Multi-scale energy exchanges between a nonlinear oscillator of Bouc-Wen type and another coupled nonlinear system
1 Université de Lyon, École Nationale des Travaux Publics de l’État, LGCB and LTDS UMR CNRS 5513, Rue Maurice Audin, 69518 Vaulx-en-Velin Cedex, France
2 Université de Lyon, École Nationale des Travaux Publics de l’État, LGCB, rue Maurice Audin, 69518 Vaulx-en-Velin Cedex, France
a e-mail: email@example.com
Received: 4 June 2013
Revised: 2 August 2013
Published online: 30 September 2013
The concept of energy exchange between coupled oscillators can be endowed for wide variety of applications such as control and energy harvesting. It has been proved that by coupling an essential nonlinear oscillator (cubic nonlinearity) to a main system (mostly linear), the latter system can be controlled in a one way and almost irreversible manner. The phenomenon is called energy pumping and the coupled nonlinear system is named as nonlinear energy sink (NES). The process of energy transfer from the main system to the nonlinear smooth or non-smooth attachment at different scales of time can present several scenarios: It can be attracted to periodic behaviors which present low or high energy levels for the main system and/or to quasi-periodic responses of two oscillators by persistent bifurcations between their stable zones. In this paper we analyze multi-scale dynamics of two attached oscillators: a Bouc-Wen type in general (in particular: a Dahl type and a modified hysteresis system) and a NES (nonsmooth and cubic). The system behavior at fast and first slow times scales by detecting its invariant manifold, its fixed points and singularities will be analyzed. Analytical developments will be accompanied by some numerical examples for systems that present quasi-periodic responses.
The endowed Bouc-Wen models correspond to the hysteretic behavior of materials or structures. This paper is clearly connected with the dynamics of systems with hysteresis and nonlinear dynamics based energy harvesting.
© EDP Sciences, Springer-Verlag, 2013