Newtonian Force Field Research Articles

On periodic motions of a gyrostat in a newtonian force field

The problem of existence of periodic motions of a gyrostat is studied. The gyrostat consists of a rigid body and a rotor, the axis of which is stationary with respect to the rigid body in a central Newtonian force field. In /1,2/, Poincaré's method of small parameter was used to show the existence of periodic motions of a gyrostat with a single fixed point in a central Newtonian force field. It was assumed that the gyrostat differs little from a dynamically symmetric one, and the constant gyrostatic moment was assumed to be sufficiently small. Moreover, it was assumed that the center of gravity of the gyrostat is sufficiently close to the fixed point, and the center of attraction is sufficiently far removed from the gyrostat.

Motion of a visco-elastic sphere in a central newtonian force field

Approximate equations are obtained, describing the evolution of translational-rotational motion of a visco-elastic sphere in a central Newtonian force field,its steady-state motions are found, and their stability is investigated. The motion described can serve as one model of tidal phenomena under the motion of a planet of the solar system /1/. It has been shown previously /2/ that in the presence of energy dissipation a deformable planet tends towards a steady-state rotation around the attracting center, under which the planet's center of mass describes a circle, while its orientation in the orbital axes in unchanged.

On optimal stabilization of gyrostat rotary motion: PMM vol. 43, no. 5, 1979, pp. 779–786

A solution of the problem of optimal stabilization of rotary motion of a gyrostat whose center of mass moves on an elliptic orbit in a central Newtonian force field is derived. A method of successive approximations for the determination of optimal control is established.

Regular precessions of a gyrostatic satellite in a circular orbit

In the present paper, the regular precessions about the mass center are studied, for a ‘Volterra-type’ gyrostatic satellite in a circular orbit under a Newtonian force field. All the possible regular precessions are determined.

The Newtonian potential of a homogeneous cube

Closed form expressions for the Newtonian force field and the Newtonian potential of a homogeneous rectangular parallelepiped were given by MacMillan (1930). Here a new concise derivation by means of multiple summations is presented. Each component of the force is written as a sum of 24 terms involving Artanh and arctan functions, whereas the expression for the potential consists of 48 terms. As a new application, the self-energy is represented in closed form, too.

The principle of least action and periodic solutions in problems of classical mechanics: PMM vol. 40, n≗ 3, 1976, pp. 399–407

The problem of existence of periodic solutions of equations of natural mechanical system motions is considered in the case when region D of all possible motions is bounded. Periodic libration solutions are derived for systems with many degrees of freedom. The trajectory of such solution is diffeomorphic to segment [0, 1], its ends lie at the boundary of D, and the representative point oscillates along that curve. Existence of libration solutions is proved in the case when the region of possible motions is diffeomorphic to the direct product N × [0, 1], where N is a smooth compact manifold. Obtained results are applied in the problem of motion of a solid body with a fixed point in a Newtonian force field.

On the existence conditions for the particular jacobi integral: PMM vol. 40, n≗ 4, 1976, pp. 611–617

For certain nonholonomic and holonomic mechanical systems we have obtained the existence conditions for the particular integral being a linear bundle of the Hamiltonian and the momenta. The conditions are simplified for a certain class of holonomic systems, containing the well-known [1] case of the particular Jacobi integral. An example of the fulfillment of these conditions is a variant of the restricted problem of translation-rotational motion of a gyrostat in a Newtonian force field.

Dynamics of a double rotating system of bodies in a Newtonian force field

Dynamics of a double rotating system of bodies in a Newtonian force field

Some problems relating to the stability of motion of a body of variable composition, with one stationary point, in a newtonian force field

Some problems relating to the stability of motion of a body of variable composition, with one stationary point, in a newtonian force field

On stabilization of the rotational motion of a solid with flywheels in a newtonian force field: PMM vol. 38, n≗4, 1974, pp. 628–635

A solution of the problem of optimal stabilization (in a specific sense) of the rotational motion of a gyrostat (a solid with two flywheels) in a central Newtonian force field is given within the framework of analytical control theory [1].

On periodic motions of a rigid body in a central Newtonian field: PMM vol. 38, n≗2, 1974, pp. 224–227

In the problem of the motion of a rigid body with one fixed point in a central Newtonian force field (in particular, in the de Brun field [1]).The existence of a family of periodic solutions is proved by the Poincaré method of small parameters. It is assumed that the body differs negligibly from a dynamically symmetric one and that its center of gravity is sufficiently close to the fixed point. The proof is carried out by using the techniques of Hamiltonian systems.

Intermediate-thrust arcs and their optimality in a central, time-invariant force field

This paper presents the general equations of the intermediate-thrust arcs in a general, time-invariant, central force field. Two families of planar arcs, namely, the family of Lawden's spirals in the equatorial plane of an oblate planet and the family of intermediate-thrust arcs in a gravitational field of the form μ/r n , have been considered in detail. The Kelley-Contensou condition has been used to test their optimality condition. It is shown that, in the first case, there exist portions of the arcs at a finite distance satisfying the condition, while, in the second case, the entire family satisfies the condition forn ≥ 3. Hence, in a perturbed Newtonian gravitational force field, the intermediate-thrust arcs, under certain favorable conditions, can be part of an optimal trajectory.

