Analysis of the 1:1 resonant energy exchanges between coupled oscillators with rheologies

The paper is composed of three main parts: The first part presents a two-degree-of-freedom coupled oscillators with rheology. One of the oscillators is intended to be the main structure, and the second one is a nonlinear energy sink. The rheology of the system is represented via a set of internal variables that are governed by either differential inclusions or differential equations or direct algebraic relations between system variables. A step-by-step methodology is explained to trace system behaviors around a 1:1 resonance at different timescales. Invariant of the system at fast timescale is detected, while possible periodic and strongly modulated regimes around its invariant are traced at slow time scales. The second part of the paper considers a set of several degree-of-freedom main oscillators which are coupled to several nonlinear energy sinks. The overall system can house several rheologies. Explained methodology of the first part is expanded to this general case for tracing system responses at different time scales around 1:1 resonances. The third part of the paper presents two practical examples: The proposed methodology is used to detect invariants of systems and their equilibrium and singular points. This methodology provides some tools for designing equilibrium and singular points, i.e., periodic and strongly modulated regimes which lead to the design of nonlinear energy sinks for passively controlling and/or energy harvesting of the main oscillators.