Plenary Lecture

Finite Element Simulation of the Dynamics of
Tethered Satellite Systems

Professor Hans Troger
Co-authors: Georg Wiedermann, Martin Krupa, Alois Steindl
Institute of Mechanics and Mechatronics
Technische Universitat Wien, Austria
E–mail: hans.troger@tuwien.ac.at

Abstract: Tethered satellite systems, that is two or more satellites in orbit, connected by thin cables up to a length of 100 km, have gained great importance over the past two decades and are now a well established new space technology with great application potential for future space missions ([1]). In the period of preparation of the two tether experiments during the Space Shuttle flights in 1992 and 1996 many practical and theoretical investigations have been carried out, best documented in [2], [3] and [4].
There exist numerous important practical applications, ranging from energy production, making use of the magnetic field of the Earth, to orbit raising or deorbiting of satellites just by cutting the tether. Some have been already tested in several flights in orbit around the Earth. These practical applications of tethered satellite systems follow from an important property of the system in orbit. If the system’s mass center is moving on a circular orbit, the system possesses a local vertical configuration. For constant tether length and if the tether is not too long this vertical configuration is a stable relative equilibrium of the system. Of course, theoretically this requires a perfectly spherical Earth. Within technical accuracy this assumption can be considered to be fulfilled.
To start and to finish a mission, deployment and retrieval of a subsatellite from a mother spaceship moving on a circular orbit must be performed and is a delicate operation because both processes lead to unstable motions with respect to the stable radial relative equilibrium of such a system for constant tether length. Therefore simulation tools for the implementation of stabilizing control are needed. In the literature various different formulations of the equations of motion are given. In [1] they are derived from balance principles, whereas we also derive them from a variational principle, yielding them in weak form, which is especially well suited for the numerical discretisation by Finite Elements.
Basically, the equations of motion of a tethered satellite system are given by a set of coupled partial and ordinary differential equations, which due to large displacements from the local vertical are strongly non-linear and, especially important for the numerical integration, the equations have the mathematically unpleasant property to be stiff. Under a stiff system of differential equations one usually understands a system, the solution of which consists of a smooth slowly changing part and a transient part which is fast changing with large gradient. If the system is damped, this transient part approaches the smooth part very rapidly. Obviously, motions are present in such a system which are changing on different time scales. For the discretization by means of Finite Elements we introduce first, following [5] a moving, but non-rotating reference frame, which moves on a prescribed orbit in the neighborhood of the system. We only require that the moving reference frame stays close enough to the system to avoid numerical problems due to small changes of large quantities. Hence the problem is split into two parts, describing the so-called nearfield and farfield dynamics. Following [5] we perform, first, a discretization in time. This reverse order of discretisation compared with the traditional approach is motivated in [5] by the argument that proceeding this way one is able to avoid having the details of the spatial discretisation obscure the time integration of finite rotations which has to be performed for the motion of the endbodies. The discretisation in time results in a nonlinear functional that is solved by means of the Newton-Raphson-method by a sequence of linearized problems. To calculate a step in the Newton Raphson method we have to discretize all quantities which are still continuous in space. This discretization is performed by means of Finite Elements. The continuous tether is divided into a number of elements and the displacement inside an element is interpolated by ansatz-functions. Inserting cubic ansatz-functions and calculating the projections we finally obtain a system of linearized equations. The structure of the matrix of the linear system is of the form of an arrow matrix, where the arrow shape is a consequence of the changing mass composition of the system due to deployment or retrieval. For constant tether length the arrow shape disappears and the matrix has block diagonal form. In this paper we want to discuss a selected number of interesting problems which arise in the treatment of such systems. We start with modelling. Then we focus on some problems related to the variational formulation of the equations of motion. Special attention is devoted to the numerical solution procedure by Finite Elements. We compare the usually used displacement coordinates with an alternative set of variables called Minakov’s variables introduced in [1] which turn out to be superior for the effective numerical integration of the mathematically stiff system of partial and ordinary differential equations if the tether is physically very stiff.
Finally we shortly consider a string moving with geostationary angular velocity in its radial relative equilibrium configuration around the Earth, reaching from the surface of the Earth far beyond the geostationary height ([6]). If properly done it could be used as track for an Earth to space elevator. Besides the question of feasibility from a technological point of view, which is answered positive by Carbon nanotubes also the question concerning the stability is relevant.

Brief Biography of the Speaker:
Date and place of birth: 1943, Villach, Austria; June 1966 Dipl.Ing. Mechanical Engineering at TU-Vienna; 1970 Dr.techn. (PhD) at TU-Vienna; 1970–1979 Assistant professor at TU-Vienna; since 1979 Full Professor for Mechanics at TU Vienna; 1985–1987 Dean of the Faculty of Mechanical Engineering. 1990 Honorary Doctorate (Dr.h.c.) of the Technical University of Budapest; 2000 Schrodinger prize awarded by the Austrian Academy of Sciences; 2002 Elected as Member of the Austrian Academy of Sciences
Member in Advisory Board of 5 International Scientific Journals, 200 research papers including review articles and book chapters, 7 books (editor), 2 books (author).