This paper presents an approximate mathematical model for the analysis of non-stationary transverse vibrations of a three-layer viscoelastic plate composed of two load-bearing face layers and a deformable core. The governing equations are derived within the framework of the three-dimensional linear elasticity theory under plane deformation assumptions and reduced to a form suitable for engineering calculations. A frequency equation for harmonic vibrations is obtained and solved numerically using Maple 17. The analysis is performed for plates with steel and aluminum face layers combined with different core materials, including polymer, fiberglass, wood plastic, and textolite, for several core thicknesses. The numerical results are presented as frequency-wave number relationships and used to evaluate the influence of geometric and physical-mechanical parameters on the dynamic response of the plate. It is shown that the lowest vibration frequency increases with increasing wave number and core thickness. For identical geometric parameters, plates with aluminum face layers exhibit slightly higher frequencies than plates with steel face layers, whereas the core material significantly affects the frequency level due to the combined influence of stiffness and density. The proposed model can be used for rapid frequency assessment of layered structural elements subjected to transverse dynamic loading. Viscoelastic dissipation in the layers is introduced through the elastic-viscoelastic correspondence principle, and the formal range of applicability of the model is identified in terms of the dimensionless wave number k h 0 and the face-to-core thickness ratio. The proposed model is verified against a Semi-Analytical Finite Element (SAFE) plane-strain 3-D elasticity benchmark, against which it agrees substantially better than the classical Kirchhoff-Love thin-plate theory within the entire claimed applicability range.
Yakhshiboev et al. (Thu,) studied this question.