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

Abstract

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. This work investigates the time response of a Duffing oscillator with time-varying parameters (excitation frequency, linear stiffness, and mass) by the multiple scale method and numerical method. When the excitation frequency sweeps in the multisolution range where two energy orbits coexist, the characteristics (maximum, jump-up, and jump-down points) of the response are dependent on the start frequency, the end frequency, and the sweep rate. The results can be explained by the transformation of the basin of attraction during the sweep. When the excitation frequency is fixed in the multisolution range, how can the state be transferred from one orbit to the other, especially from low-energy orbit to high-energy orbit? Most methods in the literature are qualitative and lack an in-depth explanation of the mechanism. In this study, a natural frequency temporary modulation method is proposed based on quantitative analysis, which can provide the guidelines in the design of nonlinear systems to work on preferred energy orbit. The principle of the method can be explained by the comparison between the trajectory of the oscillator and the transformation of the basin of attraction.