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

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 Sufficient Conditions of Stability of the Permanent Rotations of a Heavy Triaxial Gyrostat

Qualitative Theory of Dynamical Systems, 2014

In this work by means of geometric-mechanics methods we study the permanent rotations of a particular type of heavy triaxial gyrostat. Also, we use the Energy-Casimir method to obtain sufficient conditions of stability of the permanent rotations found previously. Among the various aspects related to these problems that are discussed in the literature, we can highlight the following: 1. Equilibria and stabilities in rigid bodies and gyrostats, either with fixed point or in orbit (see [14],[15],[5],[1],[17],[2],[13],[20],[9]).

On the Stability of a Rotating Blade with Geometric Nonlinearity

Journal of Applied Nonlinear Dynamics, 2012

The stability of a rotating, nonlinear blade under a time-varying torque is investigated. The nonlinear blade is presented based on the geometric nonlinearity. Compared to the nonlinear model, the linear model of a rotating blade with effective load is investigated. The two models of rotating blades are reduced to the ordinary differential equations through the Galerkin method, and gyroscopic systems with parametric excitations are obtained. From such gyroscope systems, the stability of the two models is studied by the generalized harmonic balance method, and the approximate, analytical solutions for the two models are obtained. The stability regions in parameter maps are presented for a better understanding of the stability of such parametric, time-varying systems. The analytical and numerical solutions are illustrated. This study provides an efficient and accurate way to determine the stability of gyroscope systems with time-varying excitations.

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.

On the instability of steady motion

Meccanica, 2011

This paper deals with the instability of steady motions of conservative mechanical systems with cyclic coordinates. The following are applied: Kozlov's generalization of the first Lyapunov's method, as well as Rout's method of ignoration of cyclic coordinates. Having obtained through analysis the Maclaurin's series for the coefficients of the metric tensor, a theorem on instability is formulated which, together with the theorem formulated in Furta (J. Appl. Math. Mech. 50(6):938-944, 1986), contributes to solving the problem of inversion of the Lagrange-Dirichlet theorem for steady motions. The cases in which truncated equations involve the gyroscopic forces are solved, too. The algebraic equations resulting from Kozlov's generalizations of the first Lyapunov's method are formulated in a form including one variable less than was the case in existing literature.

On the dynamical motion of a gyro in the presence of external forces

Advances in Mechanical Engineering, 2017

This work shed light on the motion of a symmetric rigid body (gyro) about one of its principal axes in the presence of a Newtonian force field besides a gyro moment in which its second component equals null. It is assumed that the body center of mass is shifted slightly relative to the dynamic symmetry axis. The governing equations of motion are investigated taking into account some initial conditions. The desired solutions of these equations are achieved in framework of the small parameter method. The periodic solutions for the case of irrational frequencies are investigated. Euler’s angles have been used to interpret the motion at any time. The geometrical representations of the obtained solutions and the phase plane schemas of these solutions are announced during several plots. Discussion of the results is presented to reinforce the importance of the considered gyro moment and the Newtonian force field. The significance of this problem is due to the framework of its several appli...