Modal approximations to damped linear systems

TL;DR

This paper introduces spectral inclusion theorems using quasi Cassini ovals for finite-dimensional damped second order systems, providing sharper estimates for system damping and overdampedness conditions.

Contribution

It presents novel spectral inclusion sets for quadratic eigenvalue problems, improving upon standard methods and applicable to a broad class of damped systems.

Findings

Quasi Cassini ovals outperform Gershgorin circles in spectral inclusion.

Sharper estimates for system overdampedness.

Applicable to modally damped and proportionally damped models.

Abstract

We consider a finite dimensional damped second order system and obtain spectral inclusion theorems for the related quadratic eigenvalue problem. The inclusion sets are the 'quasi Cassini ovals' which may greatly outperform standard Gershgorin circles. As the unperturbed system we take a modally damped part of the system; this includes the known proportionally damped models, but may give much sharper estimates. These inclusions are then applied to derive some easily calculable sufficient conditions for the overdampedness of a given damped system.