Exact statistics for linear time delayed oscillators subjected to Gaussian excitation

Large classes of stochastic systems of interest to acoustics and vibrations have time delays. The presence of these time delays hinders efforts to provide physical insight into the system behavior, because of the difficulty of finding exact solutions. This paper addresses this issue by presenting exact steady-state solutions for the probability density functions (pdfʹs) for linear time delayed oscillators subjected to Gaussian excitation of arbitrary correlation. A key parameter influencing these pdfʹs is shown to be the ratio of mean generalized kinetic energy to the mean generalized potential energy. In a single damped harmonic oscillator this ratio is unity which implies an equi-partition of generalized energy. Time delays cause deviations in the equi-partition of generalized energy which creates regions of preferential total phase and causes the amplitude pdf to shift from a Rayleigh distribution to a Hoyt distribution.