Low frequency acoustic radiation and vibration response of locally excited fluid-loaded structures

Abstract

The fluid loading of a thin elastic plate by static fluid can be characterized by the values of two parameters, one a ratio Ω of frequency to coincidence frequency (or equivalently, a Mach number M = Ω 1 2 ), the other a “fluid loading at coincidence” parameter ε = ϱc 0/ mω g . Fluid loading effects on the structural response are of crucial importance when Ω is close to 1 and when Ω is small—specifically when Ω = O(ε 3) or smaller, the parameter ε being small in all applications. This paper deals with the latter low frequency, heavy fluid loading limit, giving asymptotic solutions for a range of problems involving infinite, semi-infinite and finite thin elastic plates and membranes. The paper begins with a study of the infinite thin elastic plate under point force, line force and line moment excitation and gives a description of the surface and acoustic wave fields on the structure at all ranges from the excitation. This section provides some new results, and previous results for the drive admittances of heavily fluid-loaded plates are recovered. There follows an examination of the effects of ribs on an infinite plane structure (taken, in section 4, as a membrane for simplicity). The question of principal interest, in respect to later sections dealing with finite structures, is whether or not the phenomenon of resonance, associated with the finite spacing between ribs, can arise in the low frequency limit. It is in fact shown that in that limit there is generally substantial transmission of structural energy across a rib, regardless of the mechanical impedance of the rib, and then that there is no possibility of near-resonant response. However it is established that near-resonant response can indeed occur under certain specified circumstances involving finite rib impedance. Next the interaction of incident structural waves with the edge of a semi-infinite plate is discussed. A variety of baffled and unbaffled configurations, and of edge conditions, is of interest in this kind of problem. Here one specific case—that of an unbaffled plate with a free edge—is considered; this case is regarded as typical, as will be seen from future work in which many other cases will be briefly examined. It is shown that, except within a small distance of the edge, the vibration field consists of the incident wave plus a wave reflected with no change of amplitude, but with a phase change dependent upon the edge constraints. Fluid loading effects are completely specified by this phase shift, or equivalently by an “end-correction” giving the virtual origin from which the wave appears to be reflected. The acoustic field scattered from the edge is also calculated and shown to be very weak. This then suggests an approximate method for dealing with finite elastic plates in the low frequency heavy fluid loading limit; one merely calculates the vacuum dynamics of the structure, allowing for fluid loading effects in (i) the equivalent free wavenumber of the plate waves, (ii) the description of the response in the neighbourhood of concentrated excitation, where infinite plate results can be used for the forced motion, and (iii) the imposition of boundary conditions with phase shifts incorporated from the semi-infinite interaction analysis. The method is applied to a strip plate or membrane, giving quickly results which are identical with those derived elsewhere [1] from a thorough analysis of the finite fluid-loaded plate problem. In particular, it is shown that the strip plate has resonances at which the amplitude is limited only by small radiation losses and by mechanical losses, and that the plate has modes with the same shapes (except near the edges) as those in the absence of fluid loading. It is further shown here that the drive admittance of the finite heavily fluid-loaded line driven plate is purely imaginary (implying structural energy conservation), and that for particular choices of the plate length the admittance may be infinite (resonance), zero (antiresonance), or the same as the imaginary part of the admittance of the infinite plate (transparency). The paper ends with a brief discussion of further extensions of this approximate method which will be pursued in a subsequent study.