Nonlinear power flow analysis of the Duffing oscillator

• The Duffing oscillator is investigated from a perspective of vibrational power flows.
• Analytical/numerical approaches are employed to address the essentially nonlinear power generation and dissipation characteristics.
• The instantaneous/time-averaged power flow behaviour associated with periodic/chaotic responses is analysed.
• The effects of nonlinear phenomena such as sub-/super-harmonic resonances, non-uniqueness of solutions, bifurcation and chaos, on power flows are revealed.
• The influences of different parameters, averaging time as well as initial conditions on power flow variables are examined.

Power flow characteristics of different forms of the Duffing oscillator, subject to harmonic excitations, are studied in this paper to reveal the distinct power input and dissipation behaviour arising from its nonlinearity. Power flow variables, instead of the displacement and velocity responses, are used to examine the effects of nonlinear phenomena including sub-/super-harmonic resonances, non-uniqueness of solutions, bifurcations and chaos. Both analytical harmonic balance approximations and Runge–Kutta numerical integrations are adopted to effectively address instantaneous/time-averaged power flows of the system with periodic/chaotic motions without losing the essential nonlinear characteristics. It is demonstrated that only the in-phase velocity component with the same frequency as the excitation contributes to the time-averaged input power (TAIP). It is shown that super-/sub-harmonic resonances may result in substantial increases in TAIP and the nonlinearity leads to varying time-averaged power flow levels sensitive to the initial conditions. The study reveals that bifurcations may cause large jumps in time-averaged input power. However, for bifurcations of periodic to chaotic motions encountered in the low-frequency range, the corresponding variations in TAIP of the double-well potential systems are small. For a chaotic response, the associated TAIP is insensitive to the initial conditions but tends to an asymptotic value as the averaging time increases, and thus can be used as a measure to quantify chaotic responses. The paper concludes some inherently nonlinear power flow characteristics which differ greatly from those of the linear systems, and provides useful information for applications.