Vibration energy flow analysis of the van der pol oscillator

This paper investigates the forced Van der Pol oscillator from a viewpoint of vibration flow and energy exchange. The method of averaging and the method of harmonic balance are used to derive analytical formulations of time-averaged power flow variables associated with steady-state periodic and quasi-periodic responses, respectively. Effects of bifurcations on time-averaged power flow are investigated both by analytical methods and numerical integrations. It is found that the time-averaged input power by the external harmonic force may become negative at some excitation frequencies. Correspondingly, the damping nonlinearity may lead to positive time-averaged dissipated power. The frequency band where there is negative time-averaged dissipated power is formulated through analytical approximations. It is also found that bifurcations of the forced oscillator from periodic to quasi-periodic responses may not lead to jump phenomenon in time-averaged input power. The work provides some new understanding of the power flow characteristics of the Van der Pol oscillator and demonstrates potential benefits of using such systems for vibration energy harvesting.