Calculation of Eigenvalue and Eigenvector Derivatives for Algebraic Flutter and Divergence Eigenproblems

Abstract

A single procedure is presented for the determination of eigenvalue and eigenvector derivatives of general eigensystems, including those arising in flutter and divergence analysis. The method requires only the knowledge of the eigensolution under consideration and no information about the transposed problem. In the procedure, the eigenproblem is not presented as such, but as a nonlinear algebraic system of equations with the eigenvalue and the eigenvector as unknowns and the eigenderivatives coming from the solution of a linear system of equations that is determined trivially by differentiati ng the nonlinear equations with respect to a structural parameter. The technique can be extended easily to derivatives of any order and always requires solutions of the same linear system with different right-hand sides. The coefficient matrix of this linear system always maintains the structure of the matrix of the eigenproblem and three methods of solution are presented to take full advantage from its symmetry and sparseness, if present. Application to flutter and divergence problems is emphasized, as the basic idea of the method unifies the determination of a solution and of its derivatives in one single approach.