The Fourier method is used to find the analytical solutions to two-dimensional quasistatic problems of stationary polyharmonic vibrations and dissipative heating of a linearly viscoelastic cylinder. The influence of the cylinder thickness and the width of the loading area on the thermomechanical state of the cylinder is studied based on numerical data Keywords: polyharmonic vibrations, viscoelastic cylinder, dissipative heating Vibrational heating is an important factor to take into account in the performance evaluation of polymeric structural members [3][4][5]. The dissipative temperature field of a viscoelastic cylinder with a normal self-balanced load moving over its surface was studied in [1, 2]. The present paper deals with a more general case where the load includes both normal and tangential components and may be either self-balanced or nonself-balanced. We will study the influence of the cylinder thickness and width of the loading area (at a constant force resultant) on the temperature field of the cylinder. Polyharmonic Deformation and Dissipative Heating of a Viscoelastic Cylinder: Stationary Quasistatic Problem.Let us consider, in a cylindrical coordinate system r, ϕ, z, an infinite viscoelastic hollow cylinder (r 2 ≤ r ≤ r 2 , 0 ≤ ϕ < 2π, -∞ < z < + ∞) whose inside boundary is rigidly connected to a perfectly rigid axis and outside boundary is subject to a moving load. The motion of the load is described by the circumferential coordinate ϕ = ϕ + ωt.(1.1)If the load depends only on the coordinate ϕ (and does not depend on z), we have a two-dimensional problem of elasticity (plane strain state). Therefore, we will use the polar coordinate system r, ϕ to examine the mechanical behavior of the hollow cylinder (r 1 ≤ r ≤ r 2 , 0 ≤ ϕ < 2π). Assume that the load can be represented by a Fourier series in ϕ:

An analytic solution is obtained to describe the vibrations and dissipative heating of a simply supported infinite cylindrical panel under periodic normal loads moving over its surface with a constant velocity. Special attention is focused on resonant vibrations, which result in the most intensive dissipative heating. It is additionally assumed that the material of the panel is viscoelastic, its properties are independent of temperature, and Poisson's ratio is real. The influence of thickness, radius of curvature, load velocity, and viscoelastic properties on the thermal state of the panel is analysed against the thermal state of the plate Keywords: vibrations, dissipative heating, moving load, cylindrical panelThe dissipative heating of viscoelastic thin-walled elements under harmonic deformation was studied in [1,2,5,7,8]. Dissipative heating caused by moving surface loads was addressed in [3,4]. The dynamic vibrations and dissipative heating of thin-walled elements under polyharmonic deformation caused by moving surface loads were discussed in [3,4,6], where an analytical solution was obtained and numerical results, analyzed for a plate [3] and for a rod [4,6].In the present paper, we will obtain an analytical solution describing the vibrations and dissipative heating of a cylindrical panel under a periodic system of loads moving over its surface with a constant velocity and examine the influence of thickness, radius, load velocity, and viscoelastic properties on the thermal state of the panel.

A dissipative-heating problem for an incompressible viscoelastic cylinder rolling over a rigid foundation is solved. The effect of tangential stresses on the dissipative-heating temperature is examined