Nonlinear Stability in Engineering

This research project is planned as continuation of the FWF project P10705-MAT in which the mathematically well established methods of Local Equivariant Bifurcation theory have been applied to engineering problems. Again the main focus of this new research project is to transfer concepts of Nonlinear Dynamical System theory which are mathematically well established into engineering applications. We propose the following concepts and methods to be treated in the following years: 1. Dimension Reduction by improved Galerkin methods 2. Application of Conley Index Theor to detect chaotic dynamics 3. Stability investigations of relative equilibria in Hamiltonian systems The application of these concepts and methods will be given for two important technical systems which cover all three fields. These two systems are tethered satellite systems and fluid conveying tubes. Tethered satellites systems are systems of two or more satellites connected by long thin cables and are a very promising new space concept. In continuation of work done before by means of Center Manifold theory where, however, the admissible parameter range often is too small in practical applications, in the section Dimension Reduction the Karhunen Loeve method and the Approximate Inertial Manifold technique should be worked out to improve the classical (flat) Galerkin method usually used in engineering. Recent mathematical research showed that Conley Index theory opens up a new way to prove the existence of chaotic dynamics in a mathematical sound way, where other methods like the Melnikov method are not applicable anymore. The stability investigation of relative equilibria of infinite dimensional Hamiltonian systems is a very complicated field in stability theory. It has important applications in the attitude stability of artificial satellites with flexible parts as this is the case for tethered satellite systems. Here a combination of analytical (Energy Momentum Method) and numerical concepts is necessary in order to obtain practically useful and sharp results. All these methods are mathematically more or less well developed but have not yet penetrated into engineering application. This transformation is the main goal of this project.

The main objective of project P13131-MAT, in continuation of three previous FWF projects P10705-MAT, P7003 and P5519 was the transfer of results obtained in Applied Mathematics in the field of Nonlinear Stability Theory and Dynamical Systems Theory into practical engineering applications. In P13131-MAT the emphasis was on the following three subjects: 1. Methods of Dimension Reduction 2. Chaotic dynamics 3. Stability investigations of relative equilibria in symmetric Hamiltonian systems. These mathematical concepts have been developed for engineering application to two important technical systems: Fluid conveying tubes which can be considered as a model problem in Bifurcation Theory and in the theory of self-excited oscillations. Tethered satellite systems which, for a medium time mission, can be considered as Hamiltonian systems. The mathematically well established fact that often infinite dimensional dynamical systems, even if their dynamics is very complicated (chaotic), can be described by a low finite dimensional nonlinear dynamical system with sufficient accuracy has been worked out in engineering style going beyond Center Manifold Theory, for which the possible parameter variation often is strongly limited. Instead linear (Proper Orthogonal Decomposition) and nonlinear (Approximate Inertial Manifold) Galerkin methods are introduced and shown to be very efficient and also applicable for real practical engineering applications. Concerning special aspects of chaotic dynamics the book: "Chaos and Chance" by A. Berger relating chaotic dynamics to statistic properties of a dynamic process is a major step forward in explaining these complcated concepts to engineers. The motivation for the investigation of the stability of relative equilibria in symmetric Hamiltonian systems came from a practical problem raised during the participation of a group of the Institute of Mechanics of TU-Wien in projects of the European Space Agency (ESA) and INTAS on the dynamics of tethered satellite systems. Such systems consist of two rigid bodies (satellites) connected by a thin cable up to a length of 100 km in orbit around the Earth. The stability problem of trivial and nontrivial configurations of tethered satellite systems on circular orbits, which are so-called relative equilibria, is of great practical interest. Mathematically speaking are relative equilibria, equilibria in a coordinate frame, which is moving with the symmetry group. Due to the symmetry of the system certain quantities are preserved and hence the stability problem becomes more complicated as it is the case for equilibria in non-symmetric Hamiltonian systems. The proper theory is the Reduced Energy Momentum Method (REMM) developed in Applied Mathematics. Its application in engineering has not been broadly implemented, because of the complicated mathematical concepts (Lie-groups, symplectic geometry). We have published several papers, first, explaining in a tutorial paper to engineers in engineering language the advanced mathematics and, second, showing how to apply the REMM to a practically relevant problem.