Nonlinear stability theory applied to cable lift dynamics

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.

Cable lifts and cable trains are an important means of transportation in the Austrian Alps. Despite the fact that utmost care is taken for their design, production and operation, every year several accidents occur, some due to human mistakes, but some others also due to a combination of bad design and bad operational conditions resulting sometimes into large undesired oscillations called pumping oscillations. These are not only a problem of comfort for the passengers but can also be a safety problem. The explanation of these pumping oscillations, which can reach large amplitudes, is the main subject of this work. Especially the influence of the main parameters on their occurrence should be given. In this project, supported by FWF, it has been found that these undesired pumping oscillations are not self excited oscillations, that is oscillations that occur without a periodic excitation, but follow from the periodic arrangement of the cabins or chairs, the geometry (sag) of the cable, the arrangement of the towers and the operational speed. It is essential for the explanation of the pumping oscillations to consider more than one rope field between two towers. In this work six rope fields are considered. If the essential parameters - that is the distance between the cabins or chairs, the operational speed, the location of the rolls and the cable sag in adjacent fields - are badly chosen, a resonance may occur resulting in a large amplitude oscillation. Taking the mentioned parameters into account, a simulation model based on the Finite Element program ABACUS was established to simulate the three dimensional rope oscillations. An important aim of this simulation model was to supply a program which allows to identify and to avoid the critical parameter domains already in the phase of design and hence to contribute to a safer operation of such lifts and cable trains.