Composite constructions of multilayer plates and shells are widely used in mechanical engineering and aerospace engineering. This work presents the mathematical formulation of the problem, develops methods for its solution, and provides numerical results for a new problem — the stationary stress state of infinitely long cylindrical shells on an elastic (acoustic) foundation under the action of a non-axisymmetric normal pressure wave propagating along the shell axis at a sub-resonant velocity. The shell motion equations are constructed based on the Kirchhoff–Love hypotheses. The solution methods are based on the application of the Fourier integral transform (or the fundamental solutions method) with respect to the axial coordinate, as well as the decomposition of all given and sought quantities into Fourier series with respect to the angular coordinate. The study investigates the first (lower) modes of motion and the corresponding damping coefficients. It has been established that the increase in the liquid density has a particularly significant effect on the system’s behavior at relatively low values.

Objective. The objective of this study is to determine the non-axisymmetric stationary stress state of viscoelastic cylindrical shells with liquid and to derive the dispersion equations of the shell, based on the Kirchhoff–Love hypotheses with ideal compressible liquid.

Methods. In this study, the derived integral-differential equations are solved using the freezing methods, Fourier integral transform, and the Green-Lembe and Muller transformation methods.

Results. Solutions are based on the joint application of Fourier integral transform (or the fundamental solutions method) with respect to the axial coordinate and decomposition of all given and sought quantities into Fourier series with respect to the angular coordinate. An efficient algorithm for the joint computation of integrals and Fourier series has been developed and implemented on a computer. The first (lower) mode of motion and the corresponding damping coefficient are investigated.

Conclusions. The mathematical formulation and methods for solving the non-axisymmetric problem of stationary stress states in viscoelastic cylindrical shells with liquid are presented. It has been found that an increase in the liquid density has a particularly significant effect on the system at relatively low liquid densities.