On the reduction of nonlinear structural dynamics models

The authors analyze the problem of model reduction in structural dynamics for unforced conservative systems having static, cubic nonlinearities. The framework is the classical one of partitioning the coordinate vector into subsets of so-called master and slave coordinates, with the reduced dynamic model to contain only the masters. The objective is to compare the quality of two model-reduction methods: (1) a reduction based on the leading order calculation of the nonlinear master-slave state transformation defining the manifold containing the modes to be represented in the reduced model ("NNM-based reduction") and (2) a "linear-based reduction" utilizing the exact for the linear case master-slave state transformation; the linear-based reduction is equivalent to a modal analysis, but the reduced model is defined in terms of the physical, rather than the modal, coordinates. For the simple, low-order oscillator systems considered as examples, the authors have found the NNM-based reduction, which one would expect to be superior, to suffer degradation in quality in the presence of near (and not so near) three to one internal resonance conditions. The linear-based reduction is much simpler to implement and seems to provide competitive results over wide ranges in the system parameters. This conclusion may not carry over to other; higher degree-of-freedom systems.