Proceedings of DETC’99 1999 ASME Design Engineering Technical Conferences September 12-15, 1999, Las Vegas, Nevada, USA DETC99/VIB-8157 SIMPLIFIED DISPERSION RELATIONS FOR FLOQUETWAVES IN A PLATEWITH MULTIPLE ARRAYS OF ATTACHMENTS Dimitar Gueorguiev J. Gregory McDaniel ∗ Pierre DuPont Department of Aerospace and Mechanical Engineering Boston University Boston, Massachusetts, 02215 ABSTRACT Recently, the authors developed analytical expressions for the dispersions of Floquet waves that propagate in a structure con- sisting of a plate with multiple arrays of line attachments. The dispersions of these Floquet waves, and in particular the imagi- nary parts of their wavenumbers, quantify the attenuation of vi- brational energy in space as the frequency of a local excitation is varied. Understanding how the parameters of the attached struc- tures, such as their spacings and impedances, affect the Floquet wave dispersions could provide further means to include consider- ation of energy localization or distribution in the structural design process. Such an understanding is developed in the present work by identifying those cases in which the treatment of certain arrays can be greatly simplified. In particular, limiting cases of small and large array spacings are investigated for which the treatment of particular arrays can be greatly simplified. Such simplifica- tions are not immediately obvious without access to analytical expressions for the Floquet wavenumbers, as the dynamics of all arrays are coupled through the plate. Results presented here will aid the structural design community by indicating which design changes most effectively control energy distribution and by in- dicating when simplified finite element models of multiple-array structures are possible. NOMENCLATURE d r Distance between adjacent substructures of the r th array. * Address all correspondence to this author. E Young’s modulus of the plate material. F r Force applied by the plate to the r th array. k Transform wavenumber. k d,r Spacing wavenumber of the r th array (equation 8). k f Flexural wavelength of the bare plate. m Mass per unit area of the bare plate. Q Dispersion function (equation 12). ˜ v Wavenumber transform of plate velocity. ˜ Y Bare plate admittance in the wavenumber domain. ˜ Y r Plate admittance in the wavenumber domain with the first (r − 1) arrays attached (equation 14). Z r Line impedance looking into the r th array. Z (r) Distributed impedance looking into the r th array (equation 3). ν Poisson’s ratio of the plate material. Ω Normalized frequency (equation 43). Ω bound Bounding frequency that indicates when an array may be homogenized to engineering accuracy. ζ r Normalized impedance (equation 27). 1 INTRODUCTION A variety of engineering structures consist of a homogeneous elastic master structure, such as a plate or shell, that is attached to regularly spaced substructures, such as ribs, stringers, or fins. Due to other design criteria, such structures sometimes have two or more arrays of attachments, where each array consists of regu- larly spaced substructures that present identical impedances to the 1 Copyright  1999 by ASME