On the stability of an equilibrium position and rotational motion of a gyrostat

This article is devoted to study the compulsory stability of equilibrium position and rotational motion of a rigid body containing fluid with the help of three rotors carried on the body. The control moments on the rotors using that condition which impose the stabilization of equilibrium position of the rigid body and rotational motion are obtained.     

Stationary solutions of a small gyrostat in the Newtonian field of two bodies with equal masses

The paper deals with the dynamics of a small gyrostat satellite acted upon by the Newtonian forces of two big bodies of equal masses which rotate around their center of mass. The gyrostat’s equations of motion are derived and classes of its stationary solutions, as well as their stability are studied.     

Closed form solutions for small deformations superimposed upon the symmetrical expansion of a spherical shell

For small axially symmetric deformations of isotropic incompressible hyperelastic materials which are super-imposed upon the symmetrical expansion of a spherical shell, new closed form solutions are derived without any restrictions on the strain-energy function. These solutions are used to derive the n=1 buckling criterion for thick-walled spherical shells which are subjected to uniform external pressure. They are also used to deduce an upper bound to the force deflection relation for small superimposed translational deflections of bonded pre-compressed spherical rubber bush mountings.

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Zusammenfassung Für den Fall kleiner, axialsymmetrischer Deformationen isotropischer inkompressibler hyperelastischer Materialien, die der symmetrischen Dehnung einer sphärischen Schale überlagert sind, werden Lösungen in geschlossener Form abgeleitet, ohne einschränkende Bedingungen für die Deformationsenergiefunktion. Mit Hilfe dieser Lösungen wird das Knick-Kriterium (n=1) für dickwandige sphärische Schalen gewonnen, die gleichförmigem, äusserem Druck, ausgesetzt sind. Weiterhin wird mit Hilfe der Lösungen eine obere Grenze gewonnen für die Kraft-Ablenkungs-Relation im Falle überlagerter kleiner Translationen von gebundenen, vorgespannten sphärischen Gummibuchsen.

     

Periodic oscillations of a class of non-autonomous non-linear elastic continua

A modified perturbation method for obtaining periodic solutions to a class of non-autonomous non-linear partial differential equations is developed. The classic small divisor is discussed in detail and a general method for its elimination is presented. New terminology is introduced for the purpose of discussing forcing functions that produce in a system a response that is of the same form as a non-linear periodic mode for the same system. Specific examples are examined to verify the results of this work.     

Periodic solutions of an epidemic model with saturated treatment

Based on the fact that many infectious diseases exhibit periodic fluctuations and there is a saturated phenomenon during disease treatment, we study an SIR model with periodic incidence rate and saturated treatment function. Firstly, we find that the basic reproduction number less than 1 cannot insure the global stability of disease-free equilibrium and it needs to add other conditions. Moreover, we establish sufficient conditions for the multiplicity of positive periodic solutions. We also apply the numerical method to confirm theoretical results and show the stability of the periodic solutions. We observe that there are two periodic solutions in the system where one is stable and the other one is unstable. These results will provide some guidance for control measures of disease.     

Periodic solutions of an impulsive system with two discontinuity points

We establish conditions for the existence of a 2π-periodic solution of an impulsive system with two discontinuity points at nonfixed times.     

On the small two-dimensional oscillations of a sea-line

Summary The aim of this work is to study the small in plane oscillations of a submerged pipeline with a curved equilibrium configuration. Some cases of interest in practical applications are analyzed assuming that the curve of static deflection is a catenary. The limits of the method adopted to integrate the resulting differential equation are discussed and the natural frequencies computed starting from a catenary are compared with those obtained assuming static configurations with constant radii of curvature.

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Sommario In questo lavoro si studiano le piccole oscillazioni piane di una condotta sommersa la cui configurazione di equilibrio statico abbia curvatura diversa da zero. Sono trattati alcuni casi di interesse pratico assumendo la catenaria come curva di equilibrio. Vengono discussi i limiti di applicabilità del metodo adottato per integrare l'equazione differenziale risultante. Le frequenze naturali ottenute sono confrontate con quelle che si hanno partendo da alcune configurazioni di equilibrio a curvatura costante.

