Hans Troger

Nonlinear stability and bifurcation theory

There are now well over fifty books available on nonlinear science and chaos theory. In the past year alone, six new technical journals appeared in these areas. (Some of them may even survive). Much of the activity has been in the physics and mathematics communities. Acknowledging the latter in particulra, the authors adddress mechanicians and engineers. They hope to explain to a reader, who is assumed to possess only the minimum of mathematical background acquired by undergraduate courses, how to solve in a straightfowward manner, nonlinear stability problems. They also believe that (the problems they treat) should be understandable also for readers with little or even no knowledge in mechanics. The objects addressed are nonlinear, ordinary, and partial differential equations and iterated mappings arising as models of beams, plates, shells, linkages, railroad trucks, and the like. The book concentrates on local bifurcation and stability analysis: the problem of describing static and dynamic behaviors at parameter and state variable values near those at which loss of stability first occurs from a known branch of solutions. Global behaviors such as chaos and strange attractors are not discussed.

Methods for dimension reduction and their application in nonlinear dynamics

Abstract We compare linear and nonlinear Galerkin methods in their efficiency to reduce infinite dimensional systems, described by partial differential equations, to low dimensional systems of ordinary differential equations, both concerning the effort in their application and the accuracy of the resulting reduced system. Important questions like the choice of the form of the ansatz functions (modes), the choice of the number m of modes and, finally, the construction of the reduced system are addressed. For the latter point, both the linear or standard Galerkin method making use of the Karhunen Loeve (proper orthogonal decomposition) ansatz functions and the nonlinear Galerkin method, using approximate inertial manifold theory, are used. In addition, also the post-processing Galerkin method is compared with the other approaches.

Optimal Control of Deployment of a Tethered Subsatellite

One of the most important operations during a tethered satellite system mission is the deployment of a subsatellite from a space ship. We restrict tothe simple but practically important case that the system ismoving on a circular orbit around the Earth. The main problem duringdeployment due to gravity gradient is that the two satellites do not move along the straight radial relative equilibrium position which is stable for a tether of constant length. Instead, deploymentleads to an unstable motion with respect to the radial relativeequilibrium configuration. Therefore we introduce an optimal control strategy using theMaximum Principle to achieve a force controlled deployment of the tethered subsatellite from the radial relative equilibrium position close to the space ship to the radial relative equilibrium position far away from the space ship.

Various Methods of Controlling the Deployment of a Tethered Satellite

The deployment of a subsatellite from a mother spaceship moving on a circular orbit is a delicate operation for a tethered satellite system because this process leads to an unstable motion with respect to the stable radial relative equilibrium of such a system if the tether length is kept constant. Therefore, simulation tools for the implementation of stabilizing control are needed. Usually linear control, for example Kissels law, is implemented. In this paper, we introduce an optimal control strategy to simulate the force controlled deployment of a tethered satellite from a spaceship. We compare this strategy with free deployment, deployment controlled by Kissels law and an approach making use of the concept of “targeting” used in the controlling chaos approach.

On the paradox of the free falling folded chain

SummaryIt is shown both by experiment and also by numerical simulation that for a vertically hanging folded chain the free part, if released, is falling faster than a free falling body under gravitational acceleration. A qualitative explanation of thisparadoxical phenomenon is given by showing that a downpulling force at the fold is created. In the simulation this force is also calculated quantitatively.

Stability of Relative Equilibria. Part II: Dumbell Satellites

In the second part a practically important problem, namely the stability of relative equilibria of a dumbell satellite on an orbit around the Earth is treated by means of the reduced energy-momentum method. The dumbell satellite is used to emphasize the advantages of the reduced energy-momentum method which did not become obvious in the simple example of the rotating pendulum treated in Part I, as well as, to discuss some of the finer technical details.

One and Two-Parameter Bifurcations to Divergence and Flutter in the Three-Dimensional Motions of a Fluid Conveying Viscoelastic Tube with D4-Symmetry

The loss of stability of the trivial downhanging equilibrium position of a slender circular tube conveying incompressible fluid flow is studied. The tube is clamped at its upper end and free at its lower end. Inbetween the three-dimensional transversal motion is constrained by an elastic support which is considered to be D 4-symmetric, that is, has the symmetry of the square (Figure 1). Kirchhoff’s rod theory and the Kelvin-Voigt viscoelastic law are used to derive the tube equations under the assumption of large displacement but small strain.

Optimal Control of Retrieval of a Tethered Subsatellite

The most important and complicated operations during a tethered satellite system mission are deployment and retrieval of a subsatellite from or to a space ship. The deployment process has been treated in [15]. In this paper retrieval is considered. We restrict to the practically important case that the system is moving on a circular Keplerian orbit around the Earth. The main problem during retrieval is that it results in an unstable motion concerning the radial relative equilibrium which is stable for a tether of constant length. The uncontrolled retrieval results in a strong oscillatory motion. Hence for the practically useful retrieval of a subsatellite this process must be controlled. We propose an optimal control strategy using the Maximum Principle to achieve a force controlled retrieval of the tethered subsatellite from the radial relative equilibrium position far away from the space ship to the radial relative equilibrium position close to the space ship.

Relative equilibria of tethered satellite systems and their stability for very stiff tethers

We study the existence and stability of the relative equilibria of systems of two satellites joined by a tether. Since tethers used in practice are very stiff we consider a stiff tether as a perturbation of an inextensible tether. We show that the equations for relative equilibria and the stability conditions are continuous as stiffness approaches infinity and limit on the equations and conditions relevant to an inextensible tether. We obtain a numerical bifurcation diagram for a class of relative equilibria in the case of an inextensible tether.

Generic one-parameter bifurcation in the motion of a simple robot

In the car manufacturing industry, painting and polishing work is mostly done by robots because these tasks are quite hazardous to the health of human workers, and also because they do not require a very sophisticated robot. In general the industrial product (car) passes by the robot with constant speed, and the endpoint of the robot where the painting device, usually a spray pistol, is mounted, performs a prescribed time periodic motion. For such a painting process stability problems can become important if parameters are varied. Two cases should be mentioned as examples. Firstly, if the speed of the assembly line is increased, the endpoint of the robot must also move faster and hence a loss of stability of the basic periodic motion can occur for fixed parameters of the controller. Secondly, a change of the spray pistol to one with different mass again is a variation of a parameter which could lead to an instability.