How to play a disc brake

Abstract

We consider a gyroscopic system under the action of small dissipative and non-conservative positional forces, which has its origin in the models of rotating bodies of revolution being in frictional contact. The spectrum of the unperturbed gyroscopic system forms a "spectral mesh" in the plane "frequency -gyroscopic parameter" with double semi-simple purely imaginary eigenvalues at zero value of the gyroscopic parameter. It is shown that dissipative forces lead to the splitting of the semi-simple eigenvalue with the creation of the so-called "bubble of instability" - a ring in the three-dimensional space of the gyroscopic parameter and real and imaginary parts of eigenvalues, which corresponds to complex eigenvalues. In case of full dissipation with a positive-definite damping matrix the eigenvalues of the ring have negative real parts making the bubble a latent source of instability because it can "emerge" to the region of eigenvalues with positive real parts due to action of both indefinite damping and non-conservative positional forces. In the paper, the instability mechanism is analytically described with the use of the perturbation theory of multiple eigenvalues. As an example stability of a rotating circular string constrained by a stationary load system is studied in detail. The theory developed seems to give a first clear explanation of the mechanism of self-excited vibrations in the rotating structures in frictional contact, that is responsible for such well-known phenomena of acoustics of friction as the squealing disc brake and the singing wine glass.Comment: 25 pages, 9 figures, Presented at BIRS 07w5068 Workshop "Geometric Mechanics: Continuous and discrete, finite and infinite dimensional", August 12-17, 2007, Banff, Canad