On periodic motions in magnetohydrodynamics

Summary A uniqueness theorem and an existence theorem for motions asymptotically stable in the mean and periodically depending on time, are demonstrated in magnetohydrodynamics, on the hypothesis that a motion sufficiently regular and asymptotically stable in the mean exists. In particular, similar theorems for steady motions are deduced. The procedure used is also a simple method for constructing periodic and steady motions.

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Sommario Si dimostra un teorema di unicità ed un teorema di esistenza per i moti magnetoidrodinamici periodicî nel tempo ed asintoticamente stabili in media, subordinatamente all'ipotesi che esista un moto asintoticamente stabile in media sufficientemente regolare, e si deducono, in particolare, analoghi teoremi per i moti stazionari. Il procedimento adoperato costituisce anche un semplice metodo per la effettiva costruzione dei moti periodici e dei moti stazionari.

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This work was done in the sphere of activity of the C.N.R. groups for mathematical research and, in Italian, has already been presented for publication in Rend. Acc. Sc. Fis. Mat., Napoli.     

Stability of high-frequency periodic motions of a heavy rigid body with a horizontally vibrating suspension point

The motion of a heavy rigid body one of whose points (the suspension point) executes horizontal harmonic high-frequency vibrations with small amplitude is considered. The problem of existence of high-frequency periodic motions with period equal to the period of the suspension point vibrations is considered. The stability conditions for the revealed motions are obtained in the linear approximation. The following three special cases of mass distribution in the body are considered; a body whose center of mass lies on the principal axis of inertia, a body whose center of mass lies in the principal plane of inertia, and a dynamically symmetric body.     

On the stability of periodic motions of a two-body system with flexible connection in an elliptical orbit

Nonlinear Dynamics - This paper deals with periodic motions and their stability of a flexible connected two-body system with respect to its center of mass in a central Newtonian gravitational field...     

On the stability of periodic motions of an unbalanced rigid rotor on lubricated journal bearings

A theoretical investigation is carried out on the orbital motions of a symmetrical, unbalanced, rigid rotor subjected to a constant vertical load and supported on two lubricated journal bearings. In order to determine the fluid film forces, the short bearing theory is adopted.A method is illustrated that makes it possible to determine the analytical equation of the orbit as an approximated solution of the system of non-linear differential equations of motion of the journal axis. A procedure is also described for evaluating the stability of the solution found. Diagrams of the curves delimiting, in the working plane of the rotor -m, the areas of stability of the various periodic solutions determined are provided.Finally, the results obtained are compared and combined with those provided by a direct integration of the motion equation made using the Runge-Kutta method.Nomenclature C radial clearance - D = 2R bearing diameter - E mass unbalance cecentricity - Fx, Fy fluid film force components - fi = Fi/W dimensionless fluid film force components - L bearing length - M one half rotor mass - % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbnL2yY9% 2CVzgDGmvyUnhitvMCPzgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqe% fqvATv2CG4uz3bIuV1wyUbqee0evGueE0jxyaibaieYlf9irVeeu0d% Xdh9vqqj-hEeeu0xXdbba9frFf0-OqFfea0dXdd9vqaq-JfrVkFHe9% pgea0dXdar-Jb9hs0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaaca% qabeaadaabauaaaOqaaiaad2gacqGH9aqpcaWGnbGaam4qaiabeM8a% 3naaCaaaleqabaGaaGOmaaaakiaac+cacqaHdpWCcaWGxbaaaa!471F!\[m = MC\omega ^2 /\sigma W\] dimensionless one half rotor mass - R bearing radius - T = 2 synchronous orbit period - t time - W load per bearing - X, Y, Z coordinates - x = X/C; y = y/C; z = Z/L dimensionless coordinates - oil dynamic viscosity - = E/C dimensionless mass unbalance eccentricity - = (RL/W)/(R/C) ² (L/D) ² modified Sommerfeld number - = t dimensionless time = periodic orbit frequency - = 2/ frequency ratio - journal angular velocity - (·) dimensionless time derivative     

