Campbell diagrams of weakly anisotropic flexible rotors

Abstract

We consider an axi-symmetric rotor perturbed by dissipative, conservative, and non-conservative positional forces originated at the contact with the anisotropic stator. The Campbell diagram of the unperturbed system is a mesh-like structure in the frequency-speed plane with double eigenfrequencies at the nodes. The diagram is convenient for the analysis of the traveling waves in the rotating elastic continuum. Computing sensitivities of the doublets we find that at every particular node the untwisting of the mesh into the branches of complex eigenvalues in the first approximation is generically determined by only four 2x2 sub-blocks of the perturbing matrix. Selection of the unstable modes that cause self-excited vibrations in the subcritical speed range, is governed by the exceptional points at the corners of the singular eigenvalue surfaces--`double coffee filter' and `viaduct'--which are associated with the crossings of the unperturbed Campbell diagram with the definite Krein signature. The singularities connect the problems of wave propagation in the rotating continua with that of electromagnetic and acoustic wave propagation in non-rotating anisotropic chiral media. As mechanical examples a model of a rotating shaft with two degrees of freedom and a continuous model of a rotating circular string passing through the eyelet are studied in detail.