Statistical response predictions in complex systems with dissipation
Burkhardt, John Albert
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https://hdl.handle.net/2142/19599
Description
Title
Statistical response predictions in complex systems with dissipation
Author(s)
Burkhardt, John Albert
Issue Date
1995
Department of Study
Mechanical Science and Engineering
Discipline
Mechanical Science
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
Applied Mechanics
Engineering, Mechanical
Language
eng
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 correlations of natural frequency spectra are explored. It is found that for moderate damping the modal decay rates of complex systems are distributed according to a modified chi square distribution whose degree depends on the spatial distribution of lossy material in the system and the wavelength of the disturbance. In addition the intermodal correlations of the natural frequency spectra are found to correspond with the predictions of the Gaussian orthogonal ensemble (GOE) provided the system remains reverberant.
The effect of dissipation on statistical response estimates was explored by considering the transmission function, and the resulting energy density, of irregularly shaped damped membranes. In particular, analytical expressions estimating the variation in energy densities were developed which included a statistical distribution of modal decay rates. The introduction of a distribution of modal decay rates was found to increase expected response variations. Numerical experiments were performed on irregularly shaped damped membranes which confirmed the calculated effect of a distribution of modal decay rates on energy density variations.
The possible failure of statistical response estimates due to ignored correlations between modal amplitudes was also demonstrated. It was shown that, due to modal coherence, the mean energy density in a complex system is enhanced at the source point relative to other points in the field. The enhancement factor was found to rise from a factor of two at early times to a factor of three at times on the order of the inverse of the mean modal spacing. It was additionally demonstrated that the evolution of the enhancement factor in time from a value of two to a value of three is the same in damped systems as in undamped systems provided the simple decay of the damped reverberant field is accounted for.
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