Predicting the variance of the frequency-averaged energetic response in hybrid finite element—Statistical energy analysis

In this contribution, the hybrid finite element-statistical energy analysis method is extended such that not only the mean and the ensemble variance of the harmonic system response can be computed, but also the ensemble variance of the frequency band-averaged system response. The computed variance represents the uncertainty that is due to the assumption of a diffuse field in components of the hybrid system. The developments start with a cross frequency generalization of the diffuse field reciprocity relationship between the total energy in a diffuse field and the cross spectrum of the external loading. By making extensive use of this generalization in a first-order perturbation analysis, explicit expressions are derived for the variance of the vibrational energies in the diffuse components and for the variance of the cross spectrum of the response of the deterministic components. These expressions are extensively validated against Monte Carlo analyses of systems consisting of connected plates, in which diffuse fields are simulated by randomly distributing small concentrated masses, acting as wave scatterers, across the diffuse components.