Stability and Limit Cycle Oscillation Amplitude Prediction for Simple Nonlinear Aeroelastic Systems

The prediction of the bifurcation and post-bifurcation behaviour of nonlinear aeroelastic systems is becoming
a major area of research in the aeroelastic community due to the need for improved transonic aeroelastic
prediction, the use of non-linear control systems, and new construction techniques that reduce the amount of
inherent damping. In this paper, a novel application of the Centre Manifold Theorem is used to accurately
predict bifurcation conditions and Limit Cycle Oscillation amplitudes for simple aeroelastic systems with various
types of nonlinearity. A simple aeroelastic system with hardening cubic stiffness nonlinearity is considered
and is demonstrated to display a wide variety of bifurcation phenomena. These make it dif cult for some of
the standard existing methods, such as Normal Form, Cell Mapping and Tangential Linearisation, to quantify
the Limit Cycle Oscillation amplitudes through the entire speed range of the system. Then, the proposed approach
is introduced and applied to the same system. It is shown that it can accurately predict the limit cycle
amplitudes of the system undergoing all types of bifurcation. Finally, the new technique is applied to the same
system but with softening cubic stiffness nonlinearity. It is shown that the method can accurately predict both
the static and dynamic divergence boundaries and that it can be used to draw a worst-case stability boundary,
inside which the solution is always stable.