Lorentzian-weighted frequency averaging for the evaluation of the input power into one-dimensional structural dynamic systems

This paper deals with a proposal to evaluate the frequency average based on the use of a Lorentzian function as weighting function. Due to some specific mathematical properties, the Lorentzian function allows evaluating the frequency average of a causal quantity as a result of a single deterministic calculation, without the need for computing the response at several frequencies explicitly. The proposed approach is used to evaluate the averaged input power into one-dimensional rods of which the properties are uncertain or their dynamic behaviour is perturbed by additional randomly distributed masses. For both cases, a relation between the width of the Lorentzian-weighting curve and the statistics of the natural frequencies of the system seems to arise. The accuracy of the prediction is shown by comparing the Lorentzian-weighted frequency average to the mean response obtained from Monte Carlo simulations. After a theoretical introduction to the Lorentzian averaging, numerical applications show the efficiency and the potential of the presented approach applied to the aforementioned cases.