Transmission of energy down periodically ribbed elastic structures under fluid loading

Abstract

The paper studies a model configuration in which an elastic membrane is immersed in static compressible fluid, excited by a time-harmonic line force and supported by a periodic array of line supports (ribs) of infinite mechanical impedance. At the driven rib the velocity has a prescribed value V 0 , while the velocities vanish at the locations, x = nh ( n = ± 1, ± 2,. . .), of the supporting ribs. Fluid loading provides the only coupling between adjacent bays, and the aim is to expose the dual role of that coupling (local and long range) in the transmission of energy from the excitation to infinity along the structure and to the acoustic radiation field. This transmission is characterized by the variation with n of the force F n exerted on the structure by the n th rib. An exact formal solution is obtained for F n in terms of the Green function G(x) of the unribbed fluid-loaded structure, and explicit expressions are obtained for F n when only the subsonic surface wave component, G s ( x ), is included in G(x) (though with full account of fluid loading in G s ( x ) itself). These expressions show that under ‘significant’ and ‘heavy’ fluid loading (terms made precise in the text), fluid loading in the form of subsonic surface waves provides a local bay-to-bay coupling very much like that of an imperfect mechanical isolation, and induces a pass and stop band structure of the kind familiar from other studies of wave propagation in mechanically-coupled periodic structures in the absence of fluid loading. Under ‘light’ fluid loading it is shown that there can be no strict pass bands, but frequency bands around the vacuum bay resonance frequencies are identified within which the energy decay rate along the structure is very slow. In all these calculations the fate of the power injected by the excitation is followed in all detail, whether to infinity in the structure or to infinity in the acoustic field. The acoustic component G a ( x ) is then included, and specific asymptotic expressions for G a ( x ) are used to deal with the light and heavy fluidloading cases. These expressions for G a ( x ) involve slow algebraic decay with x , and induce a generally similar decay of the F n with n . In this sense, the acoustic component G a ( x ) provides a long-range coupling between the driven rib and distant ribs which, in the stop bands, is much stronger than the exponentially weak coupling provided by the surface wave component G s ( x ). Numerical estimates are given which indicate that in both light and heavy fluid loading the acoustic component of the force F n exceeds the surface wave component once n exceeds a very modest value. The paper ends with a discussion of the possible implications for structure-borne noise control in periodic fluid-loaded structures, for the application of Statistical Energy Analysis to structures under fluid loading, and for the relevance of the ideas of Anderson localization in an irregular structure under fluid loading.