A study of uncertainty in applications of statistical energy analysis

Abstract

Statistical Energy Analysis (SEA) is an approach to vibration analysis which provides estimates of the response of large and complex structures to broadband forces in terms of gross system parameters and simple explicit formulae. The penalty for the apparent simplicity of this approach is that the accuracy of response prediction for any individual practical system is subject to uncertainty. One source of uncertainty is purely physical and associated with variations in dimensions, damping, materials, force distributions, boundary conditions, etc., from the nominal values assumed in the models. The other source of uncertainty relates to the suitability of the coupled subsystems model employed, the validity of the SEA hypothesis as applied to the model, the choice of coupling loss factors, the centre frequency and the analysis bandwidth (i.e., the number of modes involved). Since fine details of systems under analysis are not represented in the theoretical models, SEA may be considered to produce the ensemble average behaviour of a set of similar systems which have the same gross parameters but which differ from each other in detail. The practical justification for such an approach is that no physically realisable system corresponds exactly to any theoretical model, however detailed, and hence it is uneconomic and unrealistic to expend time and money on modelling of fine detail which only cause small variations about the ensemble mean. This is particularly true when many vibrational modes contribute to the total response of the system. This thesis sets out to investigate the influence of perturbations of geometric subsystem parameters on power flow, energy levels and coupling loss factor as functions of subsystems modal densities, dissipation loss factors and analysis bandwidth. Various cases of coupled, simple, multi-mode systems are investigated. The investigations are carried out by predicting the response of the coupled system, using exact analysis, and studying the sensitivity of the prediction to random perturbations of principal geometric parameters. The latter procedure is carried out by using the Monte Carlo method. The cases analyzed cover different spatially extended, coupled systems such as one-dimensional coupled systems (beams), two-dimensional coupled systems (plates) and a three-dimensional system (panel-box). Results for the mean, the variance, and the confidence intervals for the frequency average values of the quantities of interest (power flow and coupling loss factor) are obtained across ensembles of similar systems as assumed in SEA. In addition, cumulative probability distributions for these quantities are presented. An empirical relationship which relates the variance of the coupling loss factor of coupled plates system to the modal overlap factor and the number of modes is also derived. It is shown that the coupling loss factor derived from the analysis of energy flow between semi-infinite systems can grossly overestimate the actual coupling loss factor of the finite coupled systems when the average modal overlap factor of the coupled system is much less than unity. The discrepancy decreases as the modal overlap factor approaches unity. The estimates of coupling loss factor agree well with those obtained from the corresponding semi-infinite systems at high modal overlap. It is also shown that the variances of the coupling loss factor and of the power flow between subsystems associated with specific ranges of system perturbations decrease as the modal overlap factor increases. This is supported by the study of the cumulative distribution functions of the quantities above: the distributions approach normality with modal overlap. It is demonstrated on cases of coupling between a small subsystem and a large subsystem that only one subsystem needs to have a high modal overlap factor in order to obtain acceptable estimates for the coupling loss factor and the power flow. The numerical results are supported qualitatively by experimental investigations on coupled beams and coupled plates systems. The most significant practical implications of the results of this investigation are that the application of conventional SEA to coupled systems of low modal overlap factor can greatly overestimate the transmission of energy from a directly driven subsystem to an undriven coupled system and that small variations in system parameters can produce large effects on system response.

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