Application of the centre manifold theory in non-linear aeroelasticity

In this study, a frequency relation for limit cycle oscillations of a two-degree-of-freedom aeroelastic system with structural non-linearities represented by cubic restoring spring forces is derived. The centre manifold theory is applied to reduce the original system of nine-dimensional first order ordinary differential equations to a governing system in two dimensions near the bifurcation point. The principle of normal form is used to simplify the non-linear terms of the lower dimensional system. Using the frequency relation and the amplitude-frequency relationships derived from a previous study, limit cycle oscillations (LCOs) for self-excited systems can be predicted analytically. The mathematical technique proposed here has been applied to investigate LCO near a Hopf-bifurcation for an aeroelastic system with cubic restoring forces. Not only that an excellent agreement is obtained compared to the numerical results from solving the original system of eight non-linear differential equations by Runge-Kutta time integration scheme, but we also demonstrate that the use of a mathematical approach leads to a better understanding of non-linear aeroelasticity.