Tracking modal interactions in nonlinear energy sink dynamics via high-dimensional invariant manifold
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ORIGINAL PAPER
Tracking modal interactions in nonlinear energy sink dynamics via high-dimensional invariant manifold Giuseppe Habib
· Francesco Romeo
Received: 1 March 2020 / Accepted: 1 September 2020 © The Author(s) 2020
Abstract A nonlinear energy sink (NES), conceived to mitigate the vibrations of a multi-degree-of-freedom host mechanical system, is considered. The highdimensional slow invariant manifold (SIM) describing the high-amplitude slow dynamics of the system is derived and exploited to interpret its transient regimes caused by impulsive excitation. It is shown that algebraic expressions derived from the SIM formulation enable to identify the so-called interaction points, providing the conditions in which two modes of the primary system interact and share energy through the nonlinear absorber. Moreover, the mutual effect of differently activated host system modes on the NES energy dissipation mechanism is discussed. Through sections of the multidimensional SIM, modal interaction triggering resonance capture cascades (RCC) can be effectively explained. The dissipation capabilities are eventually assessed in order to evaluate the efficiency of the RCC regime. Keywords Nonlinear energy sink · Nonlinear vibration absorber · Slow invariant manifold · Impulsive dynamics · Vibration mitigation G. Habib (B) Department of Applied Mechanics, MTA-BME Lendület Human Balancing Research Group, Budapest University of Technology and Economics, Budapest, Hungary e-mail: [email protected] F. Romeo Department of Structural and Geotechnics Engineering, Sapienza University of Rome, Rome, Italy
1 Introduction The mitigation of impact-induced vibrations in structural systems represents a challenging task in a variety of engineering contexts. Irrespective of the specific application, a realistic representation of such systems implies to consider either continuous or multi-degreeof-freedom mechanical (MDOF) models. So, the mitigation task is accomplished by absorbing efficiently energy from main excited systems, the dynamics of which typically involve either infinite or large number of frequencies. Thus, from an engineering standpoint, the goal is to devise the minimum number of possibly passive vibration absorbers designed to resonate for broad frequency band. Toward this goal, nonlinear vibration absorbers have received growing attention in the past two decades for their potentialities in overcoming the narrow frequency-band capabilities of their linear counterparts, i.e., the tuned mass damper (i.e., TMD). Within this context, the nonlinear energy sink (NES), consisting of a small mass connected to the primary system by an essential nonlinear spring, has been extensively studied. As shown in a series of works [1,2], the lack of any preferential resonance frequency of the NES makes it capable of resonating with any mode of the primary structure. In the case of multimodal response, resonance capture cascades (RCC) during which the nonlinear attachment resonates with different modes sequentially, from higher to lower ones, w
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