Nonlinear Dyn DOI 10.1007/s11071-012-0391-5 ORIGINAL PAPER On energy transfer phenomena, in a nonlinear ideal and nonideal essential vibrating systems, coupled to a (MR) magneto-rheological damper Angelo Marcelo Tusset · José Manoel Balthazar · Fábio Roberto Chavarette · Jorge Luis Palacios Felix Received: 9 October 2011 / Accepted: 6 March 2012 © Springer Science+Business Media B.V. 2012 Abstract In this paper, a control strategy is applied to a nonlinear ideal and nonideal vibrating system coupled to a magneto-rheological damping structure, exhibiting chaotic behavior. In order to suppress the chaotic oscillations and to improve the transient re- sponse, a combination of passive (energy pumping) and active (state feedback) control strategies. The chaotic vibrations are suppressed, bringing the system to a desired periodic orbit. Additionally, the control damping force is transformed into the electrical con- trol signal variable. A.M. Tusset Department of Engineering Science, UTFPR-Ponta Grossa, Av. Monteiro Lobato, Km 04, s/No., CEP: 20-84016-210, Ponta Grossa, PR, Brazil e-mail: a.m.tusset@gmail.com J.M. Balthazar ( ) Department of Statistics, Applied Mathematics and Computation (DEMAC), UNESP-Rio Claro, Av. 24 A, No. 1515, Bela Vista, CEP: 13506-700, Rio Claro, SP, Brazil e-mail: jmbaltha@rc.unesp.br F.R. Chavarette Department of Mathematics, UNESP-Ilha Solteira, Av. Brasil, 56, CEP: 15385-000, Ilha Solteira, SP, Brazil e-mail: fabioch@mat.feis.unesp.br J.L.P. Felix Department of Mathematics, UNIPAMPA-Campus Bagé, Street Travessa 45, No. 1650, CEP: 96413-170, Bagé, RS, Brazil e-mail: jorge.felix@unipampa.edu.br Keywords Ideal and nonideal structure · Nonlinear dynamics · Optimal linear control · Magneto-rheological damper · Energy pumping (transfer) 1 Introduction The study of vibrating systems when the external ex- citation is inﬂuenced by the response of the system is considered a major challenge in both theoretical and practical engineering research. It is well known that a vibrating system, for which power supply is unlimited, or the excitation source is not inﬂuenced by the response is said to be an ideal system. On the other hand, if the excitation is inﬂu- enced by the response of the system or for which the power supply is limited is said to be a non-ideal sys- tem [1]. Usually, nonideal vibrating systems are considered when motors are attached to structures that need ex- citation power levels similar to the power capacity of those motors [2, 3]. The main goal of this work is to use the recent methodology of optimal control strategy, proposed by [4], in both ideal and nonideal vibrating problems, taking into account the action of a MR damping, and the application of passive control in vibration isola- tion. Passive control is applied considering that the passive actuator is part of the system acting through