The statistical properties of random frequency response functions

In many industrial situations the vibro-acoustic response of an engineering system is sensitive to manufacturing imperfections. Such imperfections are uncertain, or random, and this implies that any given system is necessarily drawn from an ensemble of possible systems. In some cases this is a "real" ensemble, as in the case of a collection of motor vehicles produced on a production line. In other cases the ensemble may be "virtual" in the sense that only one system is actually manufactured (for example, a satellite), although the properties of that one system can be considered to be drawn from a hypothetical ensemble of possibilities. In all cases the effect of the uncertainty should be considered in assessing the performance of the system: it is not sufficient for a nominally perfect system to meet the design requirements. Rather, there should be an acceptably low probability that the random system will not meet the design requirements. Any assessment of the random response of the system clearly requires knowledge of the statistics of the response, and in vibro-acoustics this normally involves a consideration of the statistics of a frequency response function (FRF). It initially appears to be a very challenging task to predict the statistical properties of FRFs, particularly when the underlying system uncertainties may be ill-defined or unknown. However, it is shown here that under broad conditions a universal behaviour arises, and random FRFs have very remarkable properties. The key result is that the average value of any function of an FRF is equal to the function evaluated at the average value of the FRF. This is known as the AE property in nuclear physics, and it is shown here that the result also applies in vibro-acoustics and has enormous utility in response prediction