The local boundedness of the perturbed motions of a gyroscope in gimbals with dissipative and accelerating forces

The motion of an unbalanced gyroscope in gimbals in a central Newtonian field of forces is considered, taking the masses of the suspension rings into account. It is assumed that there is a moment of forces of viscous friction acting on the axis of rotation of one of the rings, and there is an accelerating (electromagnetic) moment applied to the axis of rotation axis of the other ring. The equations of motion have a partial solution such that the mean velocity of the outer ring is perpendicular to the direction from the centre of gravitation S to the stationary point O, the middle plane of the inner ring contains this direction, and the gyroscope rotates about SO with an arbitrary constant angular velocity.     

The influence of dissipative and constant forces on the form and stability of steady motions of mechanical systems with cyclic coordinates

Mechanical systems with cyclic coordinates subject to dissipative forces with complete dissipation and constant forces applied only to the cyclic variables are considered. Problems of the existence of steady motions in such systems and the conditions for their stability are discussed. It is shown, in particular, that if the Rayleigh function is proportional to the kinetic energy, the stability conditions for the steady motions of the system are the same as or (under certain assumptions) similar to such conditions for steady motions of a corresponding conservative system. The example of a physical pendulum is used to show that such conclusions are generally false: dissipative and constant forces may cause destabilization of stable motions of the system.     

Synchronization and anti-synchronization coexist in two-degree-of-freedom dissipative gyroscope with nonlinear inputs

This study demonstrates that synchronization and anti-synchronization can coexist in two-degree-of-freedom dissipative gyroscope system with input nonlinearity. Because of the nonlinear terms of the gyroscope system, the system exhibits complex motions containing regular and chaotic motions. Using the variable structure control technique, a novel control law is established which guarantees the hybrid projective synchronization including synchronization, anti-synchronization and projective synchronization even when the control input nonlinearity is present. By Lyapunov stability theory with control terms, two suitable sliding surfaces are proposed to ensure the stability of the controlled closed-loop system in sliding mode, and two variable structure controllers (VSC) are designed to guarantee the hitting of the sliding surfaces. Numerical simulations are presented to verify the proposed synchronization approach.     

On the stability of dissipative systems with circulatory forces

The stability of dissipative systems with circulatory forces is revisited. It is found that a general statement recently made by J. Teichmann [6] is erroneous and the validity of a Thompson-Tait-like statement can be derived for less severe conditions (semi-autonomous systems) than those of P. C. Müller [5].

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Zusammenfassung Die Stabilität eines dissipativen Systems mit zirkulatorischen Kräften wurde unterscht. Es stellt sich heraus, dass eine allgemeine Aussage von J. Teichmann [6] falsch ist, und dass die Gültigkeit einer Stabilitätsaussage in der Art von Thompson-Tait für eine weniger streng (halbautonome Systeme) Bedingung als in P. C. Müller [5] abgeleitet werden kann.

     

Asymptotic properties of the motions of a heavy gyroscope with internal dissipation

The limiting motions of a heavy gyroscope, simulated by a system of rigid bodies, are considered when there is internal friction. The whole set of limiting motions is determined and the nature of their stability is studied in detail for cases when the carried body of the gyroscope has a) three degrees and b) one degree of freedom with respect to the supporting body. The results of an analysis of case a are extended to the motion of a gyroscope with a fluid filling. For case b, the values of the parameters are determined for which the gyroscope, apart from steady rotations, has unsteady limiting motions that are integrable motions in the special Bobylev-Steklov case.     

The dynamics of the weakly perturbed motion of a liquid-filled gyroscope and the control problem

The Cauchy problem for the motion of a dynamically symmetrical rigid body with a cavity, filled with an ideal liquid, which is perturbed from uniform rotation, is considered in a linear formulation. The problem of the simultanious solution of the equations of hydrodynamics and the mechanics of a rigid body is reduced to the solution of an eigenvalue problem which depends solely on the geometry of the cavity and the subsequent integration of a system of differential equations.     

