New theoretical developments on eigenvector derivatives with repeated eigenvalues

Many different methods have been developed since the pioneering works of Mills-Curran (1988) and Dailey (1989) on eigenvector derivatives with repeated eigenvalues. In spite of the increasing mathematical complexities witnessed in many of the newly emerged methods, some underlying fundamental theories governing the eigenvector derivatives have neither been much discussed, nor fully established to date. The present approach seeks to fill such an outstanding theoretical gap and to lay down the necessary theoretical foundation on which existing methods can be mathematically unified and further improved in numerical accuracy and computational efficiency. The particular solutions of eigenvector derivatives generally required have been derived in terms of modal properties, thereby avoiding the computationally expensive and potentially erroneous procedure of solving a set of algebraic equations of system dimension. The contributions of higher unavailable modes have been theoretically derived, enhancing the practical applicability of the proposed method to the general case where only partial eigensolutions are made. To avoid degeneration of eigenvector space in the case of repeated eigenvalues, a concept of global design variable is developed in which all intended multivariate design modifications are grouped into a single global variable to which eigenvector derivatives are derived, rendering real major applications of the proposed method to the predictions of structural design modifications. A discrete parameter model of a turbine bladed disk assembly, which is known to have many pairs of repeated eigenvalues due to its cyclic symmetry, as well as a finite element model of a cantilevered beam with large DOFs have been employed. Numerical results have demonstrated the accuracy and the practical applicability of the proposed new theoretical developments, as well as the proposed new method.