Derivation of ODEs and Bifurcation Analysis of a Two‐DOF Airfoil Subjected to Unsteady Incompressible Flow

Abstract

An airfoil subjected to two‐dimensional incompressible inviscid flow is considered. The airfoil is supported via a translational and a torsional springs. The aeroelastic integro‐differential equations of motion for the airfoil are reformulated into a system of six first‐order autonomous ordinary differential equations. These are the simplest and least number of ODEs that can present this aeroelastic system. The differential equations are then used for the bifurcation analysis of an airfoil with a structural nonlinearity in the pitch direction. Sample bifurcation diagrams showing both stable and unstable limit cycle oscillation are presented. The types of bifurcations are assessed by evaluating the Floquet multipliers. For a specific case, a period doubling route to chaos was detected, and mildly chaotic behavior in a narrow range of velocity was confirmed via the calculation of the Lyapunov exponents.