Nonlinear Stability Theory and Cable Car Dynamics

01.07.2002 - 30.06.2005
Research funding project
In continuation of the FWF project P13131-MAT we propose the application of concepts and methods of Nonlinear Stability theory, well established in Applied and Numerical Mathematics, to the analysis and the control of oscillations of the cables of cable lifts and cable cars. The aim of this project is two-fold. First, to supply results which should help to achieve a better design and safer operation of such systems. Second, to transfer important concepts of Nonlinear Stability theory into practical engineering applications. Whereas the focus in the previous project P13131-MAT was on analytical methods, we shift in this project more to numerical methods. The reason for this shift is that we want to use practically more realistic mechanical models. This more accurate modelling will result in infinite dimensional mathematical models, described by nonlinear partial differential equations. The following methods and concepts: 1. Methods of dimension reduction of infinite dimensional systems by Galerkin methods, 2. Control of infinite dimensional systems, 3. Numerical methods for dynamic bifurcation problems of infinite dimensional systems, will be applied to the analysis of the dynamics of cable lifts and cable cars, which are practically very important technical systems for the manufacturing and tourism industry in Austria. A careful modelling of the technical system will result in a set of coupled nonlinear ordinary and partial differential equations. All methods proposed for their analysis are strongly interrelated, because for the control problem and the bifurcation analysis of the infinite dimensional system, in general, first a dimension reduction by nonlinear or linear Galerkin methods must be performed. To suppress undesired oscillations of the cable of ski-lifts after a Hopf bifurcation (flatter instability) or due to external excitations requires the application of methods of bifurcation theory. They mostly will be numerical, because a stability problem of a strongly nonlinear basic state, the moving cable, which is a relative equilibrium must be treated. For the calculation of periodic or transient cable motions the focus will be on the numerical integration of the stiff system of cable equations. For this problem and the suppression of cable oscillations by feed-back control, the experience of the applicants with their successful treatment of the dynamics of tethered satellite systems, which is a very similar problem, will be very useful. However, the cable lift problem is more complicated due to the distributed discrete masses along the moving cable.

People

Project leader

Institute

Grant funds

  • FWF - Österr. Wissenschaftsfonds (National) Austrian Science Fund (FWF)

Keywords

GermanEnglish
Nichtlineare Partielle DifferentialgleichungenNonlinear partial differential equations
VerzweigungstheorieBifurcation theory
Steife SystemeStiff systems
DimensionsreduktionDimension reduction

Publications