Mean and mean-square responses of a prototypical master/fuzzy structure

Analytical estimates are made for mean and mean-square transient responses of a single degree of freedom master oscillator attached to an undamped N degree of freedom random structure. The ensemble averaged response is found to be, asymptotically for a large number of substructural degrees of freedom, in accord with the predictions of the Pierce-Sparrow-Russell theory. In particular the mean response of the master oscillator manifests an apparent damping. Corrections to this behavior are also predicted, but are found to lie well below the root-meat-square level of the fluctuations. The corrections are therefore, for practical purposes, unimportant. The ensemble average of the square of the response is also investigated and found to take a value at later times that is in accord with simple equipartition arguments advanced in an earlier paper. The power spectral density of the late time behavior is shown to be proportional to the square of the power spectrum of the early time response, and thus to be a narrower-band process than that of the early time behavior. Numerical simulations are presented which agree with the analytical predictions at moderate times, but also show an enhanced backscatter that develops over longer time scales.