Limit Cycle Oscillation Response Research Articles

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.

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.

Reduced Order Model based on SPOD for Aerothermoelastic Analysis of a Hypersonic Panel

This study has established a reduced order model (ROM) based on spectral proper orthogonal decomposition (SPOD) method in order to proceed an aerothermoelastic response analysis of a hypersonic panel. The two-way coupling between aerothermal and aeroelastic systems is applied [1]. Three aspects of POD-ROM are investigated: 1) the selection of snapshots for POD modes; 2) the comparison between classical POD [2] and SPOD [3]; 3) how to find global POD modes in a parameter space of flight altitude and Mach number. The snapshots are sampled from aerothermoelastic response data via the classical Galerkin method. The numerical results show that the dynamic response of the panel in time history is stable flat first, dynamically stable buckling, and subsequently flutter occurs with transient chaos and then limit cycle oscillation (LCO). Moreover, we find that: 1) the selection of the entire time response as the snapshots for production of POD modes reveals that the long-term buckling and LCO response will interfere the resulting POD modes. Therefore, taking the transient chaotic response as snapshots is a better choice; 2) compared with the classical POD method, the proposed SPOD method for the present aerothermoelastic system improves the accuracy and efficiency obviously; 3) the global POD modes here are essentially the natural modes of the aerothermoelastic panel and thus are related to the temperature of panel. This is of great significance to further understand the physical mechanism of the present aerothermoelastic system and develop fast calculation methods.

Coupled Newton–Krylov Time-Spectral Solver for Flutter and Limit Cycle Oscillation Prediction

Flutter and limit cycle oscillation (LCO) are important phenomena that need to be considered in aircraft design. Previous harmonic-balance-based flutter and LCO prediction methods either have low linear convergence rates or require expensive Newton steps to achieve quadratic convergence. In this paper, we propose a preconditioned, Jacobian-free, coupled Newton–Krylov (CNK) method for the time-spectral aeroelastic equations. By solving the coupled system directly, the method reduces the computational cost of each Newton step, making quadratic convergence affordable. The proposed Jacobian-free method is easier to implement and requires less memory relative to previous methods. We demonstrate the capability of the CNK solver by verifying the results against a time-accurate solver and by comparing them to other harmonic-balance-based results reported in the literature. We observe that the proposed method is more efficient than the time-accurate method in LCO response simulations. And the LCO velocities and frequencies predicted by the proposed method and the time-accurate method are within 1% of relative difference when the same mesh is used. This method can be potentially used in aircraft design.

Gust Response Computations with Control Surface Freeplay Using Random Input Describing Functions

An analytical and computational investigation of the effect of control surface freeplay on aeroelastic behavior, including random gust response and limit cycle oscillations (LCO), is presented. An efficient method to compute the total gust+LCO response is developed using random input describing functions (RIDF), in a manner analogous to harmonic input describing functions for LCO calculation. Results are obtained for an airfoil with trailing edge flap freeplay and linear potential flow aerodynamics at various gust strengths, freeplay sizes, and flow velocities. Both stable and unstable responses are detected. It is found that LCO coexists with gust response for weak gust/large freeplay combinations, and it is quenched for sufficiently strong gust/small freeplay. Time marching results are provided to validate the RIDF method and confirm the findings. The present work appears to be the first use of RIDF in aeroelasticity although it is well known to nonlinear control systems analysts. Notably the RIDF method in its present form is also applicable when steady flow aerodynamic nonlinearities are included.

Nonlinear Dynamic Responses of Shear-Deformable Composite Panels under Combined Supersonic Aerodynamic, Thermal, and Random Acoustic Loads

The skin panel structures of vehicles in supersonic flights are subjected to combined thermal, acoustic, and aerodynamic loads. These combined loads may cause a nonlinear dynamic response of the high-speed flight vehicle’s panel structures. Among these nonlinear dynamic responses, the snapthrough and limit cycle oscillation response seriously affect the fatigue failure of the panel structures. This work investigates the nonlinear dynamic responses using the numerical method, when combined supersonic aerodynamic, thermal, and random acoustic loads are applied to the panel structures simultaneously. To consider the thin and thick composite panels, the first-order shear deformation plate theory (FSDT) and the von Karman nonlinear displacement–strain relationship are applied. The aerodynamic load in the supersonic flow is modeled using the first-order piston theory. The thermal load distribution is assumed constant in the thickness direction of the composite panel. The random acoustic load is represented as stationary white-Gaussian random pressure with zero mean and uniform magnitude over the panels. The nonlinear equations of motion of the composite panel under combined loads are derived using the principle of virtual work and the finite element method. The static displacement, which is the solution of the nonlinear static equation, is calculated using the Newton–Raphson method, and the nonlinear dynamic equation is solved using the Newmark-β time integration method. Using these numerical methods, the nonlinear dynamic analyses are conducted under various loading conditions such as thermal–random acoustic loads, thermal–supersonic aerodynamic loads, and supersonic aerodynamic–thermal–random acoustic loads. Numerical results show the nonlinear dynamic response of the composite thin and thick panels such as snapthrough and limit cycle oscillation responses. Particularly, the snapthrough response is caused when the random acoustic load is applied appropriately to the thermally buckled composite plate when the aerodynamic load is not applied or applied with the relatively small magnitude of the dynamic pressure.

Discrete Adjoint Approach for Nonlinear Unsteady Aeroelastic Design Optimization

Presented is a discrete adjoint approach for computing nonlinear unsteady aeroelastic geometric design sensitivities, which can subsequently be used for nonlinear unsteady aeroelastic geometric design optimization. The methodology is applicable to compressible Reynolds-averaged Navier–Stokes computational-fluid-dynamics solvers and is based on a harmonic balance nonlinear frequency-domain technique for modeling nonlinear aerodynamics in the frequency domain. Automatic differentiation is used to derive the computer code representing the adjoint gradient of the harmonic balance computational-fluid-dynamics solver. Discrete adjoint method airfoil geometric design nonlinear unsteady aerodynamic sensitivities are first computed for a benchmark NLR 7301 airfoil transonic aeroelastic configuration, and results are compared with finite difference computations to demonstrate the accuracy of the methodology. The airfoil surface is mathematically curve-fit using an airfoil parameterization technique, and the discrete adjoint methodology provides the gradient of the flutter onset or limit-cycle oscillation reduced velocity with respect to changes in each of the parameters of the airfoil curve-fit methodology for the cost of a single adjoint solution. This gradient information is supplied to a limited memory Broyden–Fletcher–Goldfarb–Shanno (BFGS) optimization solver, which then determines values for the curve-fit parameters that maximize flutter onset or limit-cycle oscillation reduced velocity. The discrete adjoint nonlinear unsteady aeroelastic geometric design sensitivity computation methodology is then used to redesign the shape of the benchmark NLR 7301 airfoil aeroelastic configuration to optimize the aeroelastic flutter onset reduced velocity or the reduced velocity for a specified limit-cycle oscillation response amplitude.

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.

Nonlinear Aeroelastic Analysis with Inextensible Plate Theory Including Correlation with Experiment

A computational aeroelastic model is presented for a cantilevered plate rotated from a position with the clamped edge normal to the flow (0 deg) to where the clamped edge is aligned with the flow (90 deg) using a new inextensible plate theory. A large-amplitude limit-cycle oscillation has been observed beyond the linear flutter speed in both the computations and in a wind-tunnel experiment. The computations and experiments for the limit-cycle oscillation response of the several yawed plate aeroelastic models show that the inextensible plate theory produces results in good agreement with the measurements. Also, a comparison with results obtained using von Kármán’s nonlinear plate theory has been made.

Flow field around the flapping flag

The flapping flag is a canonical fluid–structure interaction problem that describes a cantilever plate with flow along its elastic axis. When the flapping flag loses stability it enters a large amplitude Limit Cycle Oscillation (LCO). While theoretical models can accurately predict the flutter velocity and frequency, there are still discrepancies between the experimental observations and the theoretical predictions of the post-critical LCO response. This note provides recent flow field visualizations in a single longitudinal plane for a cantilevered aluminum plate in axial flow during its LCO. Particle Image Velocimetry (PIV) techniques are used to show that the flow over the midspan of the plate is attached even during the violent LCO motion. This observation suggests that potential flow aerodynamic models may be able to capture the essential features in the flow field.

Underwing Missile Aerodynamic Effects on Flight-Measured Limit-Cycle Oscillations

F-16 testing revealed differences in limit-cycle oscillation response characteristics associated with subtle aerodynamic variations of the underwing missiles. Physical length differences between long and short missiles led to the accidental discovery of the limit-cycle oscillation sensitivity. In addition to the length difference, the long missiles were found to have builtup collar sections, to which the fins and canards were attached to the missile body. Two cases are presented where both long and short missiles were carried underwing. For the case with no wingtip missiles present, carriage of the long missiles showed significant limit-cycle oscillation response, while the short missiles showed very little response. For the case with wingtip missiles present, the limit-cycle oscillation response levels were slightly higher for carriage of the short missiles but were otherwise quite similar to the long-missile responses. Computational fluid dynamics on a rigid wing suggest that the differences between the long and short missiles alter the flowfield by changing the impinging shock strength and location on the lower wing surface outboard of the underwing missile. For the empty wingtip launcher case, the location of the impinging shock on the lower surface influences the strength of the shock on the upper surface. This influence on the upper shock strength could be a key contributor to the limit-cycle oscillation mechanism when wingtip aerodynamics do not inhibit the flowfield.

Aeroelastic Response Induced by Free Play, Part 1: Theory

A computational code has been developed for aeroelastic analysis. The structural bending and torsional stiffness for an all-movable horizontal tail are basedupon the experimental data published inWrightAirDevelopmentCenter technical report 54-53 (Niles, R. H., and Spielberg, I. N., “Subsonic Flutter Tests of An Unswept All-Movable Horizontal Tail,”WrightAirDevelopment Center TR-54-53,March 1954.). A linear three-dimensional time domain vortex lattice aerodynamic model is used to study the flutter and also nonlinear limit-cycle oscillations induced by a free-play gap in the actuating mechanism at the root of the tail as well as the effects of the nominal angle of attack on limit-cycle oscillation response. It is found that the angle of attack significantly affects the limit-cycle oscillation behavior and if sufficiently largemay suppress the limit-cycle oscillation. The computed flutter results are compared with the experimental data with good quantitative agreement between theory and experiment for flutter in this paper, Part 1, and for limit-cycle oscillation in Part 2 (Tang, D., andDowell, E. H., “Aeroelastic Response Induced by Free Play, Part 2: Theoretical/Experimental Correlation Analysis,” AIAA Journal, Vol. 49, No. 11, 2011, pp. 2543– 2554).

Supercritical and subcritical Hopf bifurcation and limit cycle oscillations of an airfoil with cubic nonlinearity in supersonic\hypersonic flow

In this paper, the Hopf bifurcations and limit cycle oscillations (LCOs) of an airfoil with cubic nonlinearity in supersonic\hypersonic flow are investigated. The harmonic balance method and multivariable Floquet theory are applied to analyze the LCOs of the airfoil. Four distinct cases of the LCOs response are detected in this system: (I) supercritical Hopf bifurcation, (II) a single subcritical Hopf bifurcation, (III) two subcritical Hopf bifurcations, and (IV) no Hopf bifurcation. Furthermore, the parameter variations domains separating the supercritical and subcritical Hopf bifurcations are presented using singularity theory.

Multibody Dynamic Approach of Flight Dynamics and Nonlinear Aeroelasticity of Flexible Aircraft

The flight and structural dynamics of a very flexible aircraft, especially while maneuvering, aremore tied together than those of conventional aircraft. Hence, it is necessary to study the coupled effects of flight dynamics, aeroelasticity, and control. This paper presents a multibody dynamic approach in which the flexible aircraft is modeled as a feedback-controlled multibody system under aerodynamic forces. The governing equations of the system are established by combining the equations of amultibody system, the ONERAaerodynamicmodel, and that of controls. The trim state of the flexible aircraft is solved using a dynamic relaxation method around which the perturbation equation is derived and solved for system stability. To demonstrate the advantage of multibody dynamics inmodeling the constrained system undergoing large displacements and rotations, the proposed approach is applied to the flexible aircraft in level flight and in the circling and dive–loop–climbmaneuvers. Beyond the flutter boundary, the limit cycle oscillations response is solved, in which several interesting aspects of flexible aircraft flutter are revealed. In the end, the jointflutter andattitude control is used to suppressflutter response and stabilize attitude.

Flutter/LCO suppression for high-aspect ratio wings

AbstractAn experimental high-aspect ratio wing aeroelastic model with a device to provide a controllable slender body tip mass distribution has been constructed and the model response due to flutter and limit cycle oscillations has been measured in a wind tunnel test. A theoretical model has also been developed and calculations made to correlate with the experimental data. Structural equations of motion based on nonlinear beam theory are combined with the ONERA aerodynamic stall model (an empirical extension of Theodorsen aerodynamic theory that accounts for flow separation). A dynamic perturbation analysis about a nonlinear static equilibrium is used to determine the small perturbation flutter boundary which is compared to the experimentally determined flutter velocity and flutter frequency. Time simulation is used to compute the limit cycle oscillations response when the flutter/LCO control system is ON or OFF. Theory and experiment are in good agreement for predicting the flutter/LCO suppression that can be achieved with the control device.

Flutter/Limit Cycle Oscillation Analysis and Experiment for Wing-Store Model

A delta wing experimental model with an external store has been designed and tested in the Duke University wind tunnel. The wing structure is modeled theoretically by using von Karman plate theory that allows for geometric strain-displacement nonlinearities in the plate wing structure. A component modal analysis is used to derive the full structural equations of motion for the wing/store combination system. A three-dimensional time domain vortex lattice aerodynamic model including a reduced-order model aerodynamic technique and a slender-body aerodynamic theory for the store are also used to investigate the nonlinear aeroelastic system. The effects of the store pitch stiffness (attachment stiffness), the span location of store, and the store aerodynamics on the critical flutter velocity and limit cycle oscillations (LCO) are discussed. The correlations between the theory and experiment are good for both the critical flutter velocity and frequency but not good for the LCO amplitude, especially when the store is located near the wing tip. The theoretical structural model needs to be improved to determine LCO response, and improved results are shown in the companion paper as obtained with a higher-order structural model.

Estimating the Probability of Failure of a Nonlinear Aeroelastic System

A limit-cycle oscillation (LCO) can be characterized by a subcritical or supercritical bifurcation, and bifurcations are shown to be discontinuities in the stochastic domain. The traditional polynomial-chaos-expansion method, which is a stochastic projection method, is too inefficient for estimating the LCO response surface because of the discontinuities associated with bifurcations. The objective of this research is to extend the stochastic projection method to include the construction of B-spline surfaces in the stochastic domain. The multivariate B-spline problem is solved to estimate the LCO response surface. A Monte Carlo simulation (MCS) is performed on this response surface to estimate the probability density function (PDF) of the LCO response. The stochastic projection method via B-splines is applied to the problem of estimating the PDF of a subcritical LCO response of a nonlinear airfoil in inviscid transonic flow. A probability of failure based upon certain failure criteria can then be computed from the estimated PDF. The stochastic algorithm provides a conservative estimate of the probability of failure of this aeroelastic system two orders of magnitude more efficiently than performing an MCS on the governing equations.

Flutter and Limit-Cycle Oscillations for a Wing-Store Model with Freeplay

An experimental delta-wing/store model with freeplay has been designed and tested in the Duke wind tunnel. The wing structure is modeled theoretically using von Karman plate theory that accounts for geometric strain- displacement nonlinearities in the plate wing structure. A component modal analysis is used to derive the full structural equations of motion for the wing/store system. A linear three-dimensional time-domain vortex lattice aerodynamic model including a reduced-order model aerodynamic technique and a slender-body aerodynamic theory for the store are also used to investigate the nonlinear aeroelastic system. The effects of the freeplay gap, the span location of the store, and the initial conditions on the limit-cycle oscillations (LCO) are discussed. The correlations between the theory and experiment are good for the smaller LCO amplitudes, that is, for flow velocities slightly higher than the flutter velocity, but are not good for the larger LCO amplitudes, that is, higher flow velocities. The theoretical model needs to be improved to determine LCO response for larger-amplitude motions.

In-Flight Wing Deformation Characteristics During Limit Cycle Oscillations

Oscillatory wing response data were measured on an F-16C aircraft during limit-cycle-oscillation (LCO) testing of an external store configuration. The configuration tested exhibited typical LCO response in the transonic flight regime. Deformation characteristics were measured at 11 locations on the wing and missile launchers during various LCO events. These measurements allowed viewing of the aeroelastic mode of instability for various flight conditions. Details of a linear flutter analysis model are presented, and the predicted eigenmode from the linear flutter analysis is compared to the in-flight measured mode. The measured LCO modes are also compared for level flight vs elevated load factor flight and subcritical LCO vs critical LCO conditions. At the onset of LCO, the mode shape bears a strong resemblance to the flutter mode predicted by linear flutter analysis. Further, the wing deformation characteristics during LCO vary significantly with respect to Mach number and load factor. The nonsynchronous motion of the LCO diminishes and becomes more synchronous as the Mach number increases beyond 0.90. This contradicts the trends predicted by linear flutter analyses and suggests that a change in the oscillation bounding mechanism occurs over the flight region examined.

Experimental and Theoretical Study on Aeroelastic Response of High-Aspect-Ratio Wings

An experimental high-aspect-ratio wing aeroelastic model with a slender body at the tip has been constructed, and the response due to flutter and limit-cycle oscillations (LCO) has been measured in a wind-tunnel test. A theoretical model has been developed and calculations made to correlate with the experimental data. Structural equations of motion based on nonlinear beam theory are combined with the ONERA aerodynamic stall model to study the effects of geometric structural nonlinearity and steady angle of attack on flutter and LCO of high-aspect-ratio wings. Static deformations in the vertical and torsional directions caused by a steady angle of attack and gravity are measured, and results from theory and experiment are compared. A dynamic perturbation analysis about a nonlinear static equilibrium is used to determine the small perturbation flutter boundary, which is compared to the experimentally determined flutter velocity and oscillation frequency. Time simulation is used to compute the LCO response. The results between the theory and experiment are in good agreement for static aeroelastic response, the onset of flutter, and dynamic LCO amplitude and frequency.