Assessment of the Dissipative Properties of Viscoelastic Hollow Cylindrical Bodies with Filler During the Propagation of Natural Waves

Searching by numerical simulation for structures with optimal damping properties among viscoelastic hollow cylindrical bodies with a filler is usually associated with a large amount of computation. Formulating the mechanical problem as one of natural vibrations and natural wave propagation makes it possible to evaluate the dissipative properties of such a structure independently of external force and kinematic actions, and thereby to reduce the computational cost substantially. The solution of the natural vibration problem for a piecewise homogeneous viscoelastic hollow cylindrical body with a filler yields complex natural frequencies, the real part of which represents the vibration frequency and the imaginary part the damping factor (attenuation rate). The mechanical behavior of the viscoelastic material is described by the linear Boltzmann–Volterra hereditary theory with a three-parameter Koltunov–Rzhanitsyn relaxation kernel, within which the material characteristics are represented by complex dynamic moduli—the shear modulus and the bulk modulus—that, as a rule, depend on frequency. In the natural vibration problem these moduli become functions of the real part of the sought complex natural frequency alone, which makes the standard eigenvalue procedures of commercial finite-element codes inapplicable. The paper presents an algorithm that removes this difficulty. The dispersion relation of the piecewise homogeneous cylinder is obtained analytically in the form of a complex determinant of order 12 for a two-layer and 18 for a three-layer configuration, the elements of which are Bessel and Neumann functions of complex argument; the global stiffness and mass matrices needed for the general configuration can be assembled automatically in a general-purpose finite-element code such as ABAQUS; the resulting complex characteristic equation is solved by Muller’s method—every iteration of which evaluates the determinant by Gaussian elimination with partial pivoting, so that no expansion of the determinant is required. The efficiency of the algorithm is demonstrated for a two-layer viscoelastic hollow cylindrical body with a filler, the outer load-carrying layer being made of Kh12 steel and the inner layer (the filler) of 30 L steel. The real and imaginary parts of the complex natural frequencies, of the phase velocities and of the attenuation are obtained as functions of the dimensionless wave number, of Poisson’s ratio, of the ratio of the layer radii and of the ratio of the instantaneous elastic moduli of the layers.