Which are the important modes of a subsystem?

Abstract

SUMMARYA linearly behaving vibrational substructure (or more generally a linear dynamic subsystem) attached to a main structure (or a main dynamic system) is considered. After discretization, the substructure is represented by a finite, typically large, number of degrees of freedom, N s and hence also by N s eigenmodes. In order to reduce the computational effort, it is common to apply 'modal reduction' to the subsystem such that only N r modes out of the total number of N s modes are retained, where N r >N s . The following question then arises: 'Which N r modes should be retained?' In structural dynamics, it is traditional to retain those modes associated with the lowest frequencies. In this paper, the question is answered by solving an appropriate optimization problem. The most important modes of the subsystem are shown to be those whose coupling matrices, which are defined in a particular way, have the highest norm. This leads to a simple and effective algorithm for optimal modal reduction. The new criterion for 'modal importance' is explained both mathematically and physically, and is demonstrated by numerical examples.

A model reduction strategy for computing the forced response of elastic waveguides using the wave finite element method

“…Within the CMS framework, an optimal modal reduction technique based on the study of an error norm for coupling interface forces has been proposed in refs. [6,7]. A moment matching method (i.e.…”

“…(72) is based on the fact that both L T and L T r are unitary matrices 6 . As suggested in Section 3.…”

Padé approximants and the modal connection: Towards increased robustness for fast parametric sweeps

“…Widely used for several decades, modal methods have proved to be powerful from both an experimental and a numerical perspective, providing insight into the dynamic behaviour of the system of interest, as well as a basis suitable for a range of reduced-order models (ROMs). [1][2][3][4][5][6] These approaches aim at establishing a reduced set of equations to be solved at all the frequency steps of interest in the spectrum. Despite recent efforts made to extend such methods to more general types of damping, [7][8][9][10][11][12] these are however mostly limited to be efficient for either conservative problems or problems including simple models of damping.…”

Model reduction and perturbation analysis of wave finite element formulations for computing the forced response of coupled elastic systems involving junctions with uncertain eigenfrequencies

“…This task addresses model order reduction procedures with a view to saving additional CPU time when solving the WFE formulations 2 . Several works have been carried out on that issue concerning classical substructuring approaches, where norm-wise estimators are invoked as selection criteria of component modes [13,14,15]. The feature of these approaches is that the mode selection can be performed in a single pre-processing step, without the need for computing the forced responses.…”

“…The issue to provide forward component-wise bounds of the perturbed vector u i = u 0 + δu i , using first order Taylor series expansions 15 , is the key idea of the present study. Comparisons with the results of MCS are investigated in subsection 4.3.…”