Which are the important modes of a subsystem?

A 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, Ns and hence also by Ns eigenmodes. In order to reduce the computational effort, it is common to apply ‘modal reduction’ to the subsystem such that only Nr modes out of the total number of Ns modes are retained, where Nr>Ns. The following question then arises: ‘Which Nr 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