Harmonic oscillator driven by random processes having fractal and Hurst effects

While the response of a damped harmonic oscillator to random excitation offers the basic model in mechanics, stochastic dynamics, and stochastic fatigue of structures, the response due to random forcings with fractal and Hurst characteristics is studied for the first time. We investigate two types of such forcings: Cauchy and Dagum. Given the fact that their covariance functions lack explicit Fourier transforms, the spectral analyses are not possible, and, hence, the studies have to be conducted directly in the time domain—mathematically, a much more challenging task. In comparison with conventional models, we also give responses of the oscillator under random excitations of the white noise, Ornstein–Uhlenbeck (OU), and Matérn type. In contradistinction to the latter which has some limited ability of modeling multiscale phenomena, the Cauchy and Dagum forcings allow decoupling of fractal and Hurst effects. On the basis of a second-order stochastic differential equation, we calculate the transient second-order characteristics of the response under any given type of excitation. Overall, we find that, given the same variance on input, the variance on output is strongest for Matérn, then Cauchy, then OU, then white noise, and finally Dagum forcing. Also, if the excitation correlation function is Matérn, the correlation function of response is approximately Matérn, but this is not the case with the Cauchy excitation.