Complex dynamics and targeted energy transfer in linear oscillators coupled to multi-degree-of-freedom essentially nonlinear attachments

Abstract

We study the dynamics of a system of coupled linear oscillators with a multi-DOF end attachment with essential (nonlinearizable) stiffness nonlinearities. We show numerically that the multi-DOF attachment can passively absorb broadband energy from the linear system in a one-way, irreversible fashion, acting in essence as nonlinear energy sink (NES). Strong passive targeted energy transfer from the linear to the nonlinear subsystem is possible over wide frequency and energy ranges. In an effort to study the dynamics of the coupled system of oscillators, we study numerically and analytically the periodic orbits of the corresponding undamped and unforced hamiltonian system with asymptotics and reduction. We prove the existence of a family of countable infinity of periodic orbits that result from combined parametric and external resonance interactions of the masses of the NES. We numerically demonstrate that the topological structure of the periodic orbits in the frequency-energy plane of the hamiltonian system greatly influences the strength of targeted energy transfer in the damped system and, to a great extent, governs the overall transient damped dynamics. This work may be regarded as a contribution towards proving the efficacy the utilizing essentially nonlinear attachments as passive broadband boundary controllers.

Enhancing Robustness of Instability Suppression by Means of Multi-Degree-of-Freedom Nonlinear Energy Sinks

“…The results reported in this work, which should be viewed in conjunction with earlier theoretical and experimental results, 1, 2 indicate that appropriately designed lightweight passive nonlinear absorbers with essential stiffness nonlinearities can suppress, effectively and robustly, LCO instabilities. As shown in other work, 4 the dynamical mechanism governing this suppression is a series of resonance captures that occur between the wing modes and multiple modes of the NES, which result in broadband passive targeted transfer of unwanted vibration energy from the wing to the NES, where this energy is localized and passively dissipated. Additional work is needed to study the complex and highly degenerate structure of the dynamics of the integrated structure-MDOF NES system.…”

“…3. This kind of MDOF NESs in series coupling was studied in Tsakirtzis et al, 4 basically showing that application of the MDOF NESs can induce even richer dynamics on the frequency-energy plot (FEP). 5 This means, in turn, that there exists higher possibility for a system to possess more 'baits' to seize the dynamics caused by any disturbances onto a branch in the FEP, making it stick to (or, at least, very close to) the branch until a significant amount of energy is dissipated by a damper in the process of resonance captures.…”

Magnetic Nonlinear Energy Sink for Vibration Attenuation of Unbalanced Rotor System

“…We assume that the length of the shaft is 370 mm and the diameter is 10 mm and the modulus of elasticity is 2.11 × 10 11 Pa. When 0 = /2, the disc is in the middle of the shaft, the gyroscopic effect can be ignored, and the numerical simulation can be carried out based on (11). The bending stiffness of the rotor is 9.87 × 10 4 N/m.…”

“…The mechanism of the nonlinear energy sink is energy pumping and has been studied for a long time, and various types of nonlinear energy sinks are designed and analyzed [9][10][11][12]. Nonlinear stiffness is the most important component in all NES structures.…”

“…Nonlinear stiffness is the most important component in all NES structures. Cubic stiffness is widely applied in NES designing [8,11], and many NESs with nonpolynomial nonlinearity are also presented [13]. Vibro-impact type NES or NES with nonsmooth stiffness also proved to be efficient in controlling the vibration of the primary vibration system [14][15][16].…”