Aeroelastic instability analysis of NES-controlled systems via a mixed multiple scale/harmonic balance method

The issue of passively controlling aeroelastic instability of general nonlinear multi-degree-of-freedom systems, suffering from Hopf bifurcation, is addressed. The passive device consists of an essentially nonlinear oscillator (nonlinear energy sink [NES]), having the task of absorbing energy from the main structure. The mathematical problem is attacked by a new algorithm, based on a suitable combination of the multiple scale and the harmonic balance methods. The procedure is able to furnish the reduced amplitude modulation equations, which govern the slow flow around a critical manifold, on which the equilibrium points lie. The method is applied to a sample structure already studied in literature, consisting of a two-degree-of-freedom rigid airfoil under steady wind. It is shown that NES, under suitable conditions, can shift forward the bifurcation point and, moreover, it can reduce the amplitude of the limit cycles. Relevant asymptotic results are compared, for validation purposes, with numerical simulations.