Finite motions from periodic frameworks with added symmetry

Recent work from authors across disciplines has made substantial contributions to counting rules (Maxwell type theorems) which predict when an infinite periodic structure would be rigid or flexible while preserving the periodic pattern, as an engineering type framework, or equivalently, as an idealized molecular framework. Other work has shown that for finite frameworks, introducing symmetry modifies the previous general counts, and under some circumstances this symmetrized Maxwell type count can predict added finite flexibility in the structure.In this paper we combine these approaches to present new Maxwell type counts for the columns and rows of a modified orbit matrix for structures that have both a periodic structure and additional symmetry within the periodic cells. In a number of cases, this count for the combined group of symmetry operations demonstrates there is added finite flexibility in what would have been rigid when realized without the symmetry. Given that many crystal structures have these added symmetries, and that their flexibility may be key to their physical and chemical properties, we present a summary of the results as a way to generate further developments of both a practical and theoretic interest.     

Periodic motions of coupled simple pendulums with periodic disturbances

Several theorems on the existence of oscillatory, rotary, and mixed periodic motions of n, coupled simple pendulums are proved. These theorems are very general and include subharmonic and ultraharmonic type solutions as well as harmonic type solutions and cover cases of large coupling and disturbances as well as small. The results are also extended to include more general systems of nonlinear oscillators.     

Extensional motions of spatially periodic lattices

The behavior of microrheological models for multiphase fluids that have spatially periodic structure depends on certain kinematic properties of the unit cell. Anomalous results associated with identical objects approaching too closely during the flow can be reduced if not eliminated by satisfying lattice compatibility conditions. This is straightforward for simple shearing flow but subtle for extensional flows. Using the connection between lattice compatibility and lattice reproducibility (periodic lattice behavior with the flow) we establish sufficient conditions for compatibility of arbitrary lattices in planar extensional flow. Detailed results for square and hexagonal unit cells include: initial orientations for periodic behavior; strain periods; and minimum lattice spacings D. We identify the orientation of a square unit cell that leads to periodic behavior (with the minimum period) and the largest D of any lattice in planar extensional flow. We show that no lattice exhibits periodic behavior in uniaxial extensional flow (or biaxial extensional flow) even though Adler & Brenner have established the existence of compatibility.     

On periodic motions of lattices of Toda type via critical point theory

Communicated by P. Rabinowitz     

On steady motions of a rigid body bearing a two-degree-of-freedom control momentum gyroscope with precession axis arbitrarily oriented in the carrying body

Steady motions of a rigid body with a control momentum gyroscope are studied versus the gimbal axis direction relative to the body and the magnitude of the system angular momentum. The study is based on a formula that gives a parametric representation of the set of the system steady motions in terms of the rotation angle of the gimbal. It is shown that, depending on the values of the parameters, the system has 8, 12, or 16 steady motions and the number of stable motions is 2 or 4.     

Two- and three-dimensional motions of a body controlled by an internal movable mass

Nonlinear Dynamics - The prescribed movement of a rigid body can be achieved by means of auxiliary internal masses performing special motions relative to the body. This principle of movement is...     

Stability of periodic motions of quasilinear systems

The stability of linear and quasilinear systems with periodic coefficients is analyzed. The properties of stability are established in terms of matrix-valued Lyapunov functions. An algorithm is developed to set up matrix-valued Lyapunov functions for linear quasiperiodic systems. A numerical example is given to illustrate the application of the algorithm Translated from Prikladnaya Mekhanika, Vol. 44, No. 10, pp. 101–113, October 2008.     

On the dynamics of the rigid body with a fixed point: periodic orbits and integrability

The aim of the present paper is to study the periodic orbits of a rigid body with a fixed point and quasi-spherical shape under the effect of a Newtonian force field given by different small potentials. For studying these periodic orbits, we shall use averaging theory. Moreover, we provide information on the $\mathcal{C}^{1}$ -integrability of these motions.     

Development of vortical motions of an incompressible fluid in the cavity of a rotating body

Vortical nonstationary viscous incompressible flows in the space between coaxial cylinders or hemispherical segments rotating with a constant angular acceleration about a stationary axis of symmetry are analyzed numerically for Reynolds numbers Re — 1–10. It is shown that laminar circulating motions are realized. Two vortices form in the flow. The positions of these vortices depend substantially on the geometry of the rotating cavity.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 2, pp. 47–52, March–April, 1995.