Periodic and chaotic responses of flexible bistable energy harvesters in rotational environment

For the dual-beam coupled flexible bistable energy harvester (FBEH) with time-varying potential wells, accurately fitting the magnetic forces in two directions simultaneously using polynomials is difficult, hindering the derivation of the analytical solutions. Therefore, in this study, a semi-analytical method that combines the incremental harmonic balance method (IHBM) and the arc-length method is employed to determine the periodic solutions, with stability assessed by the Floquet theory. Comprehensive analyses of the dynamic responses are conducted, encompassing jump phenomenon, multiple solutions, and bifurcation characteristics. The obtained semi-analytical solutions demonstrate an excellent approximation for the system's periodic responses. Subsequently, the chaotic responses are analyzed via the Lyapunov exponents, and the effect of nonlinear stiffness was investigated. The nonlinear dynamic behaviors and characteristics of the FBEH demonstrate the consistent dynamic response patterns of the FBEH under different initial magnetic spacings, generally following the sequence of periodic intrawell oscillations, chaotic interwell oscillations, multi-orbit asymmetric periodic responses, chaotic responses and symmetric periodic responses. The initial magnetic spacing and nonlinear stiffness coefficients significantly influenced the jump frequency and response amplitude, whereas their impact on the dynamic response patterns was relatively minor. Overall, this study enhances the theoretical understanding of the FBEH in rotational environments by providing valuable insights and references for the design of such complex nonlinear electromechanical coupling systems.