Detection of subcritical Hopf and fold bifurcations in an aeroelastic system via the Describing Function method

Abstract

The paper deals with Limit Cycle Oscillations (LCOs) in a Pitch & Plunge model of the wing dynamics, where the stiffness on the plunge displacement is assumed to be an odd fifth-order polynomial. First, it is shown that the model dynamics can be equivalently described via a Lur’e system, i.e., the feedback interconnection between a linear time-invariant subsystem, whose transfer function depends on the wind speed, and a nonlinear memoryless one. Then, the Describing Function (DF) method is used to establish the existence of Predicted Limit Cycles (PLCs) whose amplitude can be analytically expressed in terms of the wind speed, once the intersections of the Nyquist diagram of the linear subsystem with the real axis are obtained. The related bifurcation diagram permits to predict that a stable equilibrium point and two LCOs (one unstable and one stable) coexist for wind speeds smaller than the flutter velocity, at which a subcritical Hopf bifurcation occurs, and larger than the value at which a cyclic fold bifurcation is detected. The closeness of each PLC to the related LCO is measured via the so-called distortion index, which also admits a direct expression in terms of the wind speed. The comparison of the predicted bifurcation diagram with that obtained via numerical simulations confirms the quite good accuracy of the PLCs obtained via the DF method, especially when the distortion index is small.