Response variance prediction using transient statistical energy analysis

Statistical energy analysis (SEA) is a prominent method for predicting the high frequency response of complex structures under steady loading where the structure is split into subsystems and the subsystem energies are calculated. Since at high frequencies, the dynamic response of nominally identical structures can differ greatly, methods have been developed to predict both the mean and variance of the energy in the subsystems of a system across an ensemble of systems. SEA can be extended to predict the transient response of a system, either to shock or time-varying inputs and is known as transient SEA, although this formulation has so far only been interested in the mean response. In this paper, a method for predicting the variance of the transient response is derived by considering how an individual realisation can deviate from the mean. A matrix differential equation for the covariance of the subsystem energies is derived which is driven by terms representing the variability in the system. These variance terms are provided by assuming that the natural frequencies in each subsystem conform to the Gaussian orthogonal ensemble. The accuracy of the method is investigated both numerically and experimentally using systems involving coupled plates and its limitations are discussed