     

Bifurcations of relative equilibria of an oblate gyrostat with a discrete damper

We investigate relative equilibria of an oblate gyrostat with a discrete damper. Linear and nonlinear methods yield stability conditions for simple spins about the nominal principal axes. We use analytical and numerical methods to explore other equilibria, including bifurcations that occur for varying rotor momentum and damper parameters. These bifurcations are complex structures that are perturbations of the zero rotor momentum case. We use Lyapunov–Schmidt reduction to determine an analytic relationship between parameters to determine conditions for which a jump phenomenon occurs. This paper is declared a work of the U.S. Government and is not subject to copyright protection in the United States     

Hydrodynamics in weak gravitational fields small oscillations of an ideal liquid in a cylindrical vessel

The problem of determining the frequencies and forms of small natural oscillations of an ideal liquid in a cylindrical vessel under conditions close to weightlessness is examined. It is assumed that a weak homogeneous gravitational field acts parallel to the vertical generatrix forming the cylinder. In contrast to [1], where only the first antisymmetric oscillation frequency is found for a semiinfinite cylindrical vessel, the frequencies of several axiosymmetric, antisymmetric, etc. oscillations are obtained as functions of the gravitational-field intensity and other parameters of the problem. The Ritz method is employed for two different variations of the problem, equivalent to that of oscillations of an ideal liquid under conditions of weightlessness [1–5].Translated from Izvestiya Akademii Nauk SSSR. Mekhanika Zhidkosti i Gaza, No. 2, pp. 3–13, March–April, 1973.     

Periodic solutions and stability of a tethered satellite system

In this paper, several criteria on the existence of periodic solutions for a tethered satellite system (TSS) in an elliptical orbit, as well as the uniqueness of periodic solutions for the TSS in a circular orbit are presented on the basis of coincidence degree theory. In addition, the conditions on the global asymptotic stability of the equilibrium states for the TSS are also addressed in accordance with the Lyapunov stability theory and Barbashin–Krasovski theory.     

Reliable solutions to the problem of periodic oscillations of an elastoplastic beam

We consider time-periodic oscillations of a beam with a spatially inhomogeneous Prandtl–Ishlinskii constitutive law describing the elastoplastic hysteresis. The data (mass density, Prandtl–Ishlinskii distribution, external load) are assumed to be uncertain. It is shown that a unique solution exists and is stable with respect to the data variation. Considering the total dissipated energy as a measure for the accumulated material fatigue, we identify and estimate from above the ‘worst scenario’ case, where the dissipation over one period is maximal within an admissible set of data obtained from inaccurate measurements.     

On occurrence of mixed-torus bursting oscillations induced by non-smoothness

Nonlinear Dynamics - The paper devotes to the slow–fast behaviors of a higher-dimensional non-smooth system with the coupling of two scales. Some novel bursting attractors and interesting...     

A study of particle motion induced by two-dimensional liquid oscillations

The motion of single particles in liquids undergoing two-dimensional oscillations has been studied both experimentally and theoretically. By imposing combined vertical and horizontal oscillations on a liquid, particle mean motion can be controlled so that individual particles either rise against gravity or fall. The one dimensional equation of particle motion was modified to suit this situation and numerical solutions were used to predict the occurrence of the rising motion although agreement with experiment was only qualitative. A dimensional analysis approach combined with experiments allowed prediction of particle behaviour within the range of variables of this study. Extension of the single particle studies to mixtures of quartz and galena in various flowing liquids allowed prediction of the conditions under which separation and upgrading of minerals could occur and this was verified experimentally.     

Stability of small radial oscillations of a spherical layer of liquid

An isothermal spherical layer of a viscoelastic liquid described by the one-parameter Maxwell model is considered. When the model parameter is taken equal to zero, a model of a purely viscous Newtonian fluid is obtained. The stability of the spherical layer of liquid with respect to small radial perturbations of the velocity and pressure is investigated for both types of liquids. Leningrad. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 170–171, September–October, 1988.     

The flow field induced by the torsional oscillations of a spherical cell containing a fluid drop

The flow field induced by the torsional oscillations of a spherical cell containing a fluid drop is examined. It has been found that in addition to the oscillating motion of each fluid particle there exist three standing vortices in every quadrant between the drop and the container. The steady streaming into the drop can be directed either clockwise or counterclockwise, depending on the values of the parameter of the fluids inside and outside the drop. Typical flow fields are shown graphically.