Nonlinear dynamics of a track nonlinear energy sink

A track nonlinear energy sink (TNES) has been proven as an effective control strategy. However, most researches focus on numerical or experimental aspects, with relatively little analytical study of intrinsic dynamic characteristics of the TNES. This study aims to investigate the nonlinear behaviors of a harmonically excited linear structure coupled with the TNES and reveal the vibration reduction performance of the TNES from the perspective of analysis. Firstly, the motion form of the TNES system is qualitatively analyzed by the global bifurcation, the time history, the Fourier spectrum, the phase trajectory, and the Poincare map. Secondly, the amplitude-frequency response for the periodic steady-state motion is quantitatively analyzed by combining the harmonic balance with the pseudo-arc length extension method and validated through numerical solutions. Then, the stability and bifurcation feature of the periodic steady-state solution are revealed by leveraging the Floquet theorem. Finally, the damping efficiency is explored. The results demonstrate that the complex nonlinearity of the TNES system can result in the coexistence of the periodic, quasi-periodic, and chaotic motion. Saddle-node bifurcations and Hopf bifurcations are discovered in the approximate analytical solutions. Strongly modulated responses (SMR) caused by Hopf bifurcation can greatly improve the damping efficiency of the TNES. In addition, a proper increase in mass ratio can suppress the adverse effects of frequency islands or saddle-node bifurcation curves near the resonance region. This research provides the necessary theoretical basis for optimizing and designing the TNES.