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Stability of Gyroscopic Systems with Respect to Perturbations

  • Nicola GuglielmiEmail author
  • Manuela Manetta
Chapter
Part of the Springer INdAM Series book series (SINDAMS, volume 30)

Abstract

A linear gyroscopic system is of the form:
$$\displaystyle M \ddot x + G\dot x + K x = 0, $$
where the mass matrix M is a symmetric positive definite real matrix, the gyroscopic matrix G is real and skew symmetric, and the stiffness matrix K is real and symmetric. The system is stable if and only if the quadratic eigenvalue problem \(\det (\lambda ^2 M+\lambda G + K)=0\) has all eigenvalues on the imaginary axis.

In this chapter, we are interested in evaluating robustness of a given stable gyroscopic system with respect to perturbations. In order to do this, we present an ODE-based methodology which aims to compute the closest unstable gyroscopic system with respect to the Frobenius distance.

A few examples illustrate the effectiveness of the methodology.

Keywords

Stability of gyroscopic systems Robust stability Structured matrix nearness problems Matrix ODEs 

Notes

Acknowledgements

N. Guglielmi thanks the Italian M.I.U.R. and the INdAM GNCS for financial support and also the Center of Excellence DEWS.

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Gran Sasso Science InstituteL’AquilaItaly
  2. 2.Department of Mathematics and Computer ScienceEmory UniversityAtlantaUSA

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