In this paper we study the oscillations of a circular viscoelastic plate of variable stiffness with different support conditions at the boundary, including in the presence of viscoelastic bonds (in the elastic case the problem was investigated by the authors earlier). In the case of steady-state vibrations, viscoelastic bonds in the boundary conditions are characterized by two complex coefficients that depend on the frequency of the oscillations. Three auxiliary problems that do not contain these coefficients were numerically formulated on the base of Lagrange’s variational principle. The problems were solved numerically using the Ritz method. The influence of the number of coordinate functions on the accuracy of the constructed solution and its dependence on the frequency of oscillations was investigated. The solution is sought in the form of a linear combination of the three constructed solutions. Satisfaction with the boundary conditions makes it possible to establish the fractional-rational structure of the solution depending on the coefficients in the boundary conditions. The problem of reconstructing the parameters of viscoelastic bonds based on the known (measured) deflection in a set of points at a fixed frequency was also solved. A system of nonlinear algebraic equations is constructed to find the required coupling coefficients, each of which defines a conditional hyperbolic dependence in the complex space. A method for selecting a single solution is presented. Computational experiments for various segments of coefficients change (weak support, rigid support) are presented. The ranges of the most successful reconstruction are founded. The effect of neglecting the viscoelasticity of the support on the reconstruction of the load amplitude was estimated.