Statistical Energy Analysis modelling of complex structures as coupled sets of oscillators: ensemble mean and variance of energy

Expressions are derived for the ensemble means and variances of the subsystem energies of built-up systems comprising two subsystems. The approach is based on the Statistical Energy Analysis of two spring-coupled oscillators and sets of oscillators, or coupled continuous subsystems, described by Mace and Ji [The statistical energy analysis of coupled sets of oscillators, Proceedings of the Royal Society A 1824 (2007)]. The paper focuses on spring coupling, although similar results hold for more general forms of conservative coupling. Randomness is introduced into the system by assuming that the natural frequency spacings in each subsystem conform to certain statistical distributions. A “coupling coefficient parameter” is introduced which, together with the “coupling strength parameter” defined by Mace and Ji (2007), accounts for the statistics of the coupling stiffness. Various approximations and assumptions are made. It is seen that the variance of the excited subsystem depends primarily on the variance of the input power, which in turn depends on the variance of the number of modes of the excited subsystem in the frequency band of excitation and their mode shapes. The variance of the undriven subsystem, on the other hand, depends primarily on the variance of the intermodal coupling coefficients, which in turn depend on the variances of the number of in-band modes of both subsystems and their mode shapes. The cases of Poisson and Gaussian Orthogonal Ensemble natural frequency spacing statistics are considered. Numerical examples of two plates coupled by one or a number of springs are presented.