Reduced-order modelling of self-excited, time-periodic systems using the method of Proper Orthogonal Decomposition and the Floquet theory

The mathematical models of dynamical systems become more and more complex, and hence, numerical investigations are a time-consuming process. This is particularly disadvantageous if a repeated evaluation is needed, as is the case in the field of model-based design, for example, where system parameters are subject of variation. Therefore, there exists a necessity for providing compact models which allow for a fast numerical evaluation. Nonetheless, reduced models should reflect at least the principle of system dynamics of the original model.In this contribution, the reduction of dynamical systems with time-periodic coefficients, termed as parametrically excited systems, subjected to self-excitation is addressed. For certain frequencies of the time-periodic coefficients, referred to as parametric antiresonance frequencies, vibration suppression is achieved, as it is known from the literature. It is shown in this article that by using the method of Proper Orthogonal Decomposition (POD) excitation at a parametric antiresonance frequency results in a concentration of the main system dynamics in a subspace of the original solution space. The POD method allows to identify this subspace accurately and to set up reduced models which approximate the stability behaviour of the original model in the vicinity of the antiresonance frequency in a satisfying manner. For the sake of comparison, modally reduced models are established as well.