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Description
In the mid-frequency range, the high modal density of flexible (vibro-)acoustic components makes deterministic methods computationally expensive, while small spatial variations in geometry, material properties, or boundary conditions may have a (random) wave scattering effect. An efficient statistical analysis approach is therefore required, and since spatial and frequency averaging are commonly used to reduce response variability in reverberant systems, obtaining both the mean and the covariance of frequency response functions is essential. This paper therefore investigates the covariance of frequency response functions of built-up systems comprising flexible components, treated as random, connected to stiff components, modelled deterministically. Such configurations are common in practice, such as aircraft fuselages, where stiff frames support thin skin panels, or buildings, where flexible airspaces (rooms) are bounded by stiff walls and floors. The covariance is considered for different response locations, excitation locations, and excitation frequencies.To this end, a cross-frequency extension of the diffuse field reciprocity relationship is derived and applied to the analysis of such systems. This relationship links the cross-spectral reverberant forces of the random components to the system response, and is expressed in terms of the mean and covariance of the frequency response functions of the individual components. Substituting this relationship into the system-level dynamic equations yields expressions for obtaining the mean and covariance of the full system response.The theoretical framework is validated against Monte Carlo simulations of random built-up systems, demonstrating good agreement and confirming the practical potential of the proposed method for efficient uncertainty quantification in mid-frequency (vibro-)acoustic analysis.