It is known that the theory of linear and nonlinear elastic plates and shells is the most developed part of the theory of elasticity. In this area, the necessary equations are obtained and the methods to solve them are developed. At the same time, there are gaps in considering the viscoelastic properties of a material in the problems of thin-walled structures dynamic calculations. It should be noted that in some publications the viscoelastic properties of the material (i.e. the deviation of material test diagram from Hooke's law) were taken into account according to the Voigt model, not confirmed by experiments. Ignoring viscoelastic properties of the material significantly limits practical applicability of results. The first part of the paper presents the statement and method of solution to the problem of axisymmetric vibrations of a physically nonlinear viscoelastic cylindrical shell with concentrated masses. The function characterizing the deviation of stress intensity curve from the Hooke straight line is taken in the form of cubic nonlinearity. A mathematical model, solution method and computational algorithm were developed for the problem of axisymmetric oscillations of a cylindrical shell with a concentrated mass, taking into account physically nonlinear strain of the material under different boundary conditions in the frame of the Kirchhoff-Love hypothesis. For the study of the effect of a concentrated mass the Dirac delta function was introduced. With the Bubnov-Galerkin method, based on a polynomial approximation of deflections, the problem in question is reduced to the solution, in the general case, of non-decay systems of nonlinear integro-differential equations of Volterra type. To solve the resulting system with the Koltunov-Rzhanitsyn weakly singular kernel, a numerical method was applied based on the use of quadrature formulas. A unified computational algorithm has been developed to determine the deflection of a viscoelastic cylindrical shell with concentrated masses.