Multigrid method as an accelerating procedure for solving systems of linear algebraic equations with a dissipative matrix

Rostov State University. Translated from Matematicheskoe Modelirovanie, Published by Moscow University, Moscow, 1993, pp. 45–52.     

The steady motions of a satellite with a two-degree-of-freedom powered gyroscope in a central gravitational field and their stability

The positions of relative equilibrium of a satellite carrying a two-degree-of-freedom powered gyroscope, in the axes of the framework of which only dissipative forces can act, are investigated within the limits of a restricted circular problem. For the case when the “satellite - gyroscope” system possesses the property of a gyrostat and the axis of the gyroscope frame is directed parallel to one of the principal central axes of inertia of the satellite, all the equilibrium positions are found as a function of the magnitude of the angular momentum of the rotor. It is established that the minimum number of equilibrium positions is equal to 32 and, in certain ranges of values of the system parameters, it can reach 80. All the positions satisfying the sufficient conditions for stability are also determined. The number of them is either equal to 4 or 8 depending on the values of the system parameters.     

The local index density of the perturbed de Rham complex

Czechoslovak Mathematical Journal - A perturbation of the de Rham complex was introduced by Witten for an exact 1-form Θ and later extended by Novikov for a closed 1-form on a Riemannian...     

Analysis of the set of trajectories of nonlinear dynamics: Stability and boundedness of motions

For a set of differential equations with the Hukuhara derivative, we obtain sufficient conditions for various types of boundedness of trajectories and stability of the set of stationary solutions. To this end, we use scalar and vector Lyapunov functions constructed on the basis of an auxiliary matrix-valued function.     

Asymptotically uniform motions of a gyrostat about an inclined axis under potential and gyroscopic forces

The sufficient conditions for asymptotically uniform motions about an inclined axis to exist in the problem of the motion of a gyrostat under potential and gyroscopic forces are obtained using Lyapunov's first method.     

Uniform boundedness in time and large time behavior of weak solutions to a kind of dissipative system

This paper deals with the uniform boundedness (as well as the existence) and large time behavior of the weak entropy solutions to a kind of compressible Euler equation with dissipation effect. The existence and uniform boundedness in time of weak solutions are proved by using the Lax-Friedrichs scheme and compensate compactness. Time asymptotically, the density is showed to satisfy a kind of nonlinear Fokker-Planck equation and the momentum obeys to the Darcy’s law. As a by product, the exponentially decay rate is obtained.     

The effect of dissipative and gyroscopic forces on the stability of a class of linear time-varying systems

It is shown that the effect of dissipative and gyroscopic forces on a certain class of potential linear time-varying system differs considerably from the effect of these forces on a time-invariant system. Examples are considered. In the first of these, the equations of motion of a disk, attached to a rotating weightless elastic shaft, are investigated, taking external friction into account. The results obtained are compared with the results obtained previously by others when considering this problem. In the second example, certain problems of the stability of rotation of a Lagrange top on a base subjected to vertical harmonic vibrations are investigated.     

The existence proof for forced oscillations by adding dissipative forces in the example of a spherical pendulum

Theoretical and Mathematical Physics - We consider a generalization of Whitney’s problem of periodic motion of an inverted spherical pendulum in the presence of a horizontal periodic force,...     

Necessary and sufficient conditions for the boundedness of fractional maximal operators in local Morrey-type spaces

The problem of the boundedness of the fractional maximal operator _Mα$M_{\alpha }$, 0<α<n$0<\alpha <n$, in local and global Morrey-type spaces is reduced to the problem of the boundedness of the Hardy operator in weighted _Lp$L_p$-spaces on the cone of non-negative non-increasing functions. This allows obtaining sharp sufficient conditions for the boundedness for all admissible values of the parameters. Moreover, in case of local Morrey-type spaces, for some values of the parameters, these sufficient conditions coincide with the necessary ones.