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Journal of Applied Nonlinear Dynamics 2(4) (2013) 315--328 | DOI:10.5890/JAND.2013.11.001

J.O. Maaita$^{1}$, E. Meletlidou$^{1}$, A.F. Vakakis$^{2}$, V. Rothos$^{3}$

$^{1}$ Physics Department, Aristotle University of Thessaloniki, Greece

$^{2}$ Department of Mechanical Science and Engineering, University of Illinois at Urbana-Champaign, United States

$^{3}$ Department of Mathematics, Physics and Computational Sciences, Faculty of Technology, Aristotle University of Thessaloniki, Thessaloniki, Greece

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Abstract

We study a three degree of freedom autonomous system with damping, composed of two linear coupled oscillators with an essentially nonlinear lightweight attachment. In particular, we are interested in strongly nonlinear interactions between the linear oscillators and the essentially nonlinear attachment. First, we reduce our system to a non-autonomous second order nonlinear damped oscillator. Then, we introduce a slow-fast partition of the dynamics and average out the main frequency components in order to obtain a reduced system that is studied through the Slow Invariant Manifold (SIM) approach. Depending on the pa- rameters of the system we find different interesting nonlinear dynamical phenomena. With the help of the SIM approach we can study how the parameters of the original problem influence the asymptotic behavior of the orbits of the system. This is accomplished with the application of Tikhonov’s theorem. We classify the different cases of the dynamics according to the values of the parameters and the theoretically predicted asymptotic behavior of the orbits. Interesting phenomena are reported such as orbit capture, relaxation oscillations and complex structure of the basins of attraction.

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