Self-oscillations of a rotating elastic rod with damper

We consider the self-oscillations of an elastic rod located in the field of centrifugal forces of inertia and based on a viscoelastic damper. This mathematical model, with the accuracy sufficient for engineers, describes the dynamic processes in the rotating turbine blades, in the working part of the wire brush, and in other similar mechanisms. The formulation of the eigenvalue problem is based on the variational principle and is put in a complex form. This approach makes it possible to estimate the damping ability of the rod through the imaginary part of the eigen frequency (damping coefficient), as well as to easily complicate and vary the design parameters. For example, a rod can be considered with variable cross-section or variable density along its length. In the article the validity of the method results is proved by comparing them with the data available in the literature. The following should be considered as the main result: for structurally inhomogeneous structures (i.e. structures consisting of elastic and viscoelastic elements), in case of a constant damper rheology, it is possible to increase the intensity of vibration damping due to rational selection of their geometric or elastic parameters. Meanwhile the maximum of the absorbed energy, both in the first and in the second case, is determined jointly by the damping coefficients of the two lower forms of oscillations. According to the minmax principle, the damping coefficients of the 1st and 2nd forms of oscillations act alternately as the global damping coefficient. At the extreme point there is a maximum interaction of 2 lower forms of oscillations, as a result of which this synergetic effect is observed. It is obvious that in the case of forced oscillations, the selected parameters of the mechanical system will provide minimal resonant amplitudes.