combining environmental wind information, stochastic deck excitation magnitude and a nonlinear aeroelastic VIV model. The results suggest that the proposed methods can approximate the VIV occurrence probability in comparison with empirical estimations using field measurements.

Nonlinear Aeroelastic Prediction in Transonic Buffeting Flow by Deep Neural Network

Transonic buffet is an aerodynamic phenomenon of self-sustained shock oscillations. The aeroelastic problem caused by it is very complex, including two different dynamic modes: forced vibration and frequency lock-in. The vibration of the structure has a negative influence on the fatigue life of the aircraft. Especially in the region of frequency lock-in, the limit cycle oscillations occur due to the instability of the structural mode. Researchers have accurately predicted the region of frequency lock-in in transonic buffet and have clarified its mechanism by using a linear aerodynamic model. However, the nonlinear aeroelastic modeling and prediction of the transonic buffet remain to be solved. The long short-term memory (LSTM) deep neural network is suitable for predicting the time-delayed effects of unsteady aerodynamics. And it has achieved remarkable results in sequential data modeling. In the present work, a nonlinear model is developed for the aeroelastic system with NACA0012 airfoil in transonic buffeting flow and validated with the coupled computational fluid dynamics/computational structural dynamics (CFD/CSD) simulation. First, the data set and the loss function are specially designed. Then, the reduced-order model (ROM) based on the LSTM of the flow is built by using unsteady Reynolds-averaged Navier–Stokes computations data in a post-buffet state. By coupling the ROM and the single degree-of-freedom equation for the pitching angle, the nonlinear aeroelastic model is finally produced. The results show that the phenomenon of frequency lock-in and the self-sustained buffeting aerodynamics are precisely reconstructed. And the model has a strong generalization ability and can reproduce complex vibrations caused by competition between different modes. In short, the model can replace the CFD/CSD method in the current case with high efficiency and accuracy. The method can be used for modeling and prediction of other various complex aeroelastic systems.

Comparison of Blade Aeroelastic Responses between Upwind and Downwind of 10 MW Wind Turbines under the Shear Wind Condition

This paper examines the potential for reducing the cost of energy for super-scale wind turbines through the use of a downwind configuration. Using nonlinear aeroelastic modeling, the responses of 10 MW upwind and downwind wind turbine blades are simulated and compared under shear wind conditions. The study evaluates the impact of both nonlinear and linear aeroelastic models on the dynamic response of different blade sizes, highlighting the need for a nonlinear approach. Results indicate that the linear model overestimates blade deformations (18.14%) and the nonlinear model is more accurate for predicting the aeroelastic response of ultra-long blades of 86.35 m. The study also finds that the downwind turbine blade experiences smaller flapwise moment (17.53%), and blade tip flapwise deformation (33.97%) than the upwind turbine blade, with increased load and deformation fluctuation as wind shear increases.

Nonintrusive Experimental Aeroelastic Analysis of a Highly Flexible Wing

The aeroelastic response of the Delft–Pazy wing to steady and periodic unsteady inflow conditions is analyzed experimentally. The Delft–Pazy wing is a highly flexible wing model based on the benchmark Pazy wing (Avin, O., Raveh, D. E., Drachinsky, A., Ben-Shmuel, Y., and Tur, M., “Experimental Aeroelastic Benchmark of a Very Flexible Wing,” AIAA Journal, Vol. 60, No. 3, 2022, pp. 1745–1768) and exhibits wingtip displacements of more than 24% of the span in the present study. The nonintrusive measurements are performed with an integrated optical approach that provides combined measurements of the structural response of the wing and the unsteady flowfield around it. The aeroelastic loads acting on the wing are derived using physical models and validated against force balance measurements, showing a good agreement for all considered inflow conditions. The analysis of the aeroelastic response of the wing to the unsteady inflow produced by a gust generator shows that both structural and aerodynamic responses depend strongly on the frequency of the gust. The results of this study provide a characterization of the aeroelastic behavior of the Delft–Pazy wing and can serve as a reference for the development of novel and improved nonlinear aeroelastic simulation models.

Function Approximation Technique (FAT)-Based Adaptive Feedback Linearization Control for Nonlinear Aeroelastic Wing Models Considering Different Actuation Scenarios

Function Approximation Technique (FAT)-Based Adaptive Feedback Linearization Control for Nonlinear Aeroelastic Wing Models Considering Different Actuation Scenarios

Aeroelastic prediction in transonic buffeting flow with data fusion method

The frequency locking in phenomenon will occur in the transonic buffeting flow of the elastically supported airfoil. This phenomenon is usually accompanied by a large oscillation of the structure, eventually leading to the fatigue damage of the structure. Transonic aeroelasticity research based on computational fluid dynamics is extremely time-consuming. At present, the research on nonlinear aeroelastic reduced-order models attracts attention due to its important role in nonlinear limit cycle oscillation amplitude prediction. However, the training process of current nonlinear aeroelastic reduced order model requires massive smooth amplitude-modulated pseudo-random binary signals (SAPRBS), which limits its application in engineering field. This paper proposes a nonlinear aeroelasticity reduced-order model based on the saturated limit cycle oscillation signals (Harmonic Signals) that can be directly obtained in engineering field, providing a new perspective for nonlinear reduced-order models. The model only uses harmonic signals at three frequencies to accurately predict limit cycle oscillation amplitudes within a wide range of the NACA0012 transonic buffeting frequency-locking region. In addition, the model still maintains a certain generalization ability at the extrapolated frequency points.

