Classical methods of surface damping using free and constraining damping layers are discussed. The structure of a perspective integrated version of a damping coating is presented. This integral damping coating consists of two layers of a material with pronounced viscoelastic properties, between which there is a thin reinforcing layer of a high modulus material. A generalization of the Thompson-Kelvin-Voigt model is given for the description of viscoelastic properties of the material under tension-compression in the case of a complex stress state. A finite-element method was developed to determine the dynamic response of an elongated plate with the integral damping coating. This method is based on a four-layer finite element with 14 degrees of freedom: the main material is within the Kirchhoff-Love's model, the damping layers are in a flat stress state, the reinforcing layer perceives tension and compression. This model allows us to take into account the effect of transverse compression of the damping layers of the plate, which significantly increases its damping properties at high vibration frequencies. The stiffness matrices, the damping matrices, and the mass matrices of the constituent layers aim at obtaining similar complete matrices of a finite element. A system of resolving equations was obtained on the basis of the Lagrange equations of the second kind with respect to the vector of nodal displacements of the finite element model of the plate with an arbitrary dynamic load. In the case of a harmonic load with a frequency that coincides with one of the frequencies of free vibrations of the plate, a transition to a modal equation with respect to the normal coordinate corresponding to the given frequency is possible. Numerical experiments were carried out to test the developed finite element method using the example of a hingedly supported elongated plate with an integral damping coating. The numerical experiments showed a qualitative change in the composition of stresses in the damping layers of the plate at high vibration frequencies, which significantly affects its damping properties.
