On Gyroscopic Stabilization

  • This thesis deals with systems of the form \( M\ddot x+D\dot x+Kx=0\;, \; x \in \mathbb R^n\;, \) with a positive definite mass matrix \(M\), a symmetric damping matrix \(D\) and a positive definite stiffness matrix \(K\). If the equilibrium in the system is unstable, a small disturbance is enough to set the system in motion again. The motion of the system sustains itself, an effect which is called self-excitation or self-induced vibration. The reason behind this effect is the presence of negative damping, which results for example from dry friction. Negative damping implies that the damping matrix \(D\) is indefinite or negative definite. Throughout our work, we assume \(D\) to be indefinite, and that the system possesses both stable and unstable modes and thus is unstable. It is now the idea of gyroscopic stabilization to mix the modes of a system with indefinite damping such that the system is stabilized without introducing further dissipation. This is done by adding gyroscopic forces \(G\dot x\) with a suitable skew-symmetric matrix \(G\) to the left-hand side. We call \(G=-G^T\in\mathbb R^{n\times n}\) a gyroscopic stabilizer for the unstable system, if \( M\ddot x+(D+ G)\dot x+Kx=0 \) is asymptotically stable. We show the existence of \(G\) in space dimensions three and four.

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Author:Jan Homeyer
URN (permanent link):urn:nbn:de:hbz:386-kluedo-29467
Advisor:Tobias Damm
Document Type:Doctoral Thesis
Language of publication:English
Publication Date:2012/03/20
Year of Publication:2012
Publishing Institute:Technische Universität Kaiserslautern
Granting Institute:Technische Universität Kaiserslautern
Acceptance Date of the Thesis:2012/02/16
Date of the Publication (Server):2012/03/20
Number of page:77
Faculties / Organisational entities:Fachbereich Mathematik
DDC-Cassification:5 Naturwissenschaften und Mathematik / 510 Mathematik
Licence (German):Standard gemäß KLUEDO-Leitlinien vom 15.02.2012