Coupled vibrations of viscoelastic three-layer composite plates. 1. Formulation of problem
A mathematical model of damped oscillations of three-layer plates formed by two rigid anisotropic layers and a soft middle isotropic layer of a viscoelastic polymer is proposed. Each hard layer is an anisotropic structure formed by a ﬁnite number of randomly oriented orthotropic viscoelastic composites layers. The model is based on the use of the Hamiltonian variational principle, the reﬁned theory of first-order plates (Reissner-Mindlin theory), the model of complex modules and the principle of elastic-viscoelastic correspondence in the linear theory of viscoelasticity. When describing the physical relationships of hard layer materials, the inﬂuence of the vibration frequency and the ambient temperature is considered negligible, while for the soft layer of a viscoelastic polymer, the temperature- frequency dependence of the elastic-dissipative characteristics is taken into account based on experimentally determined generalized curves. As a special case of the general problem, by neglecting the deformation of the middle surfaces of the rigid layers in one of the directions of the axes of the rigid layers of a three-layer plate, the equations of longitudinal and transverse damped oscillations of a globally orthotropic three-layer beam are obtained. Minimization of the Hamilton functional allows us to reduce the problem of damped vibrations of anisotropic structures to the algebraic problem of complex eigenvalues.