Aeroelastic instability analysis of NES-controlled systems via a mixed Multiple Scale/Harmonic Balance algorithm

The issue to passively control aeroelastic instability of general nonlinear multi-d.o.f. systems, suffering Hopf bifurcation, is addressed. The passive device consists of an essentially nonlinear oscillator (Nonlinear Energy Sink), having the task to transfer energy from the main to the secondary 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 codimension-2 invariant manifold on which the motion occurs, thus revealing the passive character of the oscillations of the NES. The algorithm also provides the bifurcation equations, which govern the slow flow on the manifold, expressed in terms of the main structure amplitude and phase of motion. 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. Theory is applied to a sample structure already studied in literature, consisting of a two-d.o.f. rigid wing under steady wind. Relevant asymptotic results are compared, for validation purposes, with numerical simulations.