Accounting for Quasi-Static Coupling in Nonlinear Dynamic Reduced-Order Models

Engineering structures are often designed using detailed finite element (FE) models. Although these models can capture nonlinear effects, performing nonlinear dynamic analysis using FE models is often prohibitively computationally expensive. Nonlinear reduced-order modeling provides a means of capturing the principal dynamics of an FE model in a smaller, computationally cheaper reduced-order model (ROM). One challenge in formulating nonlinear ROMs is the strong coupling between low- and high-frequency modes, a feature we term quasi-static coupling. An example of this is the coupling between bending and axial modes of beams. Some methods for formulating ROMs require that these high-frequency modes are included in the ROM, thus increasing its size and adding computational expense. Other methods can implicitly capture the effects of the high-frequency modes within the retained low-frequency modes; however, the resulting ROMs are normally sensitive to the scaling used to calibrate them, which may introduce errors. In this paper, quasi-static coupling is first investigated using a simple oscillator with nonlinearities up to the cubic order. ROMs typically include quadratic and cubic nonlinear terms, however here it is demonstrated mathematically that the ROM describing the oscillator requires higher-order nonlinear terms to capture the modal coupling. Novel ROMs, with high-order nonlinear terms, are then shown to be more accurate, and significantly more robust to scaling, than standard ROMs developed using existing approaches. The robustness of these novel ROMs is further demonstrated using a clamped–clamped beam, modeled using commercial FE software.