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
The increasing use of composite materials has led to a greater demand for efficient curing cycles to reduce costs and speed up production cycles in manufacturing. One method to achieve this goal is in-line cure monitoring to determine the exact curing time. This article proposes a novel method through which to monitor the curing process inside closed tools by employing ultrasonic spectroscopy. A simple experiment is used to demonstrate the change in the ultrasonic spectrum during the cure cycle of an epoxy. The results clearly reveal a direct correlation between the amplitude and state of cure. The glass transition point is indicated by a global minimum of the reflected amplitude.
The dynamics of a large class of rotor systems can be modelled by a linearized complex matrix differential equation of second order, Mz + (D + iG)ż + (K + iN )z = 0, where the system matrices M, D, G, K and N are real symmetric. Moreover M and K are assumed to be positive definite and D, G and N to be positive semidefinite. The complex setting is equivalent to twice as large a system of second order with real matrices. It is well known that rotor systems can exhibit instability for large angular velocities due to internal damping, unsymmetrical steam flow in turbines, or imperfect lubrication in the rotor bearings. Theoretically, all information on the stability of the system can be obtained by applying the Routh-Hurwitz criterion. From a practical point of view, however, it is interesting to find stability criteria which are related in a simple way to the properties of the system matrices in order to describe the effect of parameters on stability. In this paper we apply the Lyapunov matrix equation in a complex setting to an equivalent system of first order and prove in this way two new stability results. We then compare the usefulness of these results with the more classical approach applying bounds of appropriate Rayleigh quotients. The rotor systems tested are: a simple Laval rotor, a Laval rotor with additional elasticity and damping in the bearings, and a number of rotor systems with complex symmetric 4 × 4 randomly generated matrices.Mathematics Subject Classification (1991). 15A24, 34A30, 34D20.
We consider gyroscopic systems M$\ddot x$(t) + hG$\dot x$(t) + Kx(t) = 0 where M > 0, GT = —G, and K < 0. It is shown how to compute a critical value of parameter $\hat h$ which (in many cases) separates stable and unstable regimes for gyroscopic stabilization. Comparison is made with some known sufficient conditions for stability or instability, and a theory is developed (including the formulation of a related Lyapunov function) which unifies several earlier results concerning systems of this kind. Numerical examples are included.
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