Robust Flutter Analysis Research Articles

On Aerodynamic Models for Flutter Analysis: A Systematic Overview and Comparative Assessment

This work reviews different analytical formulations for the time-dependent aerodynamic load of a thin aerofoil and clarifies numerical flutter results available in the literature for the typical section of a flexible wing; inviscid, two-dimensional, incompressible, potential flow is considered in all test cases. The latter are investigated using the exact theory for small airflow perturbations, which involves both circulatory and non-circulatory effects of different nature, complemented by the p-k flutter analysis. Starting from unsteady aerodynamics and ending with steady aerodynamics, quasi-unsteady and quasi-steady aerodynamic models are systematically derived by successive simplifications within a unified approach. The influence of the aerodynamic approximations on the aeroelastic stability boundary is then rigorously assessed from both physical and mathematical perspectives. All aerodynamic models are critically discussed and compared in the light of the numerical results as well, within a comprehensive theoretical framework in practice. In all cases, results accuracy depends on the aero-structural arrangement of the flexible wing; however, simplified unsteady and simplified quasi-unsteady aerodynamic approximations are suggested for robust flutter analysis whenever the wing’s elastic axis lies ahead of the aerofoil’s control point.

Advanced aeroelastic robust stability analysis with structural uncertainties

Robust flutter analysis described in this paper is based on the robust control theory framework. Therefore, a time-domain linear fractional transformation representation of the perturbed aeroelastic system is modeled. Then, the robust stability is analyzed by means of the structured singular value mu, which is defined as an alternative measure of robustness. Robust flutter analysis deals with aeroelastic (or aeroservoelastic) stability analysis taking structural dynamics, aerodynamics and/or unmodeled system dynamics uncertainties into account. Flutter is a well-known dynamic aeroelastic instability phenomenon caused by an interaction between structural vibrations and unsteady aerodynamic forces, whereby the level of vibration may trigger large amplitudes, eventually leading to catastrophic failure of the structure. The primary motivation of the robust flutter analysis is that this method allows the computation of the worst-case flutter velocity which can support, for example, the flight test program by a valuable robust flutter boundary. This paper addresses the issue of an approach for aeroelastic robust stability analysis with structural uncertainties with respect to physical symmetric and asymmetric stiffness perturbations on the wing structure by means of tuning beams.

An extension of the structured singular value to nonlinear systems with application to robust flutter analysis

The paper discusses an extension of mu (or structured singular value), a well-established technique from robust control for the study of linear systems subject to structured uncertainty, to nonlinear polynomial problems. Robustness is a multifaceted concept in the nonlinear context, and in this work the point of view of bifurcation theory is assumed. The latter is concerned with the study of qualitative changes of the steady-state solutions of a nonlinear system, so-called bifurcations. The practical goal motivating the work is to assess the effect of modeling uncertainties on flutter, a dynamic instability prompted by an adverse coupling between aerodynamic, elastic, and inertial forces, when considering the system as nonlinear. Specifically, the onset of flutter in nonlinear systems is generally associated with limit cycle oscillations emanating from a Hopf bifurcation point. Leveraging mu and its complementary modeling paradigm, namely linear fractional transformation, this work proposes an approach to compute margins to the occurrence of Hopf bifurcations for uncertain nonlinear systems. An application to the typical section case study with linear unsteady aerodynamic and hardening nonlinearities in the structural parameters will be presented to demonstrate the applicability of the approach.

Linear Fractional Transformation co-modeling of high-order aeroelastic systems for robust flutter analysis