Active flutter suppression for a flexible wing model with trailing-edge circulation control via reinforcement learning

Previous attempts at active flutter suppression have been based on driving the deflection of multiple pairs of discontinuous mechanical control surfaces. Here, we explore the effects of trailing-edge Circulation Control (CC) for flutter control on flexible wings. To avoid the problem that the nonlinear aeroelastic model is difficult to establish accurately, we trained a closed-loop control strategy based on the model-free deep reinforcement learning algorithm through aeroelastic wind tunnel testing. The results show that the strategy can intelligently select the appropriate jet intensity according to the real-time state of the flexible wing. The oscillation amplitude of flutter can be reduced by 92%. The air consumption required for unsteady CC to suppress flutter is reduced by 37% compared to steady CC. This study aims to provide an innovative control method and strategy for active flutter suppression of large aspect ratio flexible wings.

Nonlinear aeroelastic modeling and comparative studies of three degree of freedom wing-based systems

The effects of structural and aerodynamic nonlinearities are investigated on an aeroelastic system with three degrees of freedom, those being plunge, pitch, and flap motions using two different aerodynamic load representations. Stall effects are introduced using quasi-steady and unsteady aerodynamic representations. The effects of the unsteadiness of the flow and stall effects on the type of Hopf bifurcation and the limit cycle oscillations of the system are explored. A comparative study between the unsteady and quasi-steady representations from a nonlinear perspective is also carried out. Results from the linear analysis show that for this specific aeroelastic system, its linear characteristics are dependent on the aerodynamic representation and the system’s design. Additionally, the quasi-steady approximation of the aerodynamic loads results in an inaccurate prediction of the linear flutter speed. Furthermore, the nonlinear analysis shows the existence of an instability shift from the supercritical to the subcritical type, depending on the value of the stall coefficient, as well as that of the cubic nonlinearities, and is more dominantly for the case when the quasi-steady representation is considered. Results show that, even for small reduced frequencies and small angles of attack scenarios, the unsteady representation of aerodynamic loads is critical to correctly predict the flutter speed, instability type, and system’s dynamics.

Flutter Tests of the Pazy Wing

The paper presents the flutter and limit-cycle oscillation (LCO) tests of the Pazy wing, a very flexible wing model designed to study aeroelastic phenomena associated with nonlinear geometry and provide data for nonlinear aeroelastic model validation. The Pazy wing model, analyses, and data from earlier aeroelastic tests are provided by Avin et al. (“Experimental Aeroelastic Benchmark of a Very Flexible Wing,” AIAA Journal, 2022, pp. 1–24). In the current study, three velocity sweeps were performed at different angles of attack (AoA), during which the wing entered and exited limit-cycle oscillations (LCO). A motion-recovery camera system and fiber-optics Bragg grating sensors were used to track the deformations and strains over the wingspan. The wing encountered LCO when the wingtip deformation was approximately 25% of the span. The onset velocity, frequency, and mode shape varied, depending on the test’s AoA. The paper first focuses on the onset of the oscillations, where the onset velocity and oscillation mode-shapes are analyzed. The paper then continues to investigate the LCO characteristics and the nonlinear behavior of the wing. The wing models and experimental data are publicly available through the Third Aeroelastic Prediction Workshop.**

Experimental Aeroelastic Benchmark of a Very Flexible Wing

The paper presents the design, analyses, and testing of the Pazy wing, a very flexible wing model that was designed to study aeroelastic phenomena associated with geometrically nonlinear deflections and to provide data for validation of nonlinear aeroelastic simulation models. The Pazy wing is made of a main thin aluminum spar and a Nylon 12 printed ribs chassis providing the NACA0018 aerodynamic airfoil shape. The wing was covered entirely with a polyester film that serves as the skin. Static loading tests were used for model adaptation and to assure that the wing can deform to over 50% of its span without failing. Ground vibration tests were also used for model adaptation. Because there are uncertainties in the characteristics of the polyester skin under loads, both the static and ground vibration tests were used to asses the contribution of the skin to the wing’s stiffness and determine the stiffness bound with and without the skin. Wind-tunnel tests were conducted at conditions that lead to static and unsteady structural responses. The wing deformations were tracked by a motion recovery system, and the strains over the wingspan were recorded with a Bragg-grating fiber-optics system. In the static wind-tunnel tests, the aerodynamic loads were recorded by a force balance. The wind-tunnel tests included angle-of-attack sweeps in constant airspeed and velocity sweeps in fixed angle of attack. The largest wing-tip static deformation recorded in test was of 50% span. In the velocity-sweep tests, the wing showed strong oscillations at a velocity at which the wing-tip deformation was approximately 25% of the span and became stable again at higher speeds. The wing models and test data are publicly available through the 3rd Aeroelastic Prediction Workshop (https://nescacademy.nasa.gov/workshops/AePW3/public).

