On the Reduction of Nonlinear Structural Dynamics Models

Abstract

The authors analyze the problem of model reduction in structural dynamics for unforced conser vative 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 re duction 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.

Order Reduction of Parametrically Excited Nonlinear Systems: Techniques and Applications

“…It has been shown [24] that this approach is often as accurate as the 'nonlinear projection technique' and requires less computational effort. The procedure remains the same as described in Section 2.1 up to the partitioned form given by Equation (8).…”

“…In the linear approach, the contribution of the non-dominant states is considered insignificant and hence neglected. Thus, by neglecting Equation (8) and setting the nondominant states to zero(z s = 0), the entire system dynamics is approximated bẏ…”

“…Once again, we consider Equation (4) which can be transformed to Equation (8). The objective here is to approximate Equation (8) by a reduced order system given bẏ…”

“…It is the generalization of the approach presented by Burton and coworkers [7,8] for autonomous systems. Here, the original set of second order time-periodic nonlinear equations is converted into a second order nonlinear system with time invariant linear system matrices and periodically modulated nonlinearities via the L-F and other canonical transformations.…”

“…Model reduction using the nonlinear modal coordinates is advantageous because one may use fewer nonlinear normal modes than linear ones to perform equally accurate modal analysis of nonlinear systems. The order reduction may be carried out in state space [6] or in second-order (structural) form [7]. In a recent paper, Burton and Rhee [8] suggested a NNM-based order reduction method in structural form and compared it with the linear reduction procedure for time invariant systems.…”