Power transmission variance predictions in complex systems with dissipation

Abstract

The eigenstatistics of damped complex systems are examined for the purpose of improving statistical response predictions such as those formulated in statistical energy analysis and statistical room acoustics. Both the statistical distribution of modal decay rates and the effect of dissipation on the intermodal correlation of the natural frequencies described by the gaussian orthogonal ensemble (GOE) of random matrix theory are explored. It is found that the modal decay rates are distributed according to a chi‐square distribution whose degree depends on the distribution of damping in the system and the wavelength of the disturbance. The intermodal correlations of the natural frequencies are found to be unaffected by the presence of moderate damping. Level repulsion (the absence of near degeneracies) and spectral rigidity (the near regularity of the spectrum) are found to conform to the prediction of the GOE provided the system is reverberant. The variance of the power transmission function for an irregularly shaped membrane is formulated using a GOE‐type natural frequency spectrum and chi‐square distributed modal decay rates. Numerical simulations of membranes are performed which confirm the calculated effect of a distribution of decay rates on power transmission characteristics.