Stochastic Averaging of Nonlinear Oscillators: Hilbert Transform Perspective

A novel stochastic averaging technique based on a Hilbert transform definition of the oscillator response displacement amplitude is developed. Specifically, a critical step in the conventional stochastic averaging treatment involves the selection of an appropriate period of oscillation over which temporal averaging can be performed. Clearly, for oscillators with nonlinear stiffness defining such a period is not an obvious task. To this aim, an intermediate step is often introduced relating to the linearization of the nonlinear stiffness element, i.e., treating it as response amplitude dependent. Obviously, this additional approximation can potentially decrease the overall accuracy of the technique. Thus, to circumvent some of the aforementioned limitations an alternative definition of the amplitude process is considered herein based on the Hilbert transform. In comparison to a standard definition of the response displacement amplitude, the amplitude definition used herein does not require the a priori selection of an equivalent natural frequency. Notably, this feature provides enhanced flexibility in the ensuing stochastic averaging treatment, and it can potentially result in higher accuracy. Further, an extension of the Hilbert transform–based stochastic averaging is developed to account for oscillators endowed with fractional derivative terms as well. The hardening Duffing oscillator both with and without fractional derivative terms, as well as the bilinear stiffness oscillator, are considered in the numerical examples section for demonstrating the reliability of the technique. For all cases the analytical results are compared with pertinent Monte Carlo simulation data.