Order Reduction of Nonlinear Time Periodic Systems

This work reports new approaches for order reduction of nonlinear systems with time periodic coefficients. First, the equations of motion are transformed using the Lyapunov-Floquet (LF) transformation, which makes the linear part of new set of equations time invariant. At this point, either linear or nonlinear order reduction methodologies can be applied. The linear order reduction technique is based on classical technique of aggregation and nonlinear technique is based on ‘Time periodic invariant manifold theory’. These methods do not assume the parametric excitation term to be small. The nonlinear order reduction technique yields superior results. An example of two degrees of freedom system representing a magnetic bearing is included to show the practical implementation of these methods. The conditions when order reduction is not possible are also discussed. INTRODUCTION In many engineering systems, like structures subjected to periodic loadings, asymmetric rotor bearing system, etc., we encounter a large set of nonlinear differential equations with periodic coefficients. These equations are generally difficult to solve and require special analytical tools. Though these differential equations involve large number of states, for the purpose of analysis, simulation and control application only a few dominating states are important. It is possible to retain these dominating (‘master’) modes and get rid of non-dominating (‘slave’) modes.