Geometrically nonlinear aeroelastic characteristics of highly flexible wing fabricated by additive manufacturing

Additive manufacturing technology has a potential to improve manufacturing costs and may help to achieve high-performance aerospace structures. One of the application candidates would be a wind tunnel wing model. A wing tunnel model requires sophisticated designs and precise fabrications for accurate experiments, which frequently increase manufacturing cost. In this paper, manufacturing accuracy and aeroelastic characteristics of an additively manufactured wing model are evaluated numerically and experimentally. The feasibility of such wings to use in wind tunnel tests is also demonstrated. In addition, a geometrically nonlinear aeroelastic analysis model, which have been developed in the previous study, is validated for static calculations by comparing with results of the wind tunnel test for the additively manufactured highly flexible wing model.

On the application of artificial neural network for the development of a nonlinear aeroelastic model

Aeroelastic Computational Fluid Dynamics simulations have traditionally been associated to a high computational cost, making them prohibitive in a initial phase of the design. Analytic models, which may not be accurate for nonlinear aerodynamics, have traditionally been utilized in order to size those structures. Recently, some authors have proposed the use of artificial neural networks to reduce the error in the prediction of aerodynamic coefficients of bluff bodies, which have separated flow over a substantial part of its wetted surface. This article proposes a method based on neural networks for calculating the dynamic aerodynamic coefficients of a flat plate. The procedure, which is applied for different network typologies (feed-forward and long-short term memory neural networks), is, then, coupled with a structural solver in order to create an aeroelastic reduced order model. The results are compared with CFD aeroelastic simulations, showing a high reduction of computational cost (99%) without penalties in the accuracy. The instabilities are captured and the mean deformation, amplitude and frequency of the motion are predicted. In addition, the different neural network models are compared evidencing that for the aeroelastic calculation feed-forward networks are most efficient in terms of accuracy and computational cost.

Quasi-Periodic Motion and Hopf Bifurcation of a Two-Dimensional Aeroelastic Airfoil System in Supersonic Flow

In this article, the complex dynamic behavior of a nonlinear aeroelastic airfoil model with cubic nonlinear pitching stiffness is investigated by applying a theoretical method and numerical simulation method. First, through calculating the Jacobian of the nonlinear system at equilibrium, we obtain necessary and sufficient conditions when this system has two classes of degenerated equilibria. They are described as: (1) one pair of purely imaginary roots and one pair of conjugate complex roots with negative real parts; (2) two pairs of purely imaginary roots under nonresonant conditions. Then, with the aid of center manifold and normal form theories, we not only derive the stability conditions of the initial and nonzero equilibria, but also get the explicit expressions of the critical bifurcation lines resulting in static bifurcation and Hopf bifurcation. Specifically, quasi-periodic motions on 2D and 3D tori are found in the neighborhoods of the initial and nonzero equilibria under certain parameter conditions. Finally, the numerical simulations performed by the fourth-order Runge–Kutta method provide a good agreement with the results of theoretical analysis.

The suppression of nonlinear panel flutter response at elevated temperatures using a nonlinear energy sink

A nonlinear energy sink (NES) is used to suppress the nonlinear aerothermoelastic response of panels at elevated temperatures. A nonlinear aeroelastic model for a two-dimensional panel with an NES in a supersonic airflow is established using the Galerkin method, with the aerodynamic load being based on first-order piston aerodynamic theory. The model is then used to study the nonlinear dynamic response behavior of a heated panel with an NES to obtain the NES suppression region. The effects of different NES parameters and temperature elevations on the NES suppression region are examined and a design technique is developed to facilitate finding the most suitable combination of parameter values to suppress the nonlinear panel flutter response. The results show that the NES is effective at reducing the aeroelastic response amplitude within a specific range. Varying the installed position, damping, mass ratio and stiffness of the NES will lead to variations in its controlling effect upon the suppression region and the dynamic behavior of the heated panel. An increase in the temperature elevation results in a decrease in the NES suppression region. When the temperature reaches a certain point, the NES suppression region disappears and the NES no longer has any effect on the panel flutter response. Overall, it is found that an NES provides an effect and robust solution, and a wide range of different parameter combinations can be found, that will meet the requirements identified by the proposed design method.