ANALYSIS OF FLOQUET WAVE GENERATION AND PROPAGATION IN A PLATE WITH MULTIPLE ARRAYS OF LINE ATTACHMENTS

A variety of engineering structures consist of a homogeneous “master structure”, such as a plate or shell, to which are attached multiple arrays of substructures, such as ribs or stringers. The goal of this work is to understand the effects of array impedance and spacing on energy flow in the master structure. Such an understanding may ultimately lead to a design process encompassing both the attenuation of vibrational energy and the support of static loads. Here, the vibrational response of a locally excited master structure is studied analytically for a class of structures involving an arbitrary number of substructure arrays. The approach is presented in the context of an elastic plate which is reinforced by multiple arrays of line attachments and acted upon by a line force. Extension of the analysis to other geometries and loadings is straightforward. A recursive analytical procedure is presented by which Floquet wavenumbers of a structure with p arrays are computed from the wavenumbers of a structure with p−1 of the arrays attached. In this way, the Floquet wavenumbers of any multiple array structure can be computed by first considering the master structure alone and then computing the effect of attaching each array in turn. The imaginary parts of the Floquet wavenumbers quantify the attenuation of response along the structure. In addition, the spatial response is obtained analytically as a sum of two Floquet waves through simplification and transformation of the wavenumber domain solution. By way of example, a three-array structure is considered to illustrate the recursive computation of the wavenumbers and to demonstrate the correlation between the imaginary parts of the wavenumbers and the spatial attenuation of the structural response.