Bifurcation curves of a linear system attached with a bistable nonlinear energy sink

Under harmonic excitation, a bistable Nonlinear Energy Sink (BNES) manifests diverse attractors, rendering the vibration characteristics highly intricate. This study employs the first-order harmonic balance method to analyze a two-degree-of-freedom system with a BNES, predicting the count of periodic attractors within and across potential wells. Formulas for the Fold bifurcation curves are deduced to anticipate the quantity of cross-well period responses and intra-well period responses under varying frequencies and amplitudes of excitation. The interrelation between the Fold bifurcation curves and the shapes of frequency response curves (FRC) is demonstrated, with a discussion on the shape and quantity of Fold bifurcation curves under diverse system parameters. The connecting points of the inter-well FRCs and the cross-well FRCs are resolved, yielding a Pitchfork bifurcation curve. The influence of system parameters on the shape of the Pitchfork bifurcation curve is scrutinized. Utilizing the acquired bifurcation curves allows for predicting the count of periodic attractors under different harmonic excitations. Due to the impact of various local and global bifurcations, the actual number of stable attractors may deviate from the predicted value, and the system may even lack stable periodic attractors. The stability of periodic attractors is assessed using Lyapunov exponents, unveiling the accurate predictive capability of bifurcation curves within certain parameter ranges. Corresponding to bifurcation curve predictions, under specific excitations, the system may exhibit both cross-well and intra-well periodic attractors, multiple cross-well periodic attractors, or multiple intra-well periodic attractors. While comprehensively predicting all periodic attractors remains challenging, the bifurcation curves serve as a valuable tool for identifying potential multiple steady-state responses during NES optimization. The prediction of cross-well motion contributes to designing a more efficient energy-harvesting NES.