The conditions of existence of limit cycles in dynamic motion system of related objects on an elliptical orbit
Dynamics, ballistics, movement control of flying vehicles
Moscow Aviation Institute (National Research University), 4, Volokolamskoe shosse, Moscow, А-80, GSP-3, 125993, Russia
Orbital tether system (TS) is separately formed area of the promising technologies. Research work should be combined with experimental design work and provide science-based data (the results of theoretical research of problems of dynamics of flight control laws-deployment-folding design-exploratory research elements of TS) to establish programs for flight and special devices able to complete the target task .
The solution of problems of dynamics of functioning of the TS in elliptical orbits is more complicated than for circular and requires a special theoretical study. The conducted scientific studies of the problems of dynamics of the relative motion of the connected objects [2-8] showed the effectiveness of applying the mathematical apparatus of the qualitative theory of dynamical systems and bifurcation theory [9-11]. The full results of qualitative research provide an opportunity to develop a clear picture of the possible types of phase trajectories of the system, and, consequently, of the characterization of possible trajectories of the controlled motion under all values of the control parameters, under any initial conditions of motion and at any period of time.
When conducting qualitative research of dynamic systems of the controlled motion, the most difficult is the determination of the bifurcations associated with the appearance and disappearance of limit cycles, covering the phase cylinder. In this regard, in this paper we formulated and proved two theorems.
It is analyzed the dynamic system of controlled motion of related objects in the plane of the elliptical orbit without taking into consideration disturbing factors. On the basis of the mathematical apparatus of the qualitative theory of dynamical systems defined by the equilibrium state, conditions for the existence of limit cycles are formulated and proved. The bifurcation values of the control parameter, associated with the appearance and disappearance of limit cycles at the considered dynamical system are determined.
The study of equilibrium states and conditions of the existence of limit cycles of a dynamical system has allowed to establish the number of all types of the qualitative structures of the phase trajectories of the controlled motion of related objects.
The range of values of the relative angular velocity of the continuum limit, covering the phase cylinder, is set.
Keywords:tethered system, equilibrium states, limit cycles
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