Reconstructing the transient, dissipative dynamics of a bistable Duffing oscillator with an enhanced averaging method and Jacobian elliptic functions

• New analytical approach reconstructs bistable Duffing transient dissipative dynamics.
• Averaging method and Jacobian elliptic functions utilized in new approach.
• Transient vibration amplitudes and frequencies analytically derived in closed-forms.
• Trajectory reconstructions are very accurate for both intrawell and interwell regimes.
• Analytical approach is computationally efficient compared to 4-order-RK.

Systems characterized by the governing equation of the bistable, double-well Duffing oscillator are ever-present throughout the fields of science and engineering. While the prediction of the transient dynamics of these strongly nonlinear oscillators has been a particular research interest, the sufficiently accurate reconstruction of the dissipative behaviors continues to be an unrealized goal. In this study, an enhanced averaging method using Jacobian elliptic functions is presented to faithfully predict the transient, dissipative dynamics of a bistable Duffing oscillator. The analytical approach is uniquely applied to reconstruct the intrawell and interwell dynamic regimes. By relaxing the requirement for small variation of the transient, averaged parameters in the proposed solution formulation, the resulting analytical predictions are in excellent agreement with exact trajectories of displacement and velocity determined via numerical integration of the governing equation. A wide range of system parameters and initial conditions are utilized to assess the accuracy and computational efficiency of the analytical method, and the consistent agreement between numerical and analytical results verifies the robustness of the proposed method. Although the analytical formulations are distinct for the two dynamic regimes, it is found that directly splicing the inter- and intrawell predictions facilitates good agreement with the exact dynamics of the full reconstructed, transient trajectory.