Stability Of Steady Motions Research Articles

Motion of a Variable Body with a Fixed Point in a Time-dependent Force Field

The problem of motion around a fixed point of a variable body in a time-dependent force field is considered. The conditions under which the equations of motion are reduced to the classical Euler–Poisson equations describing the motions of a rigid body in the field of attraction are indicated. The problems of the existence of the first integrals and the stability of steady motions are discussed.

Stability investigation of the steady motions of an isolated system, carrying out plane motion

The paper investigates the conditional stability of steady motions of a flat model of an isolated system consisting of a rotating LB, material point, which creates its static imbalance, and two identical mathematical pendulums, mounted on the longitudinal axis of the LB and moving in the plane of the static imbalance, the relative motion of which is prevented by the viscous resistance. It was found that in the case where there is imbalance and pendulums can eliminate it with a certain reserve, there is one basic motion; in the absence of imbalance, there is a one-parameter family of basic motions; in the case of maximum imbalance, which can be eliminated by pendulums, there is one basic motion, but it generates pseudo-family of basic motions. Also, it was found that some basic motions are conditionally asymptotically stable, if they, or family, or pseudo-family of basic motions are isolated. In the absence of imbalance, the presence of a single zero root of the characteristic equation does not affect the stability of the one-parameter family of basic motions, and is responsible for the transition from one to another steady motion of the family. In the case of maximum imbalance, the presence of a single zero root of the characteristic equation does not affect the stability of the basic motion, and is responsible for the transition from one to another steady motion of pseudo-family. Transients, depending on the system parameters can be aperiodic or oscillatory-damped. It was found that the side motions are unstable.

Relative equilibrium stability of a mechanical system with deformable elements in a circular orbit

The relative equilibrium stability is analyzed for a mechanical system in an orbit. The system consists of two rigid bodies connected by a thin inextensible elastic rod. The problem of stability of steady motions is reduced to the minimization problem for the system’s potential energy consisting of the potential energy of elastic, gravitational, and centrifugal forces.

Stability of steady motions for systems with general and particular first integrals

Several stability conditions are obtained for the steady motions of systems with general and particular first integrals. An example is discussed.

The motion of a body with a plane of symmetry over a three-dimensional sphere under the action of a spherical analogue of Newtonian gravitation

The problem of the motion of a rigid body possessing a plane of symmetry over the surface of a three-dimensional sphere under the action of a spherical analogue of Newtonian gravitation forces is considered. Approaches to introducing spherical analogues of the concepts of centre of mass and centre of gravity are discussed. The spherical analogue of “satellite approach” in the problem of the motion of a rigid body in a central field, which arises on the assumption that the dimensions of the body are small compared with the distance to the gravitating centre, is studied. Within the framework of satellite approach, assuming plane motion of the body, the question of the existence and stability of steady motions is investigated. A spherical analogue of the equation of the plane oscillations of a body in an elliptic orbit is derived.

On stability of stationary motion of a nonconservative nonholonomic rheonomic system

