Numerical methods for checking the stability of gyroscopic systems
Gyroscopic mechanical systems are modeled by the second-order differential equation \begin{equation*}\displaystyle M \ddot x(t) + G\dot x(t) + K x(t) = 0, \end{equation*} where \(M\in\mathbb{R}^{n\times n}\) is a symmetric and positive definite matrix, $G \in\mathbb{R}^{n\times n}$ is a skew-symme...
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Croatian Operational Research Society
2025-01-01
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| Series: | Croatian Operational Research Review |
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| Online Access: | https://hrcak.srce.hr/file/473283 |
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| author | Ivana Kuzmanović Ivičić |
| author_facet | Ivana Kuzmanović Ivičić |
| author_sort | Ivana Kuzmanović Ivičić |
| collection | DOAJ |
| description | Gyroscopic mechanical systems are modeled by the second-order differential equation
\begin{equation*}\displaystyle M \ddot x(t) + G\dot x(t) + K x(t) = 0,
\end{equation*} where \(M\in\mathbb{R}^{n\times n}\) is a symmetric and positive definite matrix, $G \in\mathbb{R}^{n\times n}$ is a skew-symmetric ($G^T=-G$) matrix, and $K\in\mathbb{R}^{n\times n}$ is a symmetric matrix, representing the mass, gyroscopic, and stiffness matrices, respectively. The stability of such systems, which is the primary topic of this paper, is determined by the properties of the associated quadratic eigenvalue problem (QEP)\begin{equation*}
{\mathcal G}(\lambda)x=(\lambda^2M+\lambda G+K)x=0, \quad x\in\mathbb{C}^{n},\ x\not=0.
\end{equation*} In this paper, we provide an overview of various linearizations of the QEP and propose numerical methods for checking the stability of gyroscopic systems based on solving the linearized problem. We present examples that demonstrate how the use of numerical methods provides a significantly larger stability region, which cannot be detected using the considered non-spectral criteria, or verify stability in cases where non-spectral criteria are not applicable, highlighting the advantages of numerical methods. |
| format | Article |
| id | doaj-art-5e538d8e729d439dae2e174e65f8bff0 |
| institution | OA Journals |
| issn | 1848-0225 1848-9931 |
| language | English |
| publishDate | 2025-01-01 |
| publisher | Croatian Operational Research Society |
| record_format | Article |
| series | Croatian Operational Research Review |
| spelling | doaj-art-5e538d8e729d439dae2e174e65f8bff02025-08-20T02:35:45ZengCroatian Operational Research SocietyCroatian Operational Research Review1848-02251848-99312025-01-0116212713610.17535/crorr.2025.0011Numerical methods for checking the stability of gyroscopic systemsIvana Kuzmanović Ivičić0School of Applied Mathematics and Informatics, J. J. Strossmayer University of Osijek, Osijek, CroatiaGyroscopic mechanical systems are modeled by the second-order differential equation \begin{equation*}\displaystyle M \ddot x(t) + G\dot x(t) + K x(t) = 0, \end{equation*} where \(M\in\mathbb{R}^{n\times n}\) is a symmetric and positive definite matrix, $G \in\mathbb{R}^{n\times n}$ is a skew-symmetric ($G^T=-G$) matrix, and $K\in\mathbb{R}^{n\times n}$ is a symmetric matrix, representing the mass, gyroscopic, and stiffness matrices, respectively. The stability of such systems, which is the primary topic of this paper, is determined by the properties of the associated quadratic eigenvalue problem (QEP)\begin{equation*} {\mathcal G}(\lambda)x=(\lambda^2M+\lambda G+K)x=0, \quad x\in\mathbb{C}^{n},\ x\not=0. \end{equation*} In this paper, we provide an overview of various linearizations of the QEP and propose numerical methods for checking the stability of gyroscopic systems based on solving the linearized problem. We present examples that demonstrate how the use of numerical methods provides a significantly larger stability region, which cannot be detected using the considered non-spectral criteria, or verify stability in cases where non-spectral criteria are not applicable, highlighting the advantages of numerical methods.https://hrcak.srce.hr/file/473283generalized eigenvaluesgyroscopic systemslinearizationquadratic eigenvalue problemstability |
| spellingShingle | Ivana Kuzmanović Ivičić Numerical methods for checking the stability of gyroscopic systems Croatian Operational Research Review generalized eigenvalues gyroscopic systems linearization quadratic eigenvalue problem stability |
| title | Numerical methods for checking the stability of gyroscopic systems |
| title_full | Numerical methods for checking the stability of gyroscopic systems |
| title_fullStr | Numerical methods for checking the stability of gyroscopic systems |
| title_full_unstemmed | Numerical methods for checking the stability of gyroscopic systems |
| title_short | Numerical methods for checking the stability of gyroscopic systems |
| title_sort | numerical methods for checking the stability of gyroscopic systems |
| topic | generalized eigenvalues gyroscopic systems linearization quadratic eigenvalue problem stability |
| url | https://hrcak.srce.hr/file/473283 |
| work_keys_str_mv | AT ivanakuzmanovicivicic numericalmethodsforcheckingthestabilityofgyroscopicsystems |