Complex Variable Method for Eigensolution Sensitivity Analysis

The application of complex variable method (CVM) for eigenvalue and eigenvector sensitivity analysis is presented. Gradi ent -based methods usually used in structural optimization require accurate sensitivity information. CVM is accurate, robust and easy to implement as compared to other approximate sensitivity analysis methods. By CVM, the first order modal sensitivity can b e calculated using an existing structural analysis package. This is achieved by making a small complex perturbation to a design variable. The resulting eigensolutions would be complex. The sensitivity of eigensolutions with respect to that design variable can then be obtained by taking the complex part of the eigensolutions divided by the perturbation. Since there is no subtraction involved in the sensitivity calculation, the approximation is not sensitive to step size. Thus, unlike finite difference method s CVM does not suffer the effect of subtraction -cancellation error when a very small step -size is used. The computation of 1 st order modal sensitivity is not hampered by repeated eigenvalue problem, as is the case with other sensitivity analysis methods . For repeated eigenvalues, CVM automatically provides correct eigenvalue sensitivity and the associated eigenvector sensitivity for a differentiable mode. A 5 -dof system with an eigenvalue of multiplicity 3 is used to illustrate the salient features of CVM in eigen -solution sensitivity analysis. Nomