Numerical Study of Flutter of a Two-Dimensional Aeroelastic System

This paper deals with the problem of the aeroelastic stability of a typical aerofoil section with two degrees of freedom induced by the unsteady aerodynamic loads. A method is presented to model the unsteady lift and pitching moment acting on a two-dimensional typical aerofoil section, operating under attached flow conditions in an incompressible flow. Starting from suitable generalisations and approximations to aerodynamic indicial functions, the unsteady loads due to an arbitrary forcing are represented in a state-space form. From the resulting equations of motion, the flutter speed is computed through stability analysis of a linear state-space system. 1. Introduction Flutter is the dynamic aeroelasticity phenomenon whereby the inertia forces can modify the behaviour of a flexible system so that energy is extracted from the incoming flow. The flutter or critical speed is defined as the lowest air speed at which a given structure would exhibit sustained, simple harmonic oscillations. represents the neutral stability boundary: oscillations are stable at speeds below it, but they become divergent above it. Theodorsen [1] obtained closed-form solution to the problem of an unsteady aerodynamic load on an oscillating aerofoil. This approach assumed the harmonic oscillations in inviscid and incompressible flow subject to small disturbances. Wagner [2] obtained a solution for the so-called indicial lift on a thin aerofoil undergoing a transient step change in angle of attack in an incompressible flow. The indicial lift response makes a useful starting point for the development of a general time domain unsteady aerodynamics theory. A practical way to tackle the indicial response method is through a state-space formulation in the time domain, as proposed, for instance, by Leishman and Nguyen [3]. The main objective of this paper is to investigate the aeroelastic stability of a typical aerofoil section with two degrees of freedom induced by the unsteady aerodynamic loads defined by the Leishman’s state-space model. 2. Aeroelastic Model Formulation The mechanical model under investigation is a two-dimensional typical aerofoil section in a horizontal flow of undisturbed speed , as shown in Figure 1. Its motion is defined by two independent degrees of freedom, which are selected to be the vertical displacement (plunge), , positive down, and the rotation (pitch), . The structural behaviour is modelled by means of linear bending and torsional springs, which are attached at the elastic axis of the typical aerofoil section. The springs in the typical aerofoil section