Scaling law for the slow flow of an unstable mechanical system coupled to a nonlinear energy sink

Abstract

In this paper one first shows that the slow flow of a mechanical system with one unstable mode coupled to a Nonlinear Energy Sink (NES) can be reduced, in the neighborhood of a fold point of its critical manifold, to a normal form of the dynamic saddle-node bifurcation. This allows us to then obtain a scaling law for the slow flow dynamics and to improve the accuracy of the theoretical prediction of the mitigation limit of the NES previously obtained as part of a zeroth-order approximation. For that purpose, the governing equations of the coupled system are first simplified using a reduced-order model for the primary structure by keeping only its unstable modal coordinates. The slow flow is then derived by means of the complexificationaveraging method and, by the presence of a small perturbation parameter related to the mass ratio between the NES and the primary structure, it appears as a fast-slow system. The center manifold theorem is finally used to obtain the reduced form of the slow flow which is solved analytically leading to the scaling law. The latter reveals a nontrivial dependence with respect to the small perturbation parameter of the slow flow dynamics near the fold point, involving the fractional exponents 1/3 and 2/3. Finally, a new theoretical prediction of the mitigation limit is deduced from the scaling law. In the end, the proposed methodology is exemplified and validated numerically using an aeroelastic aircraft wing model coupled to one NES.

Effect of stochastic forcing on the dynamic behavior of a self-sustained oscillator coupled to a non-linear energy sink

“…The zeroth-order approximation assumes that the left fold point is actually reached. However numerical simulations show that when approaching the left fold point the trajectory deviates slightly from the critical manifold (this behavior is analytically described in [13]). Then the slow flow undergoes a fast and nearly horizontal jump to the right attracting part of M 0 on a point called arrival point 2 .…”

“…In the deterministic case the mitigation limit can be predicted analytically by means of a slow-fast partition of the slow flow dynamics [6,13].…”

“…The previous preliminary results confirm that the effect of the noise cannot be only studied with the zeroth-order approximation. Indeed, as in the deterministic case the trajectory of the slow flow dynamics moves away from the critical manifold when it passes near the left fold point [13,40]. The zeroth-order approximation [6] assumed that the trajectory goes along the critical manifold and reaches the left jump before jumping to the right attractive part of the critical manifold.…”

“…[8,9,10]. The theoretical prediction of observed operating regimes has been performed by means of multiple time scales approaches [11,12] and improved more recently using the center manifold reduction technique [13]. Also in the context of self-oscillations induced by fluid-structure interactions, the mitigation of vortex-induced vibrations caused by the non-linear interaction of a laminar flow and a rigid circular cylinder has been first investigated by Tumkur and al.…”

Robust optimization of nonlinear energy sinks used for mitigation of friction-induced limit cycle oscillations

“…The design and optimization of NES constitute a very dynamic field of research. Recently, a lot of work has for example been produced concerning the study of new configurations and technologies allowing/enhancing TET (Tian et al (2021); Xiong et al (2021)), the analysis of the underlying dynamics of the coupled system (Bergeot (2021); Habib and Romeo (2021); Bergeot et al (2020)), the deterministic optimization of NES (Geng et al (2021); Khazaee et al (2019)), the stochastic optimization of NES Pidaparthi andMissoum (2018, 2019), the coupling of the NES to an energy harvester (Fang et al (2021); Karama et al (2021); Karličić et al (2021)) or the use of a NES as an energy harvester (Zhou et al (2014); Li and Li (2021)), etc. Among the many research subfields about NES, the present work takes place within the framework of robust optimization of NES when the latter have uncertain parameters and are used to mitigate Limit Cycle Oscillations (LCO).…”