A studyof vibrations of a cylinder with a viscoelastic coating

The paper presents the problem of steady-state longitudinal-radial vibrations of an elastic hollow cylinder with a viscoelastic coating. The properties of the coating change along the radial coordinate only and are described by the variable Lamé parameters and density. The ends of the cylinder are under sliding interface conditions, and the outer side surface is subject to periodic loads. Within the framework of the model of a standard viscoelastic body and following the principle of correspondence, the variable Lamé parameters are replaced by the complex functions of the radial coordinate and vibration frequency. The solution was obtained using two approaches. Within the framework of the first approach, the solution is constructed using the method of separation of variables and is reduced to solving a set of boundary value problems for canonical systems of first-order differential equations with variable coefficients. The solution to each of these problems is obtained numerically using the shooting method. The second approach is based on the finite element method implemented in the FlexPDE package. We compared the obtained solutions for the given laws of variation of the Lamé parameters and a fixed frequency by analyzing the graphs of the real and imaginary parts of radial displacement and stress components. The convergence of the solution found using the finite element method is shown as a function of the number of nodes for the values of the radial displacement functions measured at three points. The graphs of the amplitude-frequency characteristics measured on the outer surface are plotted for different values of the relaxation time. The effect of variable properties of the coating on the displacement function is estimated. The advantages of each approach are described, and the areas of their practical application are revealed.