Trapping vibratory energy of main linear structures by coupling light systems with geometrical and material non-linearities

• We study time multi-scale energy exchanges between a main oscillator and a NES.
• NES presents smooth geometrical and non-smooth hysteresis (Bouc–Wen) behaviors.
• Invariant manifold and its stability borders are detected.
• Equilibrium and singular points are traced or periodic regimes and strongly modulated responses are detected.
• NES can control the main structure against external forces.

Cubic potential and hysteresis behavior (Bouc–Wen type) of a non-linear energy sink are used to localize the vibratory energy of a linear structure. A general methodology is presented to deal with time evolutionary energy exchanges between two oscillators. Invariant manifold of the system and its stability borders are detected at fast time scale while traced equilibrium and singular points at slow time scale let us predict possible behaviors of the system during its pseudo-stationary regime(s). The paper is followed by an example that considers the Dahl model for representing the hysteresis behavior of the non-linear energy sink. All analytical developments and results are compared with those obtained by direct integration of system equations. Obtained analytical developments can be endowed for designing non-linear energy sink devices with hysteresis behavior to localize vibratory energy of main structures for the aim of passive control, energy harvesting and/or both of them.