Untwisting the Campbell diagrams of weakly anisotropic rotor systems

Abstract

A brake can be modeled as an axi-symmetric rotor perturbed by dissipative, conservative, and non-conservative positional forces originated at the frictional 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 is generically determined by only four 2×2 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 surface–`double coffee-filter'–which is typical also in the problems of electromagnetic and acoustic wave propagation in non-rotating anisotropic chiral media. As a mechanical example a model of a rotating shaft is studied in detail.