The Effect of Slow Invariant Manifold and Slow Flow Dynamics on the Energy Transfer and Dissipation of a Singular Damped System with an Essential Nonlinear Attachment

We study the effect of slow flow dynamics and slow invariant manifolds on the energy transfer and dissipation of a dissipative system of two linear oscillators coupled with an essential nonlinear oscillator with a mass much smaller than the masses of the linear oscillators. We calculate the slow flow of the system, the slow invariant manifold, the total energy of the system, and the energy that is stored in the nonlinear oscillator for different sets of the parameters and show that the bifurcations of the SIM and the dynamics of the slow flow play an important role in the energy transfer from the linear to the nonlinear oscillator and the rate of dissipation of the total energy of the initial system. 1. Introduction Mechanical structures where the nonlinear attachments have small masses in comparison to the structures to which they are attached and the systems are non conservative act as nonlinear energy sinks (NESs) and absorb, through irreversible transient transfer, energy from the linear parts due to resonance captures in vicinities of resonance manifolds of the underlying conservative systems for certain ranges of parameters and initial conditions [1–3]. Systems, where the masses of the nonlinear attachments are small in comparison to the linear oscillator, are singular and their dynamics are governed by different time scales. Such systems can be treated with the use of methods such as singularity analysis or multiple scale analysis [4, 5]. As it has been shown in previous works the slow flow of the system and the slow invariant manifolds (SIMs) obtained in the singular limit play a very important role in the dynamics under consideration [3, 6–8]. Furthermore, many works [1, 3, 9] have shown that energy transfer from a linear oscillator to its coupled nonlinear oscillator as well as the rate of dissipation of the energy is directly connected to the bifurcations of the SIM and the dynamics of the slow flow of the system. In this work we study a dissipative system of two linear oscillators coupled with an essential nonlinear oscillator. In previous works [6, 7] we studied the SIM of the system and made a classification of its structure. Specifically, the structure of the SIM may be classified in three cases. First case when the SIM has always one stable branch. Second case when the SIM has always three branches, two of them stable and the one that lies between them unstable. In the above cases the SIM has no bifurcations. The third case is when the SIM Bifurcates. We also showed that the slow flow of the system has rich dynamics. It may oscillate,