Invariant manifold-based order reduction of time-periodic structural dynamic systems

Order reduction of parametrically excited nonlinear structural systems represented by a set of second order equations is considered. First, the system is converted into a second order system with time invariant linear system matrices and periodically modulated nonlinearities via the Lyapunov-Floquet transformation. Then a master-slave separation of degrees of freedom is used and a relation between the slave coordinates and the master coordinates is constructed. The proposed method reduces to finding a time-periodic nonlinear invariant manifold relation in the modal coordinates of the transformed system. In the process, closed form expressions for 'true internal' and 'true combination' resonances are obtained for various nonlinearities which are generalizations of those previously reported for time-invariant systems. No limits are placed on the size of the time-periodic terms thus making this method extremely general even for strongly excited systems. A two degree-of-freedom inverted pendulum with a periodic follower force is used as an illustrative example. The nonlinear-based reduced models are compared with linearbased reduced models in the presence and absence of nonlinear resonances.