Existence and Normalization of Complex Modes for Post Experimental Use in Modal Analysis

Abstract

SummaryThe eigen characteristics of a dynamical system offer a vector sub-space suitable for performing canonical transformations on the system of equations of the structural dynamics problems being considered. While the concept is mathematically fully developed for both damped and undamped systems, practitioners --at all levels --tend to indiscriminately use the system's normal modes as a basis for applications containing nonproportional damping. Such a practice in most cases is a reasonable approximation and results in small, if not infinitesimal, errors. However, with the increase of sophistication and accuracy requirements in certain applications of modal analysis, these approximations must be fully analyzed and understood.In this paper, emphasis is directed to some specific applications for which it is generally a common practice to use normal, or undamped, modes as a vector subspace for use with nonproportionally damped system. These are: Detailed background on complex modes is presented. Commonly used transformations are examined and error models are derived and quantified.

A Measure of Non-Proportional Damping

“…This normalisation is, of course, not unique (see, for instance, Ibrahim and Sesteri [10] and, Balme`s [11]). It can be made unique by imposing additional conditions as, for instance, to require consistency with the normalisation (7) of the modal matrix of the undamped model, i.e.…”

On the Relation Between Complex Modes and Wave Propagation Phenomena

“…It is important to note that this elliptical behavior of modes can only be observed when dealing with single-wave-type propagation problems. 3. THE SPECTRAL ELEMENT METHOD Doyle [9] presented spectrally formulated "nite and semi-in"nite elements for Timoshenko beams, known as spectral elements.…”

“…However, it can also arise from gyroscopic e!ects, aerodynamic e!ects, non-linear structural behavior, experimental noise e!ects, aliasing, leakage, mass loading, high modal density, and identi"cation errors [2,3]. Many other authors have reviewed and investigated complex mode shapes [4}6].…”

Model correlation and identification of experimental reduced models in vibroacoustical modal analysis

“…This paper will be limited to the case where system matrices are obtained from identified complex modes, with a particular focus on the so-called properness condition. This condition, which is not addressed in the review papers referenced above, has been mentioned with various names in several publications [26][27][28][29]. It is automatically verified by the exact complex modes of the system.…”