Natural small oscillations of a flat viscoelastic spiral spring

Curved pipe systems are widely used in mechanical engineering, the nuclear industry, offshore oil production, and aerospace engineering. The purpose of the work is to study small vibrations of a viscoelastic helical spring. Small vibrations of a thin curved rod, the elastic line of which is a flat curve and one of the main directions of the cross-section of which lies in the plane of the curve, break down into two types: vibrations with displacements in the plane of the curve and with displacements perpendicular to the plane of the curve. The viscoelastic properties of materials are taken into account using complex elastic moduli. Asymptotic expansions are constructed for the eigenfunctions and eigenfrequencies corresponding to both types of oscillations of a repeatedly twisted flat spiral spring with fixed ends. A technique has been developed for obtaining resolving equations corresponding to the boundary conditions.