Introduction to the research project

This project aims towards the development of new mathematical and experimental techniques to systematically investigate the dependence of vibrations on model parameters (e.g. bearing or material coefficients). This aim is tried reached using different strategies.

When mechanical systems become more complex and contain parts with nonlinear characteristics, vibrations show a complicated dependence on parameters like bearing properties, damping coefficients and geometry. This dependence can include bifurcations, where small changes in parameters (e.g. by fluctuations in production or working environment) might result in dramatic changes in behavior. For investigating such systems, we normally turn to detailed computer models or simplified mathematical models, but in some cases these approaches fail: The computational burden of conducting a parameter study using the computer model might be too big, or the behavior of the physical system might be too complex to allow for a simple mathematical model. In the context of industrial production, time and other resources might also fall short. Thus it is desirable to have efficient methods for experimentally and mathematically determining the nonlinear dependency of vibrations on physical parameters.

From the purely mathematical point of view, techniques for dimension reduction are being developed. These allow to obtain an understanding of the dynamic behavior on a macroscopic scale by disregarding large amounts of unimportant information on the microscopic scale. The method fills the gap between time simulations of complex numerical models, such as nonlinear finite element models (FEM), and stability and bifurcation analyses with much simpler analytical models.

From the experimental side, the aim is tried reached by developing techniques and software for experimental bifurcation analysis by contol-based continuation. Control-based continuation allows for bypassing mathematical models, and systematically explore how vibrations of a mechanical system changes with parameters, even tracking unstable vibrations. Using a a path-following algorithm to track branches of bifurcation diagrams, the system is guided to desired states by a controller that does not influence the natural dynamics of the system. Currently we explore these ideas for a simple mechanical oscillator with a strong impact nonlinearity.