Energy Dissipation during Vibrations of Nonuniform Composite Structures: 1. Formulation of the Problem

Abstract

In this paper, a mathematical model is suggested for decaying vibrations of laminated plates formed by a finite number of arbitrarily oriented orthotropic viscoelastic layers of polymer composites arranged into an anisotropic structure with a layer of stiff isotropic viscoelastic material applied on one of its outer surfaces. The model is based on the Hamilton variation principle, the refined Mindlin–Reissner theory for plates, and the elastic–viscoelastic correspondence principle of the linear viscoelasticity theory. In the description of the physical relationships between the materials of the layers forming structural orthotropic polymeric composites, the influence of vibration frequency and ambient temperature is assumed to be negligible, whereas for the stiff viscoelastic polymeric layer, the dependence of elastic dissipation and stiffness properties on temperature and frequency is considered by means of experimentally determined generalized curves. Equations of motion are obtained for the Timoshenko beam with a layer of stiff isotropic viscoelastic polymer on one of its outer surfaces as a specific case of the general problem by neglecting mid-surface strain in the direction of one of the plate axes. Minimization of the Hamiltonian makes it possible to describe the decaying vibrations of anisotropic structures with an algebraic problem of complex eigenvalues. A system of algebraic equations is formed by the Ritz method using the Legendre polynomials as coordinate functions. First, the real solutions are obtained. To derive complex natural frequencies of the system, the obtained real natural frequencies are taken as the input values, and the complex natural frequencies are calculated applying the third-order iteration method.