How Does Krein Signature Determine Veering and Crossing of Eigencurves of Non-Conservative Rotating Continua?

Abstract

Frequency loci crossing and veering phenomena are closely related to wave propagation and instabilities in fluids and structures. In engineering applications the crossings of the eigencurves are typically observed in gyroscopic or potential systems in the presence of symmetries, such as rotational or spherical one. The examples are perfect solids of revolution that serve for modeling turbine wheels, disk and drum brakes, tires, clutches, paper calenders and other rotating machinery. We consider an axi‐symmetric flexible 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. Computing sensitivities of the doublets we find that at every particular node the unfolding of the mesh into the branches of complex eigenvalues in the first approximation 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 surfaces—‘double coffee filter’ and ‘viaduct’—which are sharply associated with the crossings of the unperturbed Campbell diagram with the definite symplectic (Krein) signature. 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.