A Novel Approximate Analysis of the Jump Points and Global Stability of the Duffing Oscillator

A novel analysis method for determining the jump points and global stability of the amplitude-frequency response (AFR), both for softening and hardening Duffing systems, is presented in this paper. A higher-order amplitude polynomial equation for calculating the jump points is proposed, using a first-order approximate HBM (harmonic balance method). Although similar equations have been given in previous studies, the form given in this paper is more concise. Simulation results show that the jump points determined by the polynomial equation are in good agreement with the direct numerical simulation and the results of the 9th-order HBM. The approximate expressions of the boundary curves (BCs), that can determine the unstable response region of the primary response, are deduced by using the Floquet theory. It can be used for discriminating the stability quantitatively in a moderately damped and strongly nonlinear system.