Libration suppression of tethered space system with a moving climber in circular orbit

This paper studies the dynamics and libration suppression of a tethered system with a moving climber in circular orbits. The tethered system is modeled by a two-piece dumbbell model that consists of one main satellite, one climber and one end-body connected by two straight, massless and inextensible tethers. A new tension control strategy to suppress the libration of the tethered system due to the moving climber is proposed by reeling in-out tether at the end-body without thrust. The control strategy is implemented with the sliding mode control to suppress the libration angle of the climber to zero by the end of climber’s transfer phase. The numerical results show that the proposed control strategy is very effective in suppressing the libration of the climber in the three-body tethered system with tension control only. Furthermore, cases with limited tension control are examined. It reveals that a longer tether between the climber and the end-body is required to supplement the limited tension in suppressing the libration of the climber.     

Studying the stability of equilibrium solutions in a planar circular restricted four-body problem

The Newtonian circular restricted four-body problem is considered. We obtain nonlinear algebraic equations determining equilibrium solutions in the rotating frame and find six possible equilibrium configurations of the system. Studying the stability of equilibrium solutions, we prove that the radial equilibrium solutions are unstable, while the bisector equilibrium solutions are stable in Lyapunov’s sense if the mass parameter satisfies the conditions μ ∈ (0, μ₀, where μ₀ is a sufficiently small number, and μ ≠ μ_j, j = 1, 2, 3. We also prove that, for μ = μ₁ and μ = μ₃, the resonance conditions of the third order and the fourth order, respectively, are satisfied and, for these values of μ, the bisector equilibrium solutions are unstable and stable in Lyapunov’s sense, respectively. All symbolic and numerical calculations are done with the Mathematica computer algebra system. Published in Neliniini Kolyvannya, Vol. 10, No. 1, pp. 66–82, January–March, 2007.     

Three-dimensional stability theory of deformable bodies. Stability of construction elements

Conclusion In [8, 9] and in the present paper we analyzed the possibilities of using the approximate approach [15, 18] in the three-dimensional stability theory of deformable bodies as applied to effects of internal and surface instability and to stability of thinwalled structural elements. The analysis mentioned has been performed by comparing for standard problems the results obtained by the approximate approach [15, 18] with the results for the similar problems, obtained within the three-dimensional linearized stability theory of deformable bodies (for example [2–5, 7, 10, 19]), constructed with the accuracy usually adopted in mechanics. The following conclusions are drawn as a result of the analysis.Applied to effects of internal and surface instability, the approximate approach leads to result in disagreement with the corresponding results of the three-dimensional linearized stability theory of deformable bodies.As applied to the study of stability of thin-walled structural elements, the use of the approximate approach is justified if we restrict ourselves to a calculational accuracy of critical loads corresponding to that of the Kirchhoff-Love hypothesis.In connection with the discussion above, numerous publications carried out on the basis of the approximate approach require further study to clarify the validity limits of the results obtained.Institute of Mechanics, Academy of Sciences of the Ukrainian SSR, Kiev. Translated from Prikladnaya Mekhanika, Vol. 22, No. 2, pp. 3–17, February, 1986.     

Stability of equilibrium of a double-layer system with a deformable interface and a prescribed heat flux on the external boundaries

The stability of mechanical equilibrium of a system of two horizontal immiscible-liquid layers with similar densities is studied. The problem is solved for a prescribed heat flux on the external boundaries. Within the framework of a generalized Boussinesq approximation, which takes the interface deformation correctly into account, the onset of convection caused by heating the system from above or below is considered. Two long-wave instability modes attributable to the presence of the deformable interface and the given heat flux on the external boundaries are detected. The system response to monotonic and oscillatory disturbances with finite wavelengths is investigated. A complete stability map is constructed.     

Hamiltonian structure for rigid body with flexible attachments in a circular orbit

Hamiltonian structure of a rigid body in a circular orbit is established in this paper. With the reduction technique, the Hamiltonian structure of a rigid body in a circular orbit is derived from Lie-Poisson structure of semidirect product, and Hamiltonian is derived from Jacobi's integral. The above method can be generalized to establish the Hamiltonian structure of a rigid body with a flexible attachment in a circular orbit. At last, an example of stability analysis is given. The project supported by National Natural Science Foundation of China and Aeronautic Science Foundation.     

Elastoplastic equilibrium of a weighable isotropic half-plane with a circular hole

The elastoplastic state of a weighable isotropic half-plane with a circular hole is studied. Complex Kolosov-Muskhelishvili functions which describe the elastic state of the half-plane are constructed. The unknown interface between the plastic and elastic regions is studied with allowance for the single-valuedness of the elastic displacements. The problem is also solved by the small-parameter method, and the two solutions are compared. Institute of Applied Mathematics and Mechanics, National Academy of Sciences of Ukraine, Donetsk, Ukraine. Translated from Prikladnaya Mekhanika, Vol. 35, No. 3, pp. 93–98, March, 1999.     

Stability of plane oscillations of a satellite in a circular orbit

We study motions of a rigid body (a satellite) about the center of mass in a central Newtonian gravitational field in a circular orbit. There is a known particular motion of the satellite in which one of its principal central axes of inertia is perpendicular to the orbit plane and the satellite itself exhibits plane pendulum-like oscillations about this axis. Under the assumption that the satellite principal central moments of inertia A, B, and C satisfy the relation B = A + C corresponding to the case of a thin plate, we perform rigorous nonlinear analysis of the orbital stability of this motion.In the plane of the problem parameters, namely, the oscillation amplitude ε and the inertial parameter, there exist countably many domains of orbital stability of the satellite oscillations in the linear approximation. Nonlinear orbital stability analysis was carried out in thirteen of these domains. Isoenergetic reduction of the system of equations of the perturbed motion is performed at the energy level corresponding to the unperturbed periodic motion. Further, using the algorithm developed in [1], we construct the symplectic mapping generated by the equations of the reduced system, normalize it, and analyze the stability. We consider resonance and nonresonance cases. For small values of the oscillation amplitude, we perform analytic investigations; for arbitrary values of ε, numerical analysis is used.Earlier, numerical analysis of stability of plane pendulum-like motions of a satellite in a circular orbit was performed in several special cases in [1–4].     

Problem of identifying unidimensional deformable elements in a dynamic regime

Northwest Polytechnic Correspondence Institute, Leningrad. Translated from Prikladnaya Mekhanika, Vol. 24, No. 8, pp. 73–79, August, 1988.     

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...     

Toward a theory of circular rolling of three-component elastically deformable systems

International Applied Mechanics -     

Characteristics of the movement of a controlled mechanical system with elastically connected mobile elements

Institute of Technical Mechanics, Academy of Sciences of the Ukrainian SSR, Dnepropetrovsk. Translated from Prikladnaya Mekhanika, Vol. 26, No. 7, pp. 71–76, July, 1990.