On the stability of certain motions of a rigid body with elastic rods and liquid: PMM vol. 36, n≗ 1, 1972, pp. 43–59

We consider the motion of a free rigid body with three pairs of elastic rods and with a cavity containing a liquid, in two cases: in a central Newtonian force field and in the absence of external forces. By an application of Rumiantsev's theorem [1] we obtain sufficient stability conditions for relative equilibrium in a circular orbit and to uniform rotations of this system. We show that the presence of a liquid with a free surface in the cavity and the connection of elastic rods to the body have a destabilizing effect on the stability of the corresponding unperturbed motions of the unaltered system. We also point out sufficient stability conditions in the case when less than three pairs of rods are attached to the body. For a large Young's modulus the stability conditions obtained lead (in the absence of the liquid) to the well-known sufficient conditions for the stability of a rigid body. Stability conditions for the case when one pair of rods is attached to the body and when there is no liquid are compared with the stability conditions obtained in [2, 3]. In connection with the assertion made in [2, 3] regarding the novelty of the method used, we remark that this method was previously developed by Rumiantsev and was applied to the solution of a number of problems on the stability of the steady-state motions of a rigid body with a liquid filling [4].

On Nonholonomic Parametrization

This paper is concerned with the problem of obtaining the differential equation of the trajectory of dynamical system. Time is eliminated by means of some first integral linear with respect to the velocities. As an example, the stability of rotation of a heavy asymmetric rigid body with a fixed point under the action of the Newtonian central force field is considered.

Optimal stabilization of rotation of a gyrostat in the Newtonian force field: PMM vol. 34, n≗5, 1970, pp. 965–972

Optimal stabilization of rotation of a gyrostat in the Newtonian force field: PMM vol. 34, n≗5, 1970, pp. 965–972

On the stability op rotational motion of a variable composition body with a gyroscope in a newtonian force field: PMM vol. 34, n≗3, 1970, pp. 564–566

On the stability op rotational motion of a variable composition body with a gyroscope in a newtonian force field: PMM vol. 34, n≗3, 1970, pp. 564–566

On a mechanical model simulating vibratory motion of a rigid body in a Newtonian central field of forces

On a mechanical model simulating vibratory motion of a rigid body in a Newtonian central field of forces

Stability of the stationary motions of a satellite with a gyroscope in a central gravity field

THE stability of satellites containing rotating parts has been investigated in a large number of papers (for a bibliography, see for example [l, 2]). In papers by the author [3, 4] some stationary motions of a gyroscope mounted in the satellite on a Cardan suspension in a central Newtonian field of force, and sufficient conditions for their stability, have been found. The stationary motions of a system consisting of the housing of a satellite and of a gyroscope mounted in it on a Cardan suspension are investigated below: the centres of mass of the satellite housing, the rotor and the rings of the Cardan suspension, and also of the fixed point of the Cardan suspension are the same. It is assumed that the centre of mass of the system moves in a Keplerian circular orbit and that the angular velocity of the natural rotation of the rotor is constant. Sufficient conditions for stability are obtained. Comparison of the results obtained with the results of [4] shows that allowing for the perturbation of the motion of the satellite housing about the centre of mass of the system does not affect the stability conditions of the gyroscope.

On the Routh theorem and the Chetaev method for constructing the liapunov function from the integrals of the equations of motion: PMM vol. 33, n≗5, 1969, pp. 904–912

The bifurcations of the equilibria of a gyrostat satellite with a centre of mass moving uniformly in a circular Kepler orbit around an attracting centre are investigated. It is assumed that the axis of rotation of a statically and dynamically balanced flywheel rotating at a constant relative angular velocity is fixed in the principal central plane of inertia of the gyrostat containing the axis of its mean moment of inertia and that it is not collinear with any principal central axis of inertia of the system. The problem is solved in a direct formulation, that is, the whole set of equilibria with respect to the orbital system of coordinates of the gyrostat satellite is determined using the given moments of inertia, the value of the gyroscopic moment and the direction cosines of the axis of rotation of the flywheel and the changes in this set are investigated as a function of the bifurcation parameter, that is, the magnitude of the gyrostatic moment of the system. A parametric analysis of the relative equilibria of the three possible classes of equilibria for a system in a circular orbit in a central Newtonian force field is carried out using computer algebra facilities.

On the steady motions op a gyrostat satellite: PMM vol. 33, no. 1, 1969, pp. 127–131

Two families of steady motions of a gyrostat satellite in a central Newtonian force field are considered. The plane of the (circular) orbit of the center of mass of the satelite is biased relative to the attracting center. Sufficient conditions for stability are derived. These motions complement the numerous already familiar [1] steady motions of a gyrostat satellite with the center of the circular orbit coincident with the attracting center. As in the case of the latter motions, the stability conditions in our case differ from those obtained under the restricted formulation of the problem [1] by quantities on the order of l 2/R 2 relative to the principal terms ( l is the characteristic dimension of the satellite, R is the distance from the attracting center). The orbital plane bias is of the order of l 2/R 2 . These quantities are very small indeed when one is dealing with real artificial earth satillites. The present study is carried out by the Routh method with the aid of some results obtained by Rumiantsev [1].