Capture of high energy orbit of Duffing oscillator with time-varying parameters

This work investigates the time response of a Duffing oscillator with time-varying parameters (excitation frequency, linear stiffness, and mass) by approximate analytical and numerical methods. When the excitation frequency sweep covers the multisolution range, the characteristics of the response (maximum response, jump-up frequency, and jump-down frequency) mainly depend on the frequency sweep rate. If the frequency sweep is ended in the multisolution range, the sweep rate determines the energy orbit that the final response will capture. The results can be explained by comparing the state spaces of the oscillator with the change of basin of attraction of the high-energy orbit during the sweep. Furthermore, if the excitation is fixed at a specific frequency in the multisolution range, a method of natural frequency temporary modulation is proposed for the capture of the high-energy orbit. For practical realization, this method is completed by two ways, that is, the linear stiffness temporary modulation and mass temporary modulation. The modulation schedules of time-varying linear stiffness and mass are determined quantitatively, and it is proved that they could help capture the high-energy orbit similar to the excitation frequency sweep. The developed methods and results of this work can provide the guidelines to design nonlinear systems to work on preferred energy orbit.