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Wave propagation in rotating continua under non-conservative perturbations: resonant deformation of the spectral mesh and combination resonance.

Subject Area Mechanics
Term from 2007 to 2011
Project identifier Deutsche Forschungsgemeinschaft (DFG) - Project number 46629384
 
Final Report Year 2011

Final Report Abstract

In 1638 Galileo Galilei remarked that “a glass of water may be made to emit a tone merely by the friction of the finger-tip upon the rim of the glass”. In the end of the XIXth century, shortly after Rayleigh qualitatively described the onset of bending waves in the singing wine glass by the friction, applied in the circumferential direction, and pointed out the proximity of the main audible frequency of the glass to the one of the spectrum of its free vibrations, Sperry and Lanchester invented disc and drum brakes. The singing wine glass and the squealing automotive brake are phenomena of the acoustics of friction, which everyone encounters almost every day. Despite of a seeming simplicity, their mathematical modeling is not easy, and experiments with them are not satisfactorily reproducible. Several fundamental reasons are responsible for this. First, the rotational symmetry of the continua makes the spectral mesh with the double eigenvalues (1:1 semi-simple resonance) at the nodes generic for the unloaded bodies of revolution. Then, the definite Krein signature of the eigenvalues in the subcritical rotation speed range makes destabilization by simply varying mass and stiffness impossible. Decades ago, practitioners empirically realized that dissipative and non-conservative perturbations are required to produce complex eigenvalues with the positive real parts, which trigger self-excited vibrations in the subcritical speed range of the rotating continua in frictional contact. Nevertheless, the variety of the proposed mechanical models of rotors and their frictional contact did not reveal the universal mechanism of the subcritical flutter instability. In different models one can observe seemingly different behavior of eigenvalues that become unstable in almost unpredictable manner. The main result of our work is the discovery of a unique mathematical object for a sufficiently general class of weakly anisotropic systems that governs the selection of the unstable modes. There exist two surfaces that represent imaginary and real parts of the eigenvalues of the rotor system as functions of the rotational speed and the parameter of a Hamiltonian perturbation such as mass or stiffness modification. In the presence of the dissipation and non-conservative forces that are usually produced by the frictional contact these surfaces become singular at a number of points connected by the branch cuts where the surfaces possess self-intersections. We have found that generically there are only two types of such singularities. One of them has a form of the two-glued coffee filters and another one resembles a viaduct. Both surfaces are well known in the modern optics and acoustics and are related to the electromagnetic and acoustical wave propagation in stationary anisotropic media such as the birefringent chiral crystals. The fundamental consequence for the stability of the weakly anisotropic rotors is the fact that the type of the surfaces persists due to change of the structure of the matrices of the dissipative and nonconservative positional forces. This means that the behavior of eigenvalues is described similarly for many seemingly different models of rotor-stator interaction. The existence of these singular surfaces is a robust and general qualitative phenomenon. Nevertheless, the orientation of the singularities, their dimensions, inclinations and other geometric parameters are sensitive to the variation of the perturbing matrices and thus to the details of the contact between the rotor and stator. Moreover, these changes in the geometry of the singular surfaces can lead to the catastrophic bifurcations of the domain of asymptotic stability. This is the main reason for the poor reproducibility of the experiments with automotive brake squeal: from one run of the experiment to another the tribological properties of the rotor and stator uncontrollably change, causing bifurcation of the stability domain. As a result, the same values of other well-tunable parameters may enter the domain of instability instead of stabilization and vice versa. We have developed an efficient perturbation approach that allows one to approximate the singular surfaces using information about the perfect isotropic rotor. Our results explicitly relate the elements of the perturbing matrices and geometrical properties of the eigenvalue surfaces. This can be employed both for the structural optimization of the elements of a brake and for the interpretation of the experimental and numerical data. Our studies of MHD dynamo and helical magnetorotational instability show that making the eigenvalues with the definite Krein signature unstable in the systems described by the linear operators which are selfadjoint in a Krein space by the perturbations that destroy this property is a topic that finds many applications not only in the dynamics of rotors but also in hydro- and magnetohydrodynamics, plasma physics and other modern physical disciplines.

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