Behavior of derivatives of eigenvalues and eigenvectors in curve veering and mode localization and their relation to close eigenvalues

The problem to measure the phenomena of eigenvalue curve veering and mode localization is addressed in this paper. The second derivative of an eigenvalue and the first derivative of an eigenvector are taken as the measures, numerically showing curve veering and mode localization. Based on the measurement, close eigenvalues, as a key factor for the occurrence of the phenomena, are defined. Two eigenvalues are considered to be close, if their difference is small enough to cause the occurrence of the phenomena. The curve veering and mode localization can be noted by comparison of the derivatives with a critical value and hence the associated eigenvalues are close. Weakly coupled springs are given as an example. (C) 2002 Elsevier Science Ltd. All rights reserved.