## 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 |
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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 |

Tag: | Gyroscopic |

Number of page: | 77 |

Faculties / Organisational entities: | Fachbereich Mathematik |

DDC-Cassification: | 5 Naturwissenschaften und Mathematik / 51 Mathematik / 510 Mathematik |

MSC-Classification (mathematics): | 15-XX LINEAR AND MULTILINEAR ALGEBRA; MATRIX THEORY |

Licence (German): | Standard gemäß KLUEDO-Leitlinien vom 15.02.2012 |