Wolfhard Kliem

Bio: Wolfhard Kliem is an academic researcher from Technical University of Denmark. The author has contributed to research in topics: Linear system & Eigenvalues and eigenvectors. The author has an hindex of 8, co-authored 25 publications receiving 172 citations.

On gyroscopic stabilization

Abstract: The mechanisms of transition between divergence, flutter, and stability for a class of conservative gyroscopic systems with parameters are studied. Two results are obtained which state sufficient conditions for gyroscopic stabilization of conservative systems with an even dimension and a negative definite stiffness matrix. A number of examples are given to demonstrate the feasibility of the results.

Indefinite damping in mechanical systems and gyroscopic stabilization

Abstract: This paper deals with gyroscopic stabilization of the unstable system Mẍ + Dẋ + Kx = 0, with positive definite mass and stiffness matrices M and K, respectively, and an indefinite damping matrix D. The main question is for which skew-symmetric matrices G the system Mẍ + (D + G)ẋ + Kx = 0 can become stable? After investigating special cases we find an appropriate solution of the Lyapunov matrix equation for the general case. Examples show the deviation of the stability limit found by the Lyapunov method from the exact value.

Bifurcations of Eigenvalues of Gyroscopic Systems With Parameters Near Stability Boundaries

Abstract: This paper deals with stability problems of linear gyroscopic systems Mx + Gx + Kx = 0 with finite or infinite degrees-of-freedom, where the system matrices or operators depend smoothly on several real parameters. Explicit formulas for the behavior of eigenvalues under a change of parameters are obtained. It is shown that the bifurcation (splitting) of double eigenvalues is closely related to the stability, flutter, and divergence boundaries in the parameter space. Normal vectors to these boundaries are derived using only information at a boundary point: eigenvalues, eigenvectors, and generalized eigenvectors, as well as first derivatives of the system matrices (or operators) with respect to parameters. These results provide simple and constructive stability and instability criteria. The presented theory is exemplified by two mechanical problems: a rotating elastic shaft carrying a disk, and an axially moving tensioned beam.

A Precise Bound For Gyroscopic Stabilization

Abstract: We consider gyroscopic systems M(t) + hG(t) + Kx(t) = 0 where M > 0, GT = —G, and K < 0. It is shown how to compute a critical value of parameter which (in many cases) separates stable and unstable regimes for gyroscopic stabilization. Comparison is made with some known sufficient conditions for stability or instability, and a theory is developed (including the formulation of a related Lyapunov function) which unifies several earlier results concerning systems of this kind. Numerical examples are included. Wir betrachten gyroskopische Systeme M(t) + hG(t) = Kx(t) = 0, wobei M > 0, GT = —G und K < 0 ist. Es wird gezeigt, wie man einen kritischen Wert des Parameters berechnen kann, der (in vielen Fallen) die stabilen und unstabilen Gebiete der gyroskopischen Stabilisierung trennt. Es werden Vergleiche mit bekannten hinreichenden Bedingungen fur Stabilitat oder Instabilitat vorgenommen, und eine Theorie wird entwickelt, die verschiedene fruhere Resultate uber derartige Systeme vereint. Diese Theorie wird mit einer passenden Lyapunov-Funktion in Verbindung gebracht. Numerische Beispiele schliesen sich an.

Stability of rotor systems: a complex modelling approach

Abstract: The dynamics of a large class of rotor systems can be modelled by a linearized complex matrix differential equation of second order, \(M\ddot z+(D+iG)\dot z+(K+iN)z=0\), where the system matrices M, D, G, K and N are real symmetric. Moreover M and K are assumed to be positive definite and D, G and N to be positive semidefinite. The complex setting is equivalent to twice as large a system of second order with real matrices. It is well known that rotor systems can exhibit instability for large angular velocities due to internal damping, unsymmetrical steam flow in turbines, or imperfect lubrication in the rotor bearings. Theoretically, all information on the stability of the system can be obtained by applying the Routh-Hurwitz criterion. From a practical point of view, however, it is interesting to find stability criteria which are related in a simple way to the properties of the system matrices in order to describe the effect of parameters on stability. In this paper we apply the Lyapunov matrix equation in a complex setting to an equivalent system of first order and prove in this way two new stability results. We then compare the usefulness of these results with the more classical approach applying bounds of appropriate Rayleigh quotients. The rotor systems tested are: a simple Laval rotor, a Laval rotor with additional elasticity and damping in the bearings, and a number of rotor systems with complex symmetric \(4\times4\) randomly generated matrices.

Dissipation-induced instabilities in finite dimensions

Abstract: The goal of this work is to introduce a coherent theory of the counterintuitive phenomena of dynamical destabilization under the action of dissipation. While the existence of one class of dissipation-induced instabilities was known to Sir Thomson (Lord Kelvin), it was not realized until recently that there is another major type of these phenomena hinted at by one of Merkin's theorems; in fact, these two cases exhaust all the generic possibilities. The theory grounded on the Thomson-Tait-Chetayev and Merkin theorems and on the geometric understanding introduced in this paper leads to the conclusion that ubiquitous dissipation is one of the paramount mechanisms by which instabilities develop in nature. Along with a historical review, the main theoretical achievements are put in a general context, thus unifying the current knowledge in this area and the multitude of relevant physical problems scattered over a vast literature. This general view also highlights the striking connection to various areas of mathematics. To appeal to the reader's intuition and experience, a large number of motivating examples are provided. The paper contains some new unpublished results and insights, and, finally, open questions are formulated to provide an impetus for future studies. While this review focuses on the finite-dimensional case, where the theory is relatively complete, a brief discussion of the current state of knowledge in the infinite-dimensional case, typified by partial differential equations, is also given.

Coupling of eigenvalues of complex matrices at diabolic and exceptional points

Abstract: 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.

Coupling of eigenvalues of complex matrices at diabolic and exceptional points

Abstract: The paper presents a general theory of coupling of eigenvalues of complex matrices of 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.

On the stability of fuzzy systems

Abstract: Studies the global asymptotic stability of a class of fuzzy systems. It demonstrates the equivalence of stability properties of fuzzy systems and linear time invariant (LTI) switching systems. A necessary and sufficient condition for the stability of such systems are given, and it is shown that under the sufficient condition, a common Lyapunov function exists for the LTI subsystems. A particular case when the system matrices can be simultaneously transformed to normal matrices is shown to correspond to the existence of a common quadratic Lyapunov function. A constructive procedure to check the possibility of simultaneous transformation to normal matrices is provided.

Brake comfort – a review

Abstract: Several state of the art papers and even books on brake vibration and/or noise have been presented in the literature. Many of them have analytically and sharply accounted for the impressive amount of research undertaken on this topic. This state of the art review focuses on the still-open questions that appear crucial from the perspective of a leading brake manufacturer. The paper deals with the phenomena of brake vibration and/or noise, the experimental and theoretical methods for studying such phenomena, and the actions that are identified to be necessary to definitely solve the addressed problem. Key topics are the modelling of friction, the modelling of the dynamics of the brake as a non-linear system subjected to deterministic or random (parametric) excitation, the proper modelling of the contact between the disc and the pad, and the experimental validation of the mathematical models.