Dynamic behavior analysis of a mechanical system with two unstable modes coupled to a single nonlinear energy sink

This paper investigates a problem of passive mitigation vibratory instabilities caused by two unstable modes means single nonlinear energy sink (NES). For this purpose, linear four-degree-of-freedom (DOF) primary structure having (reproducing the typical dynamic behavior friction system) and undergoing, as it is linear, unbounded motions when unstable, coupled to NES. In work, NES involves an essentially cubic restoring force damping force. We are interested in characterizing analytically response regimes resulting from coupling mode To end, suitable rescaling governing equations system which dynamics reduced its modal coordinates, complexification-averaging method applied. The averaged appears be fast-slow with four fast variables slow ones related structure. critical manifold obtained through geometric singular perturbation theory two-dimensional parametric surface (with respect variables) evolves whole six-dimensional variable space. asymptotic analysis reveals that attachment can produce some bounded responses suggests may have simultaneous stable attractors. Numerical simulations complement study, highlighting possible competition between attractors allowing us investigate their basins attraction. each considered situation, good agreement has been observed theoretical results numerical simulations, validates proposed analysis.