Stability Of Stationary Motions Research Articles

Efficient numerical calculation of Lyapunov-exponents and stability assessment for quasi-periodic motions in nonlinear systems

Investigating the stability of stationary motions is a highly relevant aspect when characterizing dynamical systems. For equilibria and periodic motions, well established theories and approaches exist to assess their stability: in both cases stability may be assessed using eigenvalue analyses of small perturbations. When it comes to quasi-periodic motions, such eigenvalue analyses are not applicable, since these motions can not be parameterized on finite time intervals. However, quasi-periodic motions can be densely embedded on finite invariant manifolds with periodic boundaries. In this contribution, a new approach is presented, which exploits this embedding in order to derive a sequence of finite mappings. Based on these mappings, the spectrum of 1st order Lyapunov-exponents is efficiently calculated. If the linearization of the problem is regular in the sense of Lyapunov, these exponents may be used to assess stability of the investigated solution. Beyond the numerical calculation of Lyapunov-exponents, an approach is presented which allows to check Lyapunov-regularity numerically. Together, both methods allow for an efficient numerical stability assessment of quasi-periodic motions. To demonstrate, verify and validate the developed approach, it is applied to quasi-periodic motions of two coupled van-der-Pol oscillators as well as a quasi-periodically forced Duffing equation. Additionally, a “step-by-step application instruction” is provided to increase comprehensibility and to discuss the required implementation steps in an applied context.

Stability of Stationary Motions of Mechanical Systems with a Particular Goryachev–Chaplygin Integral

Stability of Stationary Motions of Mechanical Systems with a Particular Goryachev–Chaplygin Integral

Stability of Stationary Motions of Mechanical Systems with a Particular Goryachev-Chaplygin Integral

In the existing studies, the stationary motions in a mechanical system in the case of the existence of a partial Goryachev-Chaplygin integral are analyzed. Some invariant sets and their stability, including one of the stationary motions, are also studied. In this article, the stability of the remaining second rest state is finally investigated.

Stabilization of stationary motions of a satellite near the center of mass in a geomagnetic field. III

Stabilization of stationary motions of a satellite near the center of mass in a geomagnetic field. III

Устойчивость сферического движения твердого тела с неоднородной жидкостью, совершающей однородное вихревое движение

In this paper, the equations of spherical motion of a solid body with a rotating inhomogeneous incompressible fluid that fills a completely ellipsoidal cavity are obtained and investigated. The stability of rotation of a solid with an inhomogeneous fluid having a linear density distribution is considered. Arbitrary fields of density and velocity of fluid particles are represented as a power series by spatial variables with coefficients that depend only on time. Sufficient conditions for the stability of the rotation of a solid body with a fluid around the vertical axis of dynamic symmetry are presented. The obtained equations of motion make it possible to study the stability of stationary motions of the system under consideration in order to assess the effect of fluid separation on the dynamics of the body. By analogy with the motion of a solid body, it is stated that the obtained conditions are also necessary and sufficient conditions for stationary rotations of an inhomogeneous fluid in an ellipsoidal cavity.

Stabilization of stationary motions of a satellite near the center of mass in a geomagnetic field. II

Stabilization of stationary motions of a satellite near the center of mass in a geomagnetic field. II

Stabilization of stationary motions of a satellite near the center of mass in a geomagnetic field. I

Stabilization of stationary motions of a satellite near the center of mass in a geomagnetic field. I

On first integrals and stability of stationary motions of gyrostat

The paper studies the equations of motion of a gyrostat around a fixed point under the action of a moment of forces. A general form of the moment that endows the equations with the three first integrals (energy, area, and geometric integrals) is established. Conditions for the existence of an additional integral as well as for the existence of a particular integral analogous to the Hess integral are derived. Stationary motions of the gyrostat are identified, and their stability in the Lyapunov sense is analyzed using Chetaev’s method and the integrals obtained.

Nonlinear stability analysis of a collinear libration point in the planar circular restricted photogravitational three-body problem

We investigate the problem on stability of collinear libration point in the photogravitational three-body problem. That is, we study the stability of stationary motion of an infinitesimal small body (particle) in gravitational field of two bodies (primaries) acting on the particle with radiation pressure repulsive forces. The primaries move in circular coplanar orbits about their barycenter and the particle lies in the plane of primaries motion. By using the method of normal forms and applying theorems of the Kolmogorov-Arnold-Moser theory we perform nonlinear stability study of collinear point, which located on the line connecting the primaries. Rigorous conclusions on stability in the sense of Lyapunov have been obtained both for non-resonant case and for cases of third and fourth order resonances. These results can be interesting for spaceflight and astronomical applications such as studies of dynamics of artificial satellites and observations of cosmic dust.

Darcy Convection Anisotropic Problem with Multi Stability of Stationary Motions for Rectangle

Darcy Convection Anisotropic Problem with Multi Stability of Stationary Motions for Rectangle

Isoperimetric Inequalities for Moments of Inertia and Stability of Stationary Motions of a Flexible Thread

Isoperimetric Inequalities for Moments of Inertia and Stability of Stationary Motions of a Flexible Thread

Spheroid Dynamics on a Horizontal Plane with Friction

This paper focuses on a spheroid with the center of mass coinciding with the geometric center moving in a horizontal plane with friction. For a certain class of initial conditions, a qualitative analysis of the dynamics is given depending on the presence of sliding, rolling or pivoting friction. A geometric interpretation of the results is given. The dynamics of a rigid rotation body on a plane with friction was studied within the framework of classical models of friction (dry or viscous, as well as a model of an absolutely rough plane), for example, see [1, 2]. A global qualitative analysis of the dynamics was carried out within the framework of interrelated friction models [3–5] for a uniform ball [6, 7] as well as a ball [8] and an ellipsoid [9] with a displaced center of mass. The stability of stationary motions in the case of a uniform spheroid on a plane with viscous friction is discussed in [10].

THE STUDY OF THE AUTONOMOUS SYNCHRONOUS GENERATOR MODES

The importance of the problem of the static stability of the stationary mode of the power system for its operation is extremely high. The investigation of the static stability of the power system is a subject of a number of works, but the problems of static stability of the stationary points of an autonomous synchronous generator are given little attention. The article considers transient and resonant (stationary) modes of the generator under active-inductive and active-capacitive loads. Mathematical model of transients in a natural form and in the coordinate system d, q are plotted. It is discovered that the mathematical model of the transition process of an autonomous synchronous generator is identical to the mathematical model of the transition process of the synchronous machine under three-phase short circuit. Electromagnetic transients of an autonomous synchronous generator are described by a system of linear autonomous differential equations with constant coefficients. However, the equivalent circuit of a generator contains dependent sources. We investigated the stability of stationary motion of an autonomous synchronous generator at a given angular velocity of rotation of the rotor. The condition for the existence and stability of stationary points of an autonomous synchronous generator is derived. The condition for the existence of stationary points of such a generator does not depend on the active load resistance and stator windings, and inductance of the rotor. The determining of stationary points of the generator is reduced to finding roots of a polynomial of the fourth degree. The graphs of electromagnetic torque dependencies on the angular velocity of rotation of the rotor (mechanical characteristics) are plotted. The equivalent circuits, corresponding to the equations of the transition process of an autonomous synchronous generator, are featured as well.

The Routh theorem for mechanical systems with unknown first integrals

In this paper we discuss problems of stability of stationary motions of conservative and dissipative mechanical systems with first integrals. General results are illustrated by the problem of motion of a rotationally symmetric rigid body on a perfectly rough plane.

Stability of LevitronTM revisited

This note discusses some issues related to stability of stationary motion of hovering magnetized top in a homogeneous magnetic field. Stability of synchronous motion is analyzed using the simplified model in which the hovering motion of the center of mass is ignored. Stability boundaries are derived using Lyapunov direct method. In particular, it is shown that, for a given angle ? between magnetic moment dipole and principal axis of the top, there is an interval of stationary values of nutation angle ??0 for which the stationary synchronous motion is stable.

On the stability of stationary motions of a system of coaxial bodies

We consider the stability of stationary motions of a model of a spacecraft as a system of coaxial bodies with small asymmetry caused by the shift of the axes of dynamic symmetry of bodies relative to the axis of rotation. We determine the stationary motions of the system; their stability is studied with respect to both the projections of angular velocity and the position of the axis of rotation. The sufficient conditions for the stability of these stationary motions are obtained by constructing a Lyapunov function, and the necessary conditions are obtained by analyzing the corresponding linearized equations of perturbed motion.

Stabilization of stationary motions of a gyrostat with a cavity filled with viscous fluid

The problem of spherical motions around the center of mass of a single-rotor dynamically symmetrical gyrostat with a spherical cavity, entirely filled with highly viscous fluid, is studied. The motion equations are calculated, stationary motions are found and their stability is studied. Many controls, based on the principle of feedback, are found, stabilizing unstable and stable stationary motions of a gyrostat to asymptotically stable.

Stability of stationary motions of a gimbal gyro with an electric motor

A gimbal gyro on an immovable base in the gravity field and with vertical outer suspension axis is considered. The rotor is driven by an asynchronous or synchronous electric motor, and there is no friction or any controlling torques on the suspension axes. Such a systemhas a family of steady-state motions (regular precessions and the rotor uniform rotations). It is shown that a necessary and sufficient condition for stability of the steady-statemotions with respect to generalized velocities and the angle of rotation of the inner gimbal frame is the existence of an isolated minimum of the reduced potential energy with respect to the angle of rotation of the inner frame.

Nonlinear stability analysis of a regular vortex pentagon outside a circle

A nonlinear stability analysis of the stationary rotation of a system of five identical point vortices lying uniformly on a circle of radius R0 outside a circular domain of radius R is performed. The problem is reduced to the problem of stability of an equilibrium position of a Hamiltonian system with a cyclic variable. The stability of stationary motion is interpreted as Routh stability. Conditions for stability, formal stability and instability are obtained depending on the values of the parameter q = R2/R02.

On stability of stationary motions of the body with viscous filler on the rod

In the problem of motion of the dynamically symmetrical body-shell with cavity completely filled with fluid and suspended by the rod to the fixed point, stationary motions are studied by Routh's technique. To describe the interaction between the filler and shell walls, a phenomenological model of "internal friction" is used.