Damped nonlinear normal modes of bistable nonlinear energy sink for steady-state dynamic analysis and optimal design

Bistable nonlinear energy sink (BNES) possesses multiple equilibrium points, enabling large inter-well motions at low vibration energies. This characteristic results in superior and robust vibration suppression performance compared to traditional nonlinear energy sink. However, current research on BNES’s damping mechanisms predominantly focuses on free vibration problems, needing an effective tool to perfect its performance in forced scenarios. To address these issues, the damped nonlinear normal mode (dNNM) and extended energy balance method (E-EBM) are used to clarify the steady-state dynamic and provide an optimization means for BNES within a single-degree-of-freedom linear oscillator (LO-BNES). These dNNMs can be derived via extended periodic motion concept (EPMC) and alternating-frequency-time/harmonic balance method (AFT/HBM). The extended energy balance method (E-EBM) is utilized to derive the direct relationship between resonance solutions in dNNMs and excitation amplitudes. By combining these methods with numerical integration, as well as wavelet transform, a numerical simulation is carried out to reveal the evolution of steady-state dynamic in LO-BNES with the excitation increases, showing that the LO-BNES has four dNNM branches, comprising two symmetrical dNNMs surrounding trivial equilibrium points and two unsymmetrical ones around the nontrivial equilibrium points. Under low-level excitations, the unsymmetrical dNNMs are excited, resulting in intra-well vibrations for BNES. As the excitation amplitude increases, multiple internal resonances may appear in unsymmetrical modes, causing chaotic inter-well motions. With further excitation increases, several types of motions related to symmetrical dNNM emerge, i.e., the stable inter-well motions, strongly modulated responses (SMR), and high amplitude closed detached responses (CDR). To facilitate the evaluation of BNES’s performance, five crucial indicators are defined: compliance of intra-well motion, excitation thresholds of chaotic inter-well motion, stable inter-well motion, SMR, and CDR, and their analytical expressions are derived. Utilizing these formulas, parameter analysis is conducted, and selection criteria for BNES’s linear and nonlinear stiffness is also proposed: setting negative stiffness to make frequency of BNES surrounding stable equilibrium close to primary structure’s; then, identifying cubic nonlinear stiffness to ensure a minimal threshold of CDR exceeding the upper boundary of working load.