Natural vibrations of a viscoelastic three-layer cylindrical body

The natural vibrations of a viscoelastic coaxial cylindrical body are considered; the space between the shells is filled with a viscoelastic material. The relationship between stress and strain satisfies the Boltzmann–Voltaire hereditary integral. As an example of a viscoelastic material, a three-parametric relaxation kernel with a weak Rzhanitsyn–Koltunov singularity is used. The problems of small vibrations of the mechanical system under consideration are solved. Equations of small vibrations of the aggregate in displacements are obtained on the basis of Lame's differential equations of the theory of viscoelasticity with complex coefficients. The equations of vibration of the outer and inner shells, which are made of viscoelastic material, satisfy the equations of motion of the shell, subject to the Kirchhoff–Love hypotheses. The problem is solved using the Green–Lamb transformation and the complex amplitude method. The stresses and displacements of each shell and filler are expressed through special functions of the Bessel and Neumann complex argument of an arbitrary order. A frequency equation with a complex parameter is obtained, which is solved numerically using the Muller method. For structurally inhomogeneous mechanical systems, the dependences of several modes of the complex natural frequency (real and imaginary parts) on various parameters of three-layer bodies are comparatively assessed. The application of asymptotic and numerical methods for solving frequency equations with a complex-output parameter is also comparatively assessed.