On the use of the Lorentzian function for the evaluation of the frequency averaged input power into plates

Predicting the average behaviour of an ensemble of complex systems is one of the most challenging tasks of vibro-acoustics. Well-established energetic formulations exist to address the problem at high-frequencies, where a diffuse and incoherent field is hypothesised and where the input power into the structure can be evaluated by assuming spatially infinite or semi-infinite systems. These approximations may be no longer valid when dealing with mid-frequency problems, where the average response of the ensemble can still be influenced by individual modes and a more accurate evaluation of the input power can significantly improve the quality of the results. This paper proposes a new approach to compute the frequency averaged input power through a single deterministic calculation using a Lorentzian function as weighting function. Thanks to its mathematical features the frequency averaging procedure can be evaluated straightforwardly for causal systems, without the need of the response computation at several frequencies. The Lorentzian function is characterised by the parameter , which describes its width. In order to correlate to the amount of uncertainty in the system, two strategies are proposed in this paper. The first one suggests to tune on an analytical estimate of the natural frequency statistics, while in the second case, a mode tracking approach is used. The presented approaches are used to evaluate the average input power and energy density for Kirchhoff plates perturbed by randomly distributed masses. Finally, comparisons with Monte Carlo simulations illustrate the accuracy of the proposed approaches.