This article presents a new paradigm for robust flutter modeling and analysis of high-order uncertain and linear aeroelastic systems. The fundamental idea is to couple the state-of-art in robust worst-case analysis (Linear Fractional Transformation modeling and µ analysis) with the state-of-practice in aeroelasticity (fluid-structure-interaction solvers). The issue with the latter is that, although capable of providing different levels of fidelity, they are less efficient in coping with the analysis of systems subject to uncertainties. In fact, while they have the advantage of capturing directly the physical uncertainty, the analyses can only be applied to a defined parameter combination, and due to their computational cost, it is usually only possible to consider a limited set of cases. To tackle this lack of robustness, in recent works the application of analytic worst-case methods has been proposed, but the intimately related problem of constructing accurate uncertain models has not been fully addressed. In this article, a co-modeling framework is presented that leverages the main features of both fluid-structure interaction solvers and robust control-based methods. The key idea is to combine these two typically distinct steps in a single one, enabling in this way to obtain an uncertainty description which is flexible and reconciles the physical sources of uncertainty with the uncertain parameters used in the LFT model. An exemplification of the developed framework on an unconventional aircraft configuration is provided. Results show its potential to provide valuable physical insights into the problem when analyzing complex systems.

Comparison of Aeroelastic Modeling and Robust Flutter Analysis of a Typical Section

A comparison of robust flutter analyses within the µ framework is presented. The chosen test bed is the typical section with unsteady aerodynamic loads, which enables basic modeling features to be captured and so extend the gained knowledge to practical problems treated with modern techniques. The two main approaches to pose the LFT problem are investigated and features such as numerical accuracy and physical uncertainty description are assessed. A criterion is proposed to correlate the families of plants originated by the uncertainty description of the aerodynamic operator when different approximation algorithms are employed.

Application of Pattern Search in Worst-Case Flutter Solution

ROBUST flutter solution based on analysis was first proposed by Lind and Brenner [1] and Lind [2], who used both the flighttest data and nominal aeroelastic model in the method. One of the main objectives of -method is to generate worst-case flutter solutions with respect to potential modeling errors.When computing the robust flutter speed by the method, is taken as “an exact measure of robust stability for systems with structured uncertainty, and the value of determines the allowable size of uncertainty matrices for which the plant is robustly stable as demonstrated in Theorem 3.3.2” [1]. Mathematically, the norm of the smallest structured uncertainty matrix that causes instability is exactly 1= . After thework of Lind and Brenner [1] and Lind [2], Borglund [3] and Borglund and Ringertz [4] developed the -k method which is based on traditional frequency-domain flutter analysis thus allowing for detailed aerodynamic uncertainty modeling. The fairly recent -pmethod proposed byBorglund [5] generalizes the -kmethod to the Laplace domain and computes possible variation of subcritical damping and frequency of a particular aeroelastic mode with respect to deterministic uncertainty, which is quite useful in flight flutter testing. The -pmethod is successfully implemented in robustflutter analysis with respect to complex valued aerodynamic perturbations [6,7] andmixed structural/aerodynamic uncertainty [8]. However, as noted in [8], “the structured singular value has to be estimated by a computable upper bound, which can lead to some amount of conservativeness in the analysis (in particular formixed real/complex uncertainty).” It may become crucial when computing with respect to purely real valued parametric uncertainty, because discontinuity may arise to [9,10] and lead to unreasonable results. The computation of itself is just an optimization problem taking as the objective function [11]. In addition to the method, there still exist other approaches such as optimization methods in worstcase flutter solution. Kuttenkeuler and Ringertz [12] and Becus [13] searched the most critical flutter configuration by optimizing the damping or the flutter speed in the uncertainty parameter space. However, this type of optimization problem is nonconvex in general, thus the efficient local method developed by Becus [13] can not always get the global optimal point, while the globalmethod [13] that sweeps the parameter space with a varying step would become timeconsuming subject to increasing number of uncertainty parameters. Moreover, these approaches lack a structured framework as noted in [8]. Fortunately, global optimization techniques have evolved a lot in recent years especially for the pattern search algorithm [14,15], which was proved to be with guaranteed global convergence if the function to be minimized is continuously differentiable [14]. In this Note, the aeroelastic model with purely real valued uncertainties is reviewed, and the promising pattern search algorithm is introduced and applied to solve this type of optimization problem by taking the worstor best-case flutter speed or aeroelastic damping or frequency as the objective function and taking the structured uncertainty operators as design variables. Sample test cases are studied with this approach and the results are validated by the exhaustive method. A special case with purely real uncertainties shows that the method can not give reasonable worst-case flutter solution while the present approach can.

Robust flutter analysis and control of a wing

PurposeThe purpose of this paper is to present a novel approach in aeroservoelastic analysis and robust control of a wing section with two control surfaces in leading‐edge and trailing‐edge. The method demonstrates how the number of model uncertainties can affect the flutter margin.Design/methodology/approachThe proposed method effectively incorporates the structural model of a wing section with two degrees of freedom of pitch and plunge with two control surfaces on trailing and leading edges. A quasi‐steady aerodynamics assumption is made for the aerodynamic modeling. Basically, perturbations are considered for the dynamic pressure models and uncertainty parameters are associated with structural stiffness and structural damping and are accounted for in the model by a Linear Fractional Transformation (LFT) model. The control commands are applied to a first and second order electro‐mechanical actuator.FindingsDynamic performance of aeroelastic/aeroservoelastic system including time responses, system modal specifications, critical flutter speeds, and stability margins are extracted and compared with each other. Simulation results are validated through experiments and are compared to other existing methods available to the authors. Results of simulations with four structural uncertainties and first order controllers have a good agreement with experimental test results. Furthermore, it is shown that by using a high gain second order controller, the aeroservoelastic (ASE) system does not have any coupling nature in frequency response.Originality/valueIn this study, modeling, simulation, and robust control of a wing section have been investigated utilizing the μ‐Analysis method and the wing flutter phenomenon is predicted in the presence of multiple uncertainties. The proposed approach is an advanced method compared to conventional flutter analysis methods (such as V‐g or p‐k) for calculating stability margin of aeroelastic/aeroservoelastic systems.

Robust flutter analysis based on genetic algorithm

Robust flutter analysis considering model uncertain parameters is very important in theory and engineering applications. Modern robust flutter solution based on structured singular value subject to real parametric uncertainties may become difficult because the discontinuity and increasing complexity in real mu analysis. It is crucial to solve the worst-case flutter speed accurately and efficiently for real parametric uncertainties. In this paper, robust flutter analysis is formulated as a nonlinear programming problem. With proper nonlinear programming technique and classical flutter analysis method, the worst-case parametric perturbations and the robust flutter solution will be captured by optimization approach. In the derived nonlinear programming problem, the parametric uncertainties are taken as design variables bounded with perturbed intervals, while the flutter speed is selected as the objective function. This model is optimized by the genetic algorithm with promising global optimum performance. The present approach avoids calculating purely real mu and makes robust flutter analysis a plain job. It is illustrated by a special test case that the robust flutter results coincide well with the exhaustive method. It is also demonstrated that the present method can solve the match-point robust flutter solution under constant Mach number accurately and efficiently. This method is implemented in problem with more uncertain parameters and asymmetric perturbation interval.

Robust flutter analysis of a nonlinear aeroelastic system with parametric uncertainties

This paper deals with the problem of robust stability of a 2-D nonlinear aeroelastic system with structural and aerodynamic uncertainties using μ-method and value set approach. The model includes a nonlinear spring in the pitch degree-of-freedom and a nonlinear aerodynamic model. Two types of uncertainty named vanishing and nonvanishing perturbation are investigated. For vanishing perturbations, μ-method can be used directly to examine the robust stability of the nonlinear aeroelastic system by analyzing its linearized system. For nonvanishing perturbations, the functional relationship between equilibrium point and uncertain parameters are expressed as Taylor series expansions in order to consider the uncertainty of the equilibrium point in μ-analysis framework. Furthermore, the value set approach is also used to examine the robust stability of the uncertain system. The value sets of the characteristic polynomials are computed and zero exclusion condition is applied to check the stability of the entire family of the characteristic polynomials. Numerical results are presented for a set of values of the flow velocities, and the lower bound and upper bound of robust flutter speeds are obtained from the V–μ graph and the motion of the value sets.

Robust Flutter Analysis Considering Mode Shape Variations

A study was conducted to perform robust flutter analysis, considering mode shape variation problems. The study demonstrated that the assumption of a fixed modal base can lead to incorrect flutter ...

