Effects of Parametric Uncertainty on Airfoil Limit Cycle Oscillation

Introduction T HE need for revolutionizingmethods of assessing aeroelastic stabilityhas become increasinglypressing in recentyears.This is driven primarily by two factors: 1) the desire to reduce the total cost of certiŽ cation by reducing testing requirements, and 2) the emergence of unique design concepts to provide impressive performance in military applications.A common feature of these designs is that they substantially increase the potential for nonlinear behavior beyond levels that can be adequately addressed by current engineering tools and processes. These needs and concerns were the focus of a recent workshop organizedby the Air Force OfŽ ce of ScientiŽ c Research and the Air Force Research Laboratory. The workshop addressed traditional areas of concern, such as the basic physics and computational requirements of nonlinear aeroelasticity,but it also included sessions on model veriŽ cation and validation (VVnotably,itwas agreed that UQ could provide a common language for promoting communication between analysts and test personnel. This Note is intended to demonstrate the application of standard probabilityconceptsandMonteCarlo simulation(MCS) to the study of airfoil limit cycle oscillation (LCO), which results from a subcriticalHopf bifurcation induced by including a nonlinear spring in the pitch degree of freedom (DOF). Unsteady aerodynamic forces are represented by the R. T. Jones approximation of the circulatory lift. This simple aeroelastic model permits an assessment of aeroelastic performance sensitivity and variability within the context of a well-understood system; furthermore, because each MCS realization requires time integration, the use of a simple model is computationally expedient. Employing such an elementary model is justiŽ ed in this application because our primary goals are to illustrate the qualitative consequences of uncertainty in a nonlinear aeroelastic system and also to provide a simple example of how probabilistic aeroelastic analysesmight be performed in the future. The authors and a colleague have employed essentially the same procedure detailed herein to highlight the in uence of various uncertainties in panel LCO.2i4