Quasi-static modal analysis for reduced order modeling of geometrically nonlinear structures

Abstract

Abstract When modeling dynamic structures, the quasi-static modal analysis (QSMA) approach seeks to very accurately resolve the force-displacement behavior of each individual mode but neglects dynamic modal coupling, while many other reduced order modeling (ROM) methods take an opposite view, capturing static and dynamic modal coupling but using polynomials that limit the fidelity with which the force-displacement behavior of each individual mode can be captured. This work contrasts these two approaches both theoretically and by applying them to geometrically nonlinear structures of varying complexity. The comparison reveals a potential deficiency in the QSMA approach in which the secant approximation that was typically used to estimate the effective natural frequency from the load-displacement curves is found to be inaccurate for some of the strongly nonlinear structures considered in this work. Conversely, the examples demonstrate that the low-order polynomial typically employed by ROM methods such as implicit condensation and expansion (ICE) is often insufficient to describe the force-displacement behavior with adequate fidelity when a single mode is used. These examples prompt the creation of a new method that is a hybrid between QSMA and ICE, which fits a high order polynomial to the modal response curve of a single mode. Several case studies show that the resulting single-mode ICE ROM, here dubbed a SICE-ROM, can capture the resonant behavior in the vicinity of the mode very accurately, but to do so the force-displacement relationship must be captured with extremely high accuracy. These comparisons highlight the importance of static coupling in these structures, suggesting that it is often far more important than dynamic coupling even in cases where the latter had been previously thought to be very important. The geometrically nonlinear structures studied in this work include FE models of a flat beam and an exhaust cover plate and a highly nonlinear curved beam where the Riks method is used to obtain the force-displacement curves through the snap-through regime.