Asymptotic analysis of passive mitigation of dynamic instability using a nonlinear energy sink network

Abstract

The present work studies a one-DOF nonlinear unstable primary system, which undergoes harmful limit cycle oscillations, coupled to a network of several parallel Nonlinear Energy Sinks (NESs). As usual, in the framework of NES properties exploration and particularly in the context of dynamic instabilities mitigation, four steady-state response regimes are observed. They are classified into two categories depending on whether the NESs mitigate or not the instability and therefore separating harmless situations from harmful situations. An asymptotic analysis shows that the critical manifold of the system can be reduced to a one dimensional parametric curve evolving in a N dimensional space. The shape of the critical manifold and the associated stability properties provide an analytical tool to predict the nature of the possible response regimes mentioned above. In particular, the mitigation limit of the NESs, defined as the value of the chosen bifurcation parameter which separates harmful situations from harmless situations, is predicted. Using more restrictive assumptions, i.e. neglecting the nonlinearity of the primary system and assuming N identical NESs, a literal expression of the mitigation limit is obtained. Using a Van de Pol oscillator as a primary system, theoretical results are compared, for validation purposes, to the numerical integration of the system. The comparison shows a good agreement as long as we remain within the limits of use of the asymptotic approach.