Study on the vibration reduction mechanism of nonlinear energy sink with Zener system

Viscoelastic materials are widely used in vibration isolation and reduction devices due to their simple structure and excellent energy dissipation performance. However, the introduction of viscoelastic Maxwell elements typically adds a half degree of freedom to the system, thereby increasing its complexity. The coupling effects among these complex structures and their impact on system dynamics remain unclear. This paper applies the slow-fast analysis method based on the complexification-averaging method to systems containing a half degree of freedom for the first time to study the complicated behavior and its mechanism. An approximate analytical solution for the two-and-a-half degrees of freedom system is obtained using the complexification-averaging method. By applying the multi-scale method, the slow invariant manifold of the system is derived, and the necessary conditions for a strongly modulated response are obtained. The vibration mechanisms of these responses are explained by using slow-fast analysis method with a combination of the slow invariant manifold, slowly variable response curves, and phase trajectory analysis. The result shows that the fast subsystem jumping back and forth between two branches of the slow invariant manifold is the main cause of strongly modulated response.The evaluation of the energy spectrum reveals that the damping ratio and stiffness ratio of the viscoelastic elements can be adjusted to further enhance the vibration reduction efficiency of the system. Additionally, the control equations for the Fold bifurcation and Hopf bifurcation of the system are derived, and the stability of the system response is analyzed.