A new analytical solution of a bistable Duffing oscillator under large amplitude periodic vibrations

In the paper, we report a new analytical method to solve the forced bistable Duffing oscillator with large amplitude periodic vibrations. The analytical solution is given in terms of the Jacobi elliptic function of corresponding conservative system and harmonic solution of nonconservative system. First, in order to simplify the influence of elliptic function on the subsequent calculation, we use the higher order harmonics to approximate the elliptic function solution of conservative system and find that the velocity term often has higher harmonic components than the displacement term. In addition, we demonstrate the accuracy of this phenomenon by using the amplitude spectrum. Furthermore, we combine this phenomenon with the perturbation method to solve the approximate response of a forced bistable Duffing oscillator with damping and harmonic excitation. Different system parameter values are selected for analytical verification, and the analytic results are consistent with those obtained by the fixed-step fourth-order Runge–Kutta method (RK-4), which verifies the accuracy of the analytical solution. This indicates that the proposed method is suitable for solving the forced bistable Duffing oscillator with large amplitude vibrations.