The energy envolope of a randomly excited non-linear oscillator

The exact differential equation for the energy envelope of a randomly excited non-linear oscillator is approximated by a time-averaging procedure. The resulting equation shows that, if the damping is sufficiently light and the correlation time scale of the excitation process is sufficiently small, the energy envelope can be approximated as a one-dimensional Markov process, governed by an appropriate Fokker-Planck equation. The physical significance of the various terms in this latter equation is emphasized.