Abstract One considers, in this paper, the motion of a mechanical system in a nonstationary field of potential and positional forces, subject to the action of rheonomic holonomic and nonholonomic linear homogeneous constraints. Assuming that differential equations of motion of the system considered satisfy the conditions for the existence of Painleve's integral of energy, formulated in [Painleve, P., 1897. Lecons sur l'integration des equations de la Mecanique, Paris] and [Appell, P., 1911. Traite de mecanique rationnelle, T. II, Dynamique des systemes – Mecanique analitique, Gauthier-Villars, Paris] and generalized in [Covic, V., Veskovic, M., 2004. On stability of motion of a rheonomic system in the field of potential and positional forces, BAMM-1720/2004, No-2233, 93–100] and [Covic, V., Veskovic, M., 2005. Hagedorn's theorem in some special cases of rheonomic systems. Mechanics Research Communications 32 (3), 265–280], the original mechanical system is substituted by an equivalent one whose Lagrangian function, nontransformed with respect to nonholonomic constraints, does not depend on time explicitly. Using the properties of the equivalent system, which, in contrast to the original one, moves in a stationary field of potential forces and in a nonstationary field of gyroscopic forces, the definition of cyclic coordinates is generalized, as well as sufficient conditions for the existence of (cyclic) first integrals, corresponding to coordinates mentioned and linear in velocities are established. Further, the conditions for the existence of steady motion of the system considered are found. In the case of existence of such a motion of the system, the Theorem of Routh's type on stability of that motion, based on the minimum of reduced potential for which it is shown that, in contrast to known cases (see, for example, [Gantmacher, F., 1975. Lectures in Analytical Mechanics. Mir Publisher, Moscow; Neimark, J., Fufaev, N., 1972. Dynamics of Nonholonomic Systems. Amer. Math. Soc., Providence, RI; Pars, L., 1962. An Introduction to Calculus of Variations. Heinemann, London; Karapetyan, A., Rumyantsev, V., 1983. Stability of conservative and dissipative systems. In: Itogi Nauki I Tekhniki: Obschaya Mekh., vol. 6, VINITI, Moscow, pp. 3–128 (in Russian)]), it includes the influence of the positional forces field, is formulated. Thus, the Routh's Theorem on stability of steady motion of a conservative mechanical system is extended to the case of a nonconservative system.

The stability of steady motions of non-holonomic chaplygin systems

A stability theorem is established for steady motions of non-holonomic Chaplygin systems, with cyclic coordinates, acted upon by potential and dissipative forces, generalizing a previously proved theorem [1]. The theorem enables rigorous sufficient conditions for the stability of steady motions of non-holonomic systems to be derived in cases that are more general than those considered hitherto. As an example, the problem of the stability of the steady motions of a one-wheeled carriage is considered.

The motion of a rigid body in a fluid under the action of central newtonian attractive forces

The structure of the forces and moments which arise when a rigid body moves in an unbounded volume of an ideal incompressible fluid under the action of central Newtonian attractive forces is discussed, the equations of motion are written in explicit form, their first integrals are indicated and the properties of these integrals are investigated. The problem of possible simplifications in the formulation of the problem is investigated, based on an analogy with the classical “satellite approximation” and which hold when the translational velocity of the body can be regarded as being independent of its rotational motion. The question of the existence and stability of steady motions, in this case relative equilibria, is investigated in the satellite approximation.

The stability of the steady motions of a symmetrical gyrostat on an absolutely rough horizontal plane

The problem of the motion, without slipping, of a dynamically symmetrical gyrostat on a fixed horizontal plane is investigated. It is shown that Chaplygin's equations express, in projections on the axes of a semimobile system of coordinates, the theorem on the angular momentum of the gyrostat about the point of contact with the plane. All possible steady motions of a heavy symmetrical gyrostat on an absolutely rough plane including the case when the nutation angle is equal to zero are investigated using the generalized Routh theorem. The effect of the rotor on the stability of steady motions is also examined. The case when the body of the gyroscope is a solid with a circular base, in particular, a disc with a rotor, is considered. It is shown that the rotating rotor has a stabilizing influence on equilibrium of the gyrostat and a destabilizing influence on the rolling of the gyrostat about a straight line, provided the axial angular momentum of the gyrostat is zero.

Effect of convection on the stability of a liquid with a nonuniformly distributed heavy admixture

The effect of the transverse temperature gradient on the stability of steady motion of a viscous incompressible liquid in a plane vertical layer bounded by two infinite solid surfaces is studied. The motion of the liquid is caused by sedimentation of heavy solid spherical particles distributed nonuniformly across the layer and by the horizontal temperature gradient. Spectra of decrements of small normal perturbations are calculated for different particle sizes and different degrees of nonuniformity of the distribution of admixture particles. The stability of a steady flow of the liquid with an admixture decreases with increasing temperature gradient and increasing particle radius and increases with a tendency of the particles to a uniform distribution.

