On the Stability of Rotating Systems (original) (raw)

On the theory of linear gyroscopic systems

Journal of Applied Mathematics and Mechanics, 1996

The behaviour of the oscillation frequencies of Hamiltonian systems when their stiffness and inertia are changed is reviewed. The number of frequencies of the first and second kind, expressed in terms of the number of positive and negative eigenvalues of the Hamiltonian, is found. It follows from these results, in particular, that Rayleigh's classical theorem only holds for gyroscopic systems when the numbe:r of its frequencies is equal to the number of frequencies of the system without gyroscopic forces. The degree of instability of a gyroscopic system when there are small dissipative forces is found; in a gyroscopically stabilized system, it is half the degree of styptic instability.

Stability Analysis and Response Bounds of Gyroscopic Systems

Asian Research Journal of Mathematics, 2017

In this work, we develop a stability theorem for determining the stability or otherwise of a gyroscopic system. A Lyapunov function is obtained by solving the arising Lyapunov matrix equation. The Lyapunov function is then used to obtain response bounds for displacements and velocities both in the homogeneous and inhomogeneous cases. Examples are given to illustrate the efficacy of the results obtained.

Gyroscopic stabilization of non-conservative systems

Physics Letters A, 2006

Gyroscopic stabilization of a linear conservative system, which is statically unstable, can be either improved or destroyed by weak damping and circulatory forces. This is governed by Whitney umbrella singularity of the boundary of the asymptotic stability domain of the perturbed system.

Effect of small dissipative and gyroscopic forces on the stability of nonconservative systems

Doklady Physics, 2003

We analyze the effect of small forces proportional to the generalized velocity vector on the stability of a linear autonomous mechanical system with nonconservative positional forces. It is known that arbitrarily small dissipation generally destabilizes a nonconservative system . Necessary and sufficient conditions on the matrix of dissipative and gyroscopic forces under which the system is asymptotically stable are obtained. The two-dimensional system is studied in detail. The problem of the stability of the Ziegler-Herrmann-Jong pendulum is considered as a mechanical example.

On the instability of the equilibrium for a Lagrangian system with gyroscopic forces

Nonlinear Differential Equations and Applications NoDEA, 1994

We consider a Lagrangian Differential System (L.D.S.) with Lagrangian function L(q, q) = T(q, gl) + U(q), sufficiently smooth in a neighbourhood of the critical point q = 0 of the potential function U(q). The kinetic function T(q, Cl) is a non homogeneous quadratic function of the 4's, i.e. the L.D.S. contains the so-called gyroscopic forces. The potential function U(q) starts with a degenerate (but non zero), semidefinite-negative, quadratic form. Moreover, q = 0 is not a proper maximum of U, and this property has to be recognized in a suitable way. By analizing the problem of the existence of solutions of the L.D.S., which asymptotically tend to the equilibrium solution, (q, q) = (0, 0), we provide a sufficient criterium for its instability.

Stability of the permanent rotations of an asymmetric gyrostat in a uniform Newtonian field

Applied Mathematics and Computation

The stability of the permanent rotations of a heavy gyrostat is analyzed by means of the Energy-Casimir method. Su cient and necessary conditions are established for some of the permanent rotations. The geometry of the gyrostat and the value of the gyrostatic moment are relevant in order to get stable permanent rotations. Moreover, the necessary conditions are also su cient, for some configurations of the gyrostat.

Stability of Gyroscopic Circulatory Systems

AIAA Journal, 2018

This paper presents results related to the stability of gyroscopic systems in the presence of circulatory forces. It is shown that when the potential, gyroscopic, and circulatory matrices commute, the system is unstable. This central result is shown to be a generalization of that obtained by Lakhadanov, which was restricted to potential systems all of whose frequencies of vibration are identical. The generalization is useful in stability analysis of large scale multidegree-of-freedom real life systems, which rarely have all their frequencies identical, thereby expanding the compass of applicability of stability results for such systems. Comparisons with results in the literature on the stability of such systems are made, and the weakness of results that deal with only general statements about stability is exposed. It is shown that the commutation conditions given herein provide definitive stability results in situations where the well-known Bottema-Karapetyan-Lakhadanov result is inapplicable.