Finite-Element Harmonic Analysis of Sandwich Plates with Periodically Modulated Faces and a Viscoelastic Core

Constrained-layer damping in sandwich floors and wall panels is usually analysed with uniform faces and a uniform core, while phononic treatments of periodicity address stop bands rather than the first few forced resonances of a finite plate. A seven-degree-of-freedom rectangular finite element is formulated here for a three-layer sandwich plate whose face thicknesses and core shear moduli vary in the plane as separable cosines. The kinematics follow Mead–Markus: Kirchhoff faces, first-order shear in a soft Kelvin–Voigt core, and a single transverse deflection. Membrane interpolation is bilinear; bending uses the Adini–Clough–Melosh incomplete quartic. A Navier–Galerkin sine series on two single-mass plates verifies the element to within one frequency step. On the modulated plate the Galerkin first peak is $f_1=8.93\mathrm{Hz}$, and the finite-element first peak is $f_1=9.57\mathrm{Hz}$; on the uniform plate, both methods give $f_1=7.66\mathrm{Hz}$. These three frequencies belong to the materials and planform of those two plates. Three archives then isolate core periodicity, face periodicity, and the layout of a fixed total attached mass. The first resonance of the core-campaign reference stays near 12$\mathrm{Hz}$; its height follows the area-averaged storage modulus. The two mean face thicknesses, $c_{1\mathrm{m}}$ and $c_{3\mathrm{m}}$, dominate the higher peaks; at fixed sum an unequal split is stiffer than two equal faces. Relocating the attached masses onto midlines or a diagonal rearranges the centre-point frequency-response function more than reweighting the $3\times 3$ lattice. The spectra contain no phononic stop band.