Parametric analysis of a Nonlinear Energy Sink for an unstable dynamic system

This paper proposes a methodology to design a Nonlinear Energy Sink (NES) to control passively the vibration of an unstable dynamic system. The passive device is composed of purely nonlinear stiffness (cubic stiffness), a linear damper and a mass attached directly to a simplified dynamic model representation of the main system (i.e., unstable mode of vibration). Asymptotic methods (Method of Multiple Scale mixed with Harmonic Balanced Method, MMS-HBM) is used to treat nonlinear equations which results to a singular perturbed system. Such a system is studied with Geometric Singular Perturbation Theory (GSPT), also with analytical developments. Response of the dynamical system is related to information obtained through GSPT applications: slow invariant manifold, slow-flow fixed points and their stability. This information is correlated with steady-state response regime attained with the NES and, therefore, leading to analytical expressions about necessary conditions to attain different responses. Parametric investigations are then performed and design maps are created in order to predict the response regime for large combination of NES's parameters (i.e., mass ratio, damping factor and nonlinear stiffness). Results highlight that mainly mass ratio and damping factor parameters affects the response regime while nonlinear stiffness parameters affect the amplitude of the motion. Moreover, effects of initial condition and level of the instability severity on robustness of the NES are investigated.