The paper is composed of three main parts: the first part presents a two degrees 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 time scales. Invariant of the system at fast time scale 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 degrees 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.

No abstract

No abstract

No abstract

No abstract

No abstract

No abstract

Abstract. This work deals with the design and experiment of a cubic nonlinear energy sink (NES) for horizontal vibration mitigation of a bridge cable. Modal analysis of horizontal linear modes of the cable is experimentally performed using accelerometers and displacement sensors. A theoretical simplified 2-dof model of the coupled cable-NES system is used to analytically design the NES by mean of multi-time scale systems behaviours and detection its invariant manifold, equilibrium and singular points which stand for periodic and strongly modulated regimes, respectively. Numerical integration is used to confirm the efficiency of the designed NES for the system under step release excitation. Then, the prototype system is built using geometrical cubic nonlinearity as the potential of the NES. Efficiency of the prototype system for mitigation of horizontal vibrations of the cable under for step release and forced excitations is experimentally demonstrated.