Asymptotic modal analysis and statistical energy analysis of dynamical systems

Abstract

The well known text by Lyon on Statistical Energy Analysis (SEA) [R. H. Lyon, Statistical Energy Analysis of Dynamical Systems: Theory and Applications (MIT Press, Cambridge, MA, 1975)] is the standard reference on the subject. As originally conceived SEA was based upon several plausible hypotheses. Various authors have considered the underlying basis for SEA and thereby advanced our understanding of it. In the present paper, the relationship between classical modal analysis (CMA) and SEA is studied. It is shown how the results of SEA may be obtained as an asymptotic limit of CMA. The present work only considers the response of a single general linear structure under a random or sinusoidal load. However future plans will consider (a) the structural acoustic response of an enclosure bounded by a flexible wall and excited by external forces and (b) two (or more) coupled structural systems. Here the emphasis is on (1) a new derivation of the SEA results from CMA for a single structure under a random load, (2) the generalization of the usual SEA result to show that, asymptotically, all points on the structure have the same response. In the literature this is sometimes invoked as an assumption. Here the result is derived as part of the asymptotic limit of CMA. (3) A numeriCal example which displays the manner in which the asymptotic limit is approached for random loading and (4) an extension of the usual SEA result for sinusoidal loading. In view of the above, the present results might be called more appropriately Asymptotic Modal Analysis (AMA) rather than SEA.