Modal interactions between a linear oscillator and a nonlinear absorber: multiscale energy transfers

Abstract

Considerable attention has been given to the use of a nonlinear energy sink (NES) as a nonlinear absorber. The NES is an efficient passive control device, which has been the focus of extensive research. This paper uses the Complexification-Averaging/Geometric Singular Perturbation Theory (CX-A/GSPT) to investigate the multiscale energy transfers in a forced two-DOF system coupled to a grounded NES (GNES) with both nonlinear grounded stiffness and damping. Due to the coexistence of fundamental resonance and internal resonance in the system, complex modal interactions occur between the linear oscillator (LO) and the GNES. In addition, the ratio of the mass of the GNES and the LO can be seen as a perturbation. This may lead to the occurrence of multiscale dynamics and energy transfers in the system. By using the CX-A method, the slow flow equations of the system can be obtained. Further application of GSPT can obtain the critical manifold that is equivalent to the so-called slow invariant manifold (SIM). With different values of nonlinear grounded stiffness and damping, the critical manifolds have different structures. These critical manifolds can capture diverse types of system responses. Moreover, Hilbert-Huang transform (HHT) is used to analyze the time-frequency-energy relationship of system response. With different parameters, the instantaneous frequencies of the responses of the LO and GNES show a completely different change. The Hilbert spectrums confirm the occurrence of complex modal interactions and different time-scale energy transfers in the system.