An investigation of the complete post-buckling behavior of axisymmetric spherical shells

Summary The post-buckling behavior of an elastic spherical shell is studied for large axisymmetric deformations. The complete post-buckling path is given for the experimentally confirmed single dimple solution in a load-deformation diagram, making use of the methods of local bifurcation theory, singular perturbation analysis and numerical analysis. From the specific form of the post-buckling path with its predominating unstable part follows the strong imperfection sensitivity of the shell structure.

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Zusammenfassung Es wird das Nachbeulverhalten einer elastischen Kugelschale für große achsensymmetrische Deformationen untersucht. Der gesamte Nachbeulpfad wird für die experimentell bestätigte single dimple Lösung, die eine starke lokale Eindellung darstellt, angegeben, wobei Methoden der lokalen Verzweigungstheorie, der singulären Störungsrechnung und der numerischen Analysis verwendet wurden. Aus der besonderen Form des Nachbeulpfades mit seinem überwiegend instabilen Teil ist die starke Imperfektionsempfindlichkeit dieser Konstruktion deutlich erkennbar.

     

Zur korrekten Modellbildung in der Dynamik diskreter Systeme

Übersicht Zur korrekten Modellbildung bei linearen, autonomen, dynamischen Systemen wird für die Ermittlung der richtigen Anzahl von Parametern die Theorie der Verzweigungsdiagramme von Matrizen von V. I. Arnold verwendet. Als Beispiele werden die Doppelpendel mit tangentialer und richtungstreuer Endlast behandelt.

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Summary In order to obtain a correct model of a linear, autonomous, dynamical system with the right number of parameters the theory of bifurcation diagrams of matrices developed by V. I. Arnold is used. As examples two double pendula with a follower force respectively with dead loading are considered.

     

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.     

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.     

Stability of Relative Equilibria. Part I: Comparison of Four Methods

In this first part of the paper, we review methods for the investigation of stability of relative equilibria of symmetric Hamiltonian systems and explain them by means of the model problem of a rotating pendulum. For this example the modern approaches, known as energy momentum methods are compared with stability assessment by linearization and by the classical method of Routh.     

On the dynamics of circulating monocable aerial ropeways

We investigate the dynamics of a hauling rope modelled as an inextensible completely flexible string and calculate the first natural frequencies. Furthermore we discuss the sensitivity of the model on the operating speed of the ropeway. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On stability problems of the space elevator

A long dumb-bell satellite in the radial configuration represents a simple model of a Space Elevator. The dumb-bell moves with the angular velocity of the planet, but not relative to it. To calculate the stability of such a relative equilibrium we used the reduced energy-momentum method. We considered dumb-bell satellite models with rigid and elastic, but massless, rods. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the equations of motion of the folded inextensible string

In a number of excellent, even recently published, books in Theoretical Mechanics the equations of motion of a folded inextensible string, for example to explain the crack of a whip, are derived using either the energy principle, if a system of one degree of freedom is considered, or Lagrange's equations for conservative systems, if a system of more than one degree of freedom is studied. However, it will be shown in this paper that the resulting equations are incorrect because physically a nonconservative system is given. Hence, both methods mentioned above based on energy conservation must not be used. A correct derivation of the equations of motion is given by means of the linear momentum balance.     

A targeting strategy for the deployment of a tethered satellite system

^** Email: hans.troger{at}tuwien.ac.at One of the most important operations during a tethered satellitesystem mission is the deployment of a subsatellite from a spaceship.We restrict to the simple, but practically important case thatthe system is moving on a circular Keplerian orbit around theEarth. The main problem during deployment due to the local gravitygradient vector is that due to the Coriolis acceleration thetwo satellites do not move along the straight radial relativeequilibrium position in the orbital frame. Instead, a strongdeviation from the local vertical direction occurs, which afterthe deployment process is finished results in weakly dampedlarge-amplitude oscillations, which in some cases are even transientchaotic. This chaotic dynamics will be used to steer the satellitewith small control actions into the final radial relative equilibriumposition far away from the spaceship. Both deployment time andenergy input are computed and compared to other deployment strategies.