On the Asymptotic Reduction of Classical Modal Analysis for Nonlinear and Coupled Dynamical Systems
Asymptotic Modal Analysis (AMA) is a computationally efficient and accurate method for studying the response of dynamical systems experiencing banded, random harmonic excitation at high frequencies when the number of responding modes is large. In this work, AMA has been extended to systems of coupled continuous components as well as nonlinear systems. Several prototypical cases are considered to advance the technique from the current state-of-the-art. The nonlinear problem is considered in two steps. First, a method for solving problems involving nonlinear continuous multi-mode components, called Iterative Modal Analysis (IMA), is outlined. Secondly, the behavior of a plate carrying a nonlinear spring-mass system is studied, showing how nonlinear effects on system natural frequencies may be accounted for in AMA. The final chapters of this work consider the coupling of continuous systems. For example, two parallel plates coupled at a point are studied. The principal novel element of the two-plate investigation reduces transfer function sums of the coupled system to an analytic form in the AMA approximation. Secondly, a stack of three parallel plates where adjacent plates are coupled at a point are examined. The three-plate investigation refines the reduction of transfer function sums, studies spatial intensification in greater detail, and offers insight into the diminishing response amplitudes in networks of continuous components excited at one location. These chapters open the door for future work in networks of vibrating components responding to banded, high-frequency, random harmonic excitation in the linear and nonlinear regimes.
Reduced Order Modeling
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