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## 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.

Author: Jan Homeyer urn:nbn:de:hbz:386-kluedo-29467 Tobias Damm Doctoral Thesis English 2012/03/20 2012 Technische Universität Kaiserslautern Technische Universität Kaiserslautern 2012/02/16 2012/03/20 Gyroscopic 77 Fachbereich Mathematik 5 Naturwissenschaften und Mathematik / 510 Mathematik 15-XX LINEAR AND MULTILINEAR ALGEBRA; MATRIX THEORY Standard gemäß KLUEDO-Leitlinien vom 15.02.2012