A coupled system consisting of an acoustic cavity and an elastic panel is a classical problem in structural acoustics and is typically analyzed using modal approaches based on in vacuo structural modes and the rigidly walled acoustic modes which are pre-determined based on separate component models. Such modeling techniques, however, tend to suffer the following drawbacks or limitations: (a) a panel is only subjected to ideal boundary conditions such as the simply supported, (b) the coupling between the cavity and panel is considered weak, and (c) the particle velocity cannot be correctly predicted from the pressure gradient on the contacting interface, to name a few. Motivated by removing these restrictions, this paper presents a general method for the vibro-acoustic analysis of a three-dimensional (3D) acoustic cavity bounded by a flexible panel with general elastically restrained boundary conditions. The displacement of the plate and the sound pressure in the cavity are constructed in the forms of standard two-dimensional and 3D Fourier cosine series supplemented by several terms introduced to ensure and accelerate the convergence of the series expansions. The unknown expansions coefficients are treated as the generalized coordinates and determined using the Rayleigh-Ritz procedure based on the energy expressions for the coupled structural acoustic system. The accuracy and effectiveness of the proposed method are demonstrated through numerical examples and comparisons with the results available in the literature.

A Fourier series method is proposed for the acoustic analysis of a rectangular cavity with impedance boundary conditions arbitrarily specified on any of the walls. The sound pressure is expressed as the combination of a three-dimensional Fourier cosine series and six supplementary two-dimensional expansions introduced to ensure (accelerate) the uniform and absolute convergence (rate) of the series representation in the cavity including the boundary surfaces. The expansion coefficients are determined using the Rayleigh-Ritz method. Since the pressure field is constructed adequately smooth throughout the entire solution domain, the Rayleigh-Ritz solution is mathematically equivalent to what is obtained from a strong formulation based on directly solving the governing equations and the boundary conditions. To unify the treatments of arbitrary nonuniform impedance boundary conditions, the impedance distribution function on each specified surface is invariantly expressed as a double Fourier series expansion so that all the relevant integrals can be calculated analytically. The modal parameters for the acoustic cavity can be simultaneously obtained from solving a standard matrix eigenvalue problem instead of iteratively solving a nonlinear transcendental equation as in the existing methods. Several numerical examples are presented to demonstrate the effectiveness and reliability of the current method for various impedance boundary conditions, including nonuniform impedance distributions.

In this paper an analytical method is proposed to investigate the vibration and power flows of periodically reinforced plate with general boundary conditions. Both the plate and stiffening beams are modeled as 3D structural components, and the couplings at the interfaces are specified in terms of 3D elastic joints. The displacement function for each stiffening beam is expressed as a modified Fourier cosine series, and the transverse and in-plane displacements for the plate are similarly expressed as the 2D versions of the modified Fourier cosine series expansions. The unknown Fourier coefficients are calculated using the Rayleigh-Ritz technique. The key advantages of the proposed method include: 1) it is capable of dealing with arbitrary boundary and coupling conditions, 2) it allows modeling any number of reinforcing beams with arbitrary lengths, and 3) the structural intensity, power flows, and kinetic energy distributions are readily calculated analytically from the displacement functions through appropriate mathematical (differential) operations, to name a few. The power flow characteristics of periodically reinforced plates are studied against various influencing factors, such as, plate and beam boundary conditions, coupling conditions, excitation locations, and dislocations resulting from minor misplacement of a reinforcing beam. vibrations, reinforced plates, periodic structures, analytical methods Citation:Xu H A, Li W L. Vibration and power flow analysis of periodically reinforced plates.