Robust Eigenvalue Analysis Using the Structured Singular Value: The µ-p Flutter Method

Abstract

This paper introduces a new technique for robust aeroelastic analysis that extends standard linear flutter analysis to take deterministic uncertainty and variation into account. The basic principle of the proposed -p method is to exploit structured-singular-value (or ) analysis to investigate if the system uncertainties can make the flutter determinant zero for a given flutter eigenvalue p. This makes it possible to compute regions of feasible eigenvalues in the complex plane as well as extreme eigenvalues that can be used to predict damping bounds and perform robust flutter analysis. The capability to predict damping bounds at subcritical flight conditions is a very attractive feature of the new method, as flight testing is rarely taken to the flutter point. The -p formulation also opens up new possibilities to bound the magnitude of the system uncertainties based on frequency and/or damping estimates from flight testing. In the final part of the paper, the -p framework is successfully applied to perform robust aeroelastic analysis of a low-speed wind-tunnel model. Nomenclature F= flutter matrix F u = upper linear fractional transformation g = damping K = stiffness matrix k = reduced frequency L = reference length M = mass matrix N = linear fractional transformation matrix P = system matrix p = eigenvalue Q = aerodynamic matrix V = airspeed w = input from uncertainty matrix z = output to uncertainty matrix = uncertainty matrix = uncertainty parameters = modal coordinates = structured singular value = modal force = air density = maximum singular value

Structured-Singular-Value-Based Optimal Aeroelastic Uncertainty Quantification using Surrogate Models and Flight Test Data

“…The quantified uncertainty determined using the method presented here can be effectively used for uncertainty analysis at both stable and unstable flight conditions. 5,6,7 In addition, it can be used § § With a positive real part, this nominal system is unstable. online to determine subsequent least conservative robust flutter boundaries where the optimal quantified uncertainty is taken into account by leveraging real-time in-flight data (e.g., to aid the NASA Flutterometer 25 ).…”

Evaluation of Aeroelastic Uncertainty Analysis Methods

“…It is also noted that the structured singular value framework can be exploited to take flight data into account; recent methods have been developed that use measured frequencies and damping values of aeroelastic modes to validate the model at subcritical nonflutter conditions [28]. This capability is extremely attractive for aeroservoelastic clearance of operational aircraft.…”

“…Introduction of some complex bounded uncertainty would most likely result in more efficient computations. Alternative approaches for determining robust flutter boundaries with have been developed recently that do not require the stability parameter to be part of the structure or be restricted to polynomial dependency [26][27][28][29]. Some recent methods have also introduced complex aerodynamic uncertainty [26], which is undoubtedly present in real-world aeroelastic systems.…”

Industrial Application of Robust Aeroelastic Analysis

“…Recently, Borglund developed the -p method, which is, in essence, a generalization of the -k method to cover the Laplace domain [11]. Using this new method, robust analysis can be performed to investigate aeroelastic properties at any flight condition.…”