A study on design sensitivity analysis for general nonlinear eigenproblems

A general nonlinear eigenproblem is considered in this paper. It is shown that the widely used undamped, viscously or nonviscously damped eigenproblem can be considered as a special case of the more general nonlinear eigenproblem. The existing formulas of the derivatives of eigensolutions of this nonlinear eigenproblem are very complex since the normalizations of undamped eigenproblems are considered in their studies. To simplify the computation of eigensensitivity, a new normalization of the general nonlinear eigenproblem is presented. The new normalization of the general eigenproblem derived here can degenerate to the familiar mass orthogonal relationship of undamped eigenvectors. Based on this normalization, the design sensitivity analysis for the general nonlinear eigenproblem with respect to arbitrary design parameters is studied. Moreover, this method can address the eigenproblem for both repeated and distinct eigenvalues. As it can be seen, under such general normalization the derivatives of eigensolutions can be expressed in a way similar to those of undamped systems. Finally, five numerical examples are provided to illustrate the effectiveness of the derived results. It is shown that the derivatives of eigensolutions can be treated in a unified way for different structural systems (i.e., undamped systems, viscously and nonviscously damped systems, and nonlinear systems).

► Design sensitivity analysis for general nonlinear eigenproblem is presented. ► This method can deal with undamped, viscous and nonviscous damped systems. ► The method can address the eigensystems with repeated eigenvalues. ► A new normalization for general nonlinear eigenproblem is proposed.