Oleg N. Kirillov

Paradoxes of dissipation-induced destabilization or who opened Whitney's umbrella?

The paradox of destabilization of a conservative or non-conservative system by small dissipation, or Ziegler’s paradox (1952), has stimulated an ever growing interest in the sensitivity of reversible and Hamiltonian systems with respect to dissipative perturbations. Since the last decade it has been widely accepted that dissipation-induced instabilities are closely related to singularities arising on the stability boundary. What is less known is that the first complete explanation of Ziegler’s paradox by means of the Whitney umbrella singularity dates back to 1956. We revisit this undeservedly forgotten pioneering result by Oene Bottema that outstripped later findings for about half a century. We discuss subsequent developments of the perturbation analysis of dissipation-induced instabilities and applications over this period, involving structural stability of matrices, Krein collision, Hamilton-Hopf bifurcation, and related bifurcations. c

Coupling of eigenvalues of complex matrices at diabolic and exceptional points

The paper presents a general theory of coupling of eigenvalues of complex matrices of an arbitrary dimension depending on real parameters. The cases of weak and strong coupling are distinguished and their geometric interpretation in two and three-dimensional spaces is given. General asymptotic formulae for eigenvalue surfaces near diabolic and exceptional points are presented demonstrating crossing and avoided crossing scenarios. Two physical examples illustrate effectiveness and accuracy of the presented theory.

Nonconservative Stability Problems of Modern Physics

This work gives a complete overview on the subject of nonconservative stability from the modern point of view. Relevant mathematical concepts are presented, as well as rigorous stability results and numerous classical and contemporary examples from mechanics and physics. It deals with both finite- and infinite-dimensional nonconservative systems and covers the fundamentals of the theory, including such topics as Lyapunov stability and linear stability analysis, Hamiltonian and gyroscopic systems, reversible and circulatory systems, influence of structure of forces on stability, and dissipation-induced instabilities, as well as concrete physical problems, including perturbative techniques for nonself-adjoint boundary eigenvalue problems, theory of the destabilization paradox due to small damping in continuous circulatory systems, Krein-space related perturbation theory for the MHD kinematic mean field ?²-dynamo, analysis of Campbell diagrams and friction-induced flutter in gyroscopic continua, non-Hermitian perturbation of Hermitian matrices with applications to optics, and magnetorotational instability and the Velikhov-Chandrasekhar paradox. The book serves present and prospective specialists providing the current state of knowledge in the actively developing field of nonconservative stability theory. Its understanding is vital for many areas of technology, ranging from such traditional ones as rotor dynamics, aeroelasticity and structural mechanics to modern problems of hydro- and magnetohydrodynamics and celestial mechanics.

Exceptional Points in a Microwave Billiard with Time-Reversal Invariance Violation

We report on the experimental study of an exceptional point (EP) in a dissipative microwave billiard with induced time-reversal invariance (T) violation. The associated two-state Hamiltonian is non-Hermitian and nonsymmetric. It is determined experimentally on a narrow grid in a parameter plane around the EP. At the EP the size of T violation is given by the relative phase of the eigenvector components. The eigenvectors are adiabatically transported around the EP, whereupon they gather geometric phases and in addition geometric amplitudes different from unity.

Gyroscopic Stabilization in the Presence of Nonconservative Forces

This paper analyzes the stability of a linear autonomous nonconservative system with an even number of degrees of freedom in the presence of potential, gyroscopic, dissipative, and nonconservative positional forces. It is well known that, when applied separately, dissipative and nonconservative positional forces destroy gyroscopic stabilization [1, 3]. However, their combination can make a system asymptotically stable. It is found that the complexity of the choice of such a combination is associated with a Whitney umbrella singularity existing on the boundary of the gyroscopic stabilization domain of the nonconservative system. In this paper, an approximation to the boundary of the asymptotic stability domain near the singularity is explicitly found and an analytical estimate of the critical gyroscopic parameter is obtained. As an example, we analyze the stability of Hauger’s gyropendulum under the action of a follower torque. 1. Consider an autonomous nonconservative system of the form

Modeling and stability analysis of an axially moving beam with frictional contact

This paper considers a moving beam in frictional contact with pads, making the system susceptible for self-excited vibrations. The equations of motion are derived and a stability analysis is performed using perturbation techniques yielding analytical approximations to the stability boundaries. Special attention is given to the interaction of the beam and the rod equations. The mechanism yielding self-excited vibrations does not only occur in moving beams, but also in other moving continua such as rotating plates, for example.

Unfolding of eigenvalue surfaces near a diabolic point due to a complex perturbation

The paper presents a new theory of unfolding of eigenvalue surfaces of real symmetric and Hermitian matrices due to an arbitrary complex perturbation near a diabolic point. General asymptotic formulae describing deformations of a conical surface for different kinds of perturbing matrices are derived. As a physical application, singularities of the surfaces of refractive indices in crystal optics are studied.

Geometric phase around exceptional points

A wave function picks up, in addition to the dynamic phase, the geometric (Berry) phase when traversing adiabatically a closed cycle in parameter space. We develop a general multidimensional theory of the geometric phase for (double) cycles around exceptional degeneracies in non-Hermitian Hamiltonians. We show that the geometric phase is exactly {pi} for symmetric complex Hamiltonians of arbitrary dimension and for nonsymmetric non-Hermitian Hamiltonians of dimension 2. For nonsymmetric non-Hermitian Hamiltonians of higher dimension, the geometric phase tends to {pi} for small cycles and changes as the cycle size and shape are varied. We find explicitly the leading asymptotic term of this dependence, and describe it in terms of interaction of different energy levels.

In- and Out-of-plane Vibrations of a Rotating Plate with Frictional Contact: Investigations on Squeal Phenomena

Rotating plates are used as a main component in various applications. Their vibrations are mainly unwanted and interfere with the functioning of the complete system. The present paper investigates the coupling of disk (in-plane) and plate (out-of-plane) vibrations of a rotating annular Kirchhoff plate in the presence of a distributed frictional loading on its surface. The boundary value problem is derived from the basics of the theory of elasticity using Kirchhoff’s assumptions. This results in precise information about the coupling between the disk and the plate vibrations under the action of frictional forces. At the same time we obtain a new model, which is efficient for analytical treatment. Approximations to the stability boundaries of the system are calculated using a perturbation approach. In the last part of the paper nonlinearities are introduced leading to limit cycles due to self-excited vibrations.

Subcritical flutter in the acoustics of friction

We consider a gyroscopic system with two degrees of freedom 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. Explicit conditions are established for the origination of the bubble of instability and its transition from the latent to active phase, clarifying the key role of indefinite damping and non-conservative positional forces in the development and localization of the subcritical flutter instability. 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 selfexcited 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.Linearized models of elastic bodies of revolution, spinning about their symmetrical axes, possess the eigenfrequency plots with respect to the rotational speed, which form a mesh with double semi-simple eigenfrequencies at the nodes. At contact with friction pads, the rotating continua, such as the singing wine glass or the squealing disc brake, start to vibrate owing to the subcritical flutter instability. In this paper, a sensitivity analysis of the spectral mesh is developed for the explicit predicting the onset of instability. The key role of the indefinite damping and non-conservative positional forces is clarified in the development and localization of the subcritical flutter. An analysis of a non-self-adjoint boundary-eigenvalue problem for a rotating circular string, constrained by a stationary load system, shows that the instability scenarios, revealed in the general two-dimensional case, are typical also in more complicated finite-dimensional and distributed systems.