Stability and Control of Nonlinear Systems
Nowadays, many engineering systems like machines and load bearing structures are operated in the nonlinear regime, which may result in catastrophic failure due to a loss of stability, unwanted large deformations or a snapping into an unwanted operating regime. In order to ensure their proper functionality in the nonlinear regime, these phenomena must be avoided at all times. Therefore, we are conducting research concerned with fundamental aspects of nonlinear stability theory, but also with the translation of theoretical results into practical problems in the field of structural stability.
The analysis of the stability of solutions of nonlinear dynamical mechatronical systems and their sensitivity with respect to disturbances and uncertainties is one of the main research topics the Research Group is active in. By means of nonlinear stability theory, we can conclude on the post-critical behaviour of the mechatronical system. By reducing the dynamical system to its active modes and applying normal form theory, new solution types and their stability can be determined. This method applies to static problems, like beams, plates and shells under compression, and also to dynamical systems, like periodically forced or self-excited structures. For symmetrical or optimised structures, multiple eigenvalues usually occur at the stability boundaries, causing rather complicated post- critical solution scenarios and even chaotic behaviour; to treat such systems, we apply equivariant bifurcation theory. Another important topic is dimension reduction. For realistic models, the degrees of freedom can be very large, prohibiting e.g. the design of control strategies. To overcome these difficulties, we use dimension reduction, with which the dynamics is reduced to the governing modes and reliable small-scale models can be constructed, which render the system treatable and the methods of nonlinear stability theory applicable.
Structural Stability and Control
We also consider the translation of theoretical results on stability and control of nonlinear systems towards the practically important field of structural stability and control; in particular, concerning systems of structural mechanics such as rods, plates and shells. Our main emphasis here is on the study and analysis of buckling problems and on the post-buckling behaviour of these structures as well as on periodically forced or self-excited structures. Our research enables the stabilisation of these structures in the post-buckling regime, the control of their nonlinear vibrations, and the active destabilisation for the sake of e.g. producing high strokes, which would not be possible otherwise. Our results can be applied to many practical engineering problems, which range from the study of fluid conveying tube motion stability to the simulation of self-excited vibrations in transmission belts or rotating structures, to optimal control of tethered satellites and to stabilization of vehicles.