Some Techniques for Order Reduction of Nonlinear Time Periodic Systems

Abstract

In this paper, some techniques for order reduction of nonlinear systems with time periodic coefficients are introduced. The equations of motion are first trasformed using the Lyapunov-Floquet transformation such that the linear parts of the new set of equations are time-invariant. To reduce the order of this transformed system three model reduction techniques are suggested. The first approach is simply an application of the well-known linear method to nonlinear systems. In the second technique, the idea of singular perturbation and noninear projection are employed, whereas the concept of invariant manifold for time-periodic system forms the basis for the third method. A discussion of nonlinear projection method and time periodic invariant manifold technique is included. The invariant manifold based technique yields a ‘reducibility condition’. This is an important result due to the fact that various types of resonance are present in such systems. If the ‘reducibility condition’ is satisfied only then a nonlinear order reduction is possible. In order to compare the results obtained from various reduced order modeling techniques, an example consisting of two parametrically excited coupled pendulums is included. Reduced order results and full-scale dynamics are used to construct approximate and exact Poincare´ maps, respectively, because it portrays the long-term behavior of system dynamics. This measure is more convincing than just comparing the time traces over a short period of time. It is found that the invariant manifold yields the most accurate results followed by the nonlinear projection and the linear techniques.