Match Point Solution for Robust Flutter Analysis in Constant-Mach Prediction

This paper presents a method for robust flutter computation which uses flight altitude as the perturbation variable in order to obtain a match point solution. The air density and sound speed of standard atmosphere model are approximated as the polynomial function of altitude, such that the flight altitude becomes the single perturbation variable that describes the aeroelastic system. The uncertainties of generalized stiffness and damping are considered and the uncertain aeroelastic system can be formulated as linear fractional transformation (LFT) representation which is suitable for μ analysis framework. Finally, the match point solutions of robust flutter margins can be computed with structured singular value (SSV) theory. The robust flutter analysis method provided in this paper is suitable for constant-Mach flight flutter test and provides valuable reference for flight envelope expansion.

Robust Flutter Analysis of an Airfoil with Flap Freeplay Uncertainty

Modern robust flutter method is an advanced technique for flutter margin estimation. It always gives the worst-case flutter speed with respect to potential modeling errors. Most literatures are focused on linear parameter uncertainty in mass, stiffness and damping parameters, etc. But the uncertainties of some structural nonlinear parameters, the freeplay in control surface for example, have not been taken into account. A robust flutter analysis approach in μ-framework with uncertain nonlinear operator is proposed in this study. Using describing function method the equivalent stiffness formulation is derived for a two dimensional wing model with freeplay nonlinearity in its flap rotating stiffness. The robust flutter margin is calculated for the two dimensional wing with flap freeplay uncertainty and the results are compared with that obtained with nominal parameter values. It is found that by considering the perturbation of freeplay parameter more conservative flutter boundary can be obtained, and the proposed method in μ-framework can be applied in flutter analysis with other types of concentrated nonlinearities.

Assessment of Critical Fuel Configurations Using Robust Flutter Analysis

An approach to assess critical fuel configurations using robust flutter analysis is presented. A realistic aircraft model is considered to demonstrate how an available finite element model can be ...

Efficient Computation of Robust Flutter Boundaries Using the mu-k Method

A simple and efficient algorithm for robust flutter analysis is presented. First, a general linear fractional transformation formulation of the mu-k method is provided, making it straightforward to pose the uncertain flutter equation in a form suitable for structured singular value analysis. The new formulation establishes a close connection between mu-k flutter analysis and traditional frequency-domain flutter analysis, which is used to formulate an efficient algorithm for computation of robust flutter boundaries. The proposed method is successfully applied to an F-16 sample test case with uncertain external stores aerodynamics, showing that standard tools for structural dynamics and unsteady aerodynamics can be used to perform robust flutter analysis with only modest additional modeling.

Upper Bound Flutter Speed Estimation Using the mu-k Method

The use of upper-bound μ-k estimation method in the development of robust flutter analysis and flutter testing, was described. Since only the frequency-domain aerodynamic forces are required to compute μ(k), established aerodynamic methods can be used for the robust flutter analysis. A robust flutter analysis considering wing-tip aerodynamic uncertainty was developed in MATLAB® for a wind-tunnel model in low-speed airflow. The results show that the extended procedure for robust flutter analysis was successfully applied to a wind-tunnel model in low-speed airflow.

The mu-k Method for Robust Flutter Solutions

A straightforward frequency-domain method for robust flutter analysis is presented. First, a versatile uncertainty description for the unsteady aerodynamic forces is derived by assigning uncertainty to the frequency-domain pressure coefficients. The uncertainty description applies to any frequency-domain aerodynamic method, benefits from the same level of geometric detail as the underlying aerodynamic model, exploits the modal formulation of the flutter equation, and is computed by simple postprocessing of standard aerodynamic data. Next, structured singular value analysis is applied to derive an explicit criterion for robust flutter stability based on the flutter equation and a parametric uncertainty description. The resulting procedure for computation of a worst-case flutter boundary resembles a p-k or g-method flutter analysis, produces match-point flutter solutions and allows for detailed aerodynamic uncertainty descriptions. Finally, the proposed method is successfully applied to a wind-tunnel model in low-speed airflow.