Lyapunov Functions and Solutions of the Lyapunov Matrix Equation for Marginally Stable Systems

2008

Z. Angew. Math. Phys.

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This paper deals with gyroscopic stabilization of the unstable system Mẍ + Dẋ + Kx = 0, with positive definite mass and stiffness matrices M and K, respectively, and an indefinite damping matrix D. The main question is for which skew-symmetric matrices G the system Mẍ + (D + G)ẋ + Kx = 0 can become stable? After investigating special cases we find an appropriate solution of the Lyapunov matrix equation for the general case. Examples show the deviation of the stability limit found by the Lyapunov method from the exact value. (2000). 34A30, 34D20, 70J25. Mathematics Subject Classification

Section: Some Known Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Indefinite damping in mechanical systems and gyroscopic stabilization

2008

Z. Angew. Math. Phys.

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“…For n odd,

r a n k (A_{1}) = n - 1

and

A_{1} v_{n} = 0

, C must be of the form

C = A_{2} A_{1} + h v_{n} v_{n}^{T}

, see Theorem . This gyroscopic stabilization is accomplished by using a theorem on linear marginally stable systems proved earlier by the authors in [], see Theorem . The procedure includes the finding of the general solution X to the homogeneous matrix equation

A_{1} X = X^{T} A_{1}

, where

A_{1} = - A_{1}^{T}

, see Theorem .…”

Section: Discussionmentioning

confidence: 99%

“…A general theory for constructing marginally stable systems of the form

I \ddot{x} + B \dot{x} + C x = 0, B, C \in {double-struckC}^{n \times n},

has been proved in []. There we find the following general theorem based on the theory of Lyapunov functions.…”

Section: Theory Of Marginally Stable Systemsmentioning

confidence: 99%

See 1 more Smart Citation

Gyroscopic stabilization of a class of circulatory systems

2018

Z Angew Math Mech

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A few examples in the literature show that unstable circulatory systems can be stabilized by gyroscopic forces. But to our knowledge no general theory has been presented. In our paper we will generate a class of unstable circulatory systems which can be stabilized gyroscopically. For circulatory systems the stability will always be marginal. Therefore we can use a theorem on marginally stable systems which the authors have developed in an earlier paper. Examples will demonstrate the synthesis of the relevant circulatory systems and the stabilizing procedure.

Solution of the Lyapunov matrix equation for a system with a time‐dependent stiffness matrix

2003

Z Angew Math Mech

The stability of the linearized model of a rotor system with non-symmetric strain and axial loads is investigated. Since we are using a fixed reference system, the differential equations have the advantage to be free of Coriolis and centrifugal forces. A disadvantage is nevertheless the occurrence of time-dependent periodic terms in the stiffness matrix. However, by solving the Lyapunov matrix equation we can formulate several stability conditions for the rotor system. Hereby the positive definiteness of a certain averaged stiffness matrix plays a crucial role.