Steady motions of gyrostat satellites and their stability

The steady motions of a rigid body rotating about its maximum principal axis of inertia, while the radius vector lies in the direction of its minimum principal axis of inertia, is known to be stable in the sense of Lyapunov. Due in part to their stowed configuration in launch vehicles, however, satellites typically have an initial rotation about their minimum principal axis of inertia. Such rotation may be unstable in the presence of some dissipations. This paper investigates the effect of momentum wheels on the stability of steady motions. It is proved that the momentum wheels increase the effective moment of inertia of the gyrostat-satellite system about some desired axis. Stability of the steady rotation about the desired axis can be established only for the case when the moment of inertia of the axis aligned with the radius vector is smaller than that of the axis of linear momentum. A new set of stability criteria is obtained which includes the effects of the coupling between the orbital and attitude dynamics and may be useful in the design of attitude control systems for large spacecraft in low Earth orbit. >

Steady motions in controlled gyrostat systems

The equations of motion of a gyrostat system are given for the general case in which gyroscopic terms are present which are not due to the rotor motion. The stability of steady motions is investigated for free and for controlled systems. Observability and controllability are discussed, with the relative rotor speed as observation and the torque on the rotor as control variable.

An Instability Theorem for Steady Motions in Free and Restrained Dynamical Systems

The stability of steady motions in dynamical systems with ignorable coordinates is considered. In addition to the original “free” systems “restrained” systems are defined in such a way that the ignorable velocities remain constant along all motions; the stability behavior of the two systems is compared. A previously established instability theorem is generalized and three examples are given.

Erratum: “On the Stability of Steady Motions in Free and Restrained Dynamical Systems” (Journal of Applied Mechanics, 1979, 46, pp. 427–432)

Erratum: “On the Stability of Steady Motions in Free and Restrained Dynamical Systems” (Journal of Applied Mechanics, 1979, 46, pp. 427–432)

On the Stability of Steady Motions in Free and Restrained Dynamical Systems

In this paper the stability of the steady motions of dynamical systems with ignorable coordinates is considered. In addition to the original “free” systems “restrained” systems are defined in such a way that the ignorable velocities remain constant for all motions. The relation between the stability behavior of these two types of systems is examined in detail and several stability and instability theorems are given for damped and undamped systems. An illustrative example deals with the steady motions of a heavy gyrostat.

Die umkehrung der stabilitätssätze von Lagrange-Dirichlet und Routh

The Lagrange-Dirichlet theorem states that the equilibrium position of a discrete, conservative mechanical system is stable if the potential energy U(q) assumes a minimum in this position. Although everything seems to indicate that the equilibrium is always unstable in case of a maximum of the potential energy, this has yet to be proven. In all existing instability theorems the function U(q) has to satisfy additional requirements which are very restrictive. In this paper instability is proved in the case of a maximum of U(q)eC 2, without further restrictions. The instability follows directly from the existence of certain types of motions which are not found as solutions of differential equations, but as the solutions of a variational problem. Existence theorems are given for the variational problem, based on a result found by Caratheodory. In similar way an “inversion” of Routh's theorem on the stability of steady motions in systems with cyclic coordinates is also given. The result obtained here is not as general as the inversion of the Lagrange-Dirichlet theorem because the equations of motion are of a more complex type.

The stability of steady motions of nonholonomic voronets systems

The stability of steady motions of nonholonomic voronets systems

On the problem of stability of steady motions of a rigid body in a potential force field

On the problem of stability of steady motions of a rigid body in a potential force field

On the stability of steady motions of a heavy body of revolution on an absolutely rough horizontal plane

On the stability of steady motions of a heavy body of revolution on an absolutely rough horizontal plane

Stability of Structures

THE meaning of stability is not easy to define. In dynamics and mechanics we have stability of steady motion and stability of equilibrium, of position and of friction. To the civil engineer the word is usually applied to the power of a structure to withstand for an indefinite time all the loads and forces that may be brought to bear on it.