The statistical energy analysis of coupled sets of oscillators

Abstract

The paper concerns the statistical energy analysis (SEA) of two conservatively coupled oscillators, sets of oscillators and continuous subsystems under broadband excitation. The oscillator properties are assumed to be random and ensemble averages found. Account is taken of the correlation between the coupling parameters and the oscillator energies. For coupled sets of oscillators or continuous subsystems, it is assumed that the coupling power between a pair of oscillators is proportional to the difference of either their actual energies or their ‘blocked’ energies, and expressions for the ensemble averages and coupling loss factors (CLFs) are found. Various observations are made, some of which differ from those that are commonly assumed within SEA. The coupling power and CLF are governed by two parameters: the ‘strength of connection’ and the ‘strength of coupling’. The CLF is proportional to damping at low damping and independent of damping in the high damping, weak coupling limit. Equipartition of energy does not occur as damping tends to zero, except for the case of two oscillators that have identical natural frequencies. While attention is focused on spring-coupled oscillators, similar results hold for more general forms of conservative coupling. The examples of two spring-coupled rods and two spring-coupled plates are considered. Conventional SEA and the coupled oscillator results are in good agreement for weak coupling but diverge for strong coupling. For strong coupling and weak connection, the coupled oscillator results agree well with an exact wave analysis and Monte Carlo simulations.