ENSEMBLE AVERAGED VIBRATION ENERGY FLOWS IN A THREE-ELEMENT STRUCTURE

Abstract

Closed form expressions are presented for ensemble averaged vibration energies and energy flows in a three-element structure. The ensemble consist of one-dimensional elements, having random lengths with an uncertainty that is plus or minus a quarter of the wavelength. The ensemble averages are functions of only the ensemble averaged input power, the modal densities, the dissipation loss factors and the transmission factors for the couplings, found from travelling wave analysis. The ensemble averages are not related to each other according to the SEA hypothesis that energy flow is proportional to the difference of coupled elements' modal energies. However, energy balance is governed by terms directly proportional to differences of modal energies while also relating non-directly coupled elements. Numerical investigations of this indirect energy flow indicate a measure of coupling strength: i.e., the ratio of the transmission factor to the product of the two modal overlaps in both connected elements. Based on the proposed measure, series expansions are made showing that when coupling is weak and when the response in each element is reverberant, the SEA hypothesis is true. Also, the ensemble averaged energies are then determined by a standard SEA model by using travelling wave estimates of the coupling loss factors.