Model reduction of high-dimensional self-excited nonlinear systems using floquet theory based parameterization method

The parameterization method has demonstrated remarkable efficacy in the construction of reduced order models for nonlinear dynamical systems, which are mostly applied to depict the dynamics near a fixed point or a forced periodic solution perturbed from it. In this paper, we develop a method based on direct linear algebra to construct reduced order models around the limit cycles of high dimensional autonomous self-excited nonlinear systems. In particular, Taylor–Fourier series is adopted to parameterize the invariant manifold around a periodic solution of the self-excited systems. The basic theory and derivation of equations are presented in terms of matrices and Kronecker product, which benefits computer implementation and enhances computational efficiency. Two approaches to solve co-homological equations based on Floquet normal form and direct linear algebra are introduced and compared. The method based on Floquet normal form proves to be impossible to be applied in the high dimensional cases due to the extensive computing consumption. To facilitating the implementation, an iterative formula for the series expansion of generic nonlinear functions by composition of elementary function is also proposed. The reduced order models of a self-excited wing model, a rotor-stator rubbing model and a FEM rotor model are constructed. It is shown that the proposed approaches are valuable in predicting responses of high-dimensional self-excited nonlinear systems and can greatly reduce the computational costs.