Transversal vibrations of thick and thin cylindrical shells, interacting with deformable medium

On the assumption of the exact solutions in transformations of three-dimensional problems of the viscoelasticity theory, equations of the transversal vibrations of the circular cylindrical viscoelastic shell situated in the viscoelastic medium are developed. The algorithm defining the stress-strain state of the arbitrary section of the shell for any moments of time is developed. Because of absence of environment, two limiting cases are analyzed: the equations for thin-walled cylindrical shell and for the circular cylindrical beam. The obtained results are compared with known results following from the Kirchhoff-Love theory and the Timoshenko theory. The reaction of the environmental medium to transversal vibrations of the cylindrical shell proves that the beam in represented by a three-dimensional matrix, some elements of which are expressed through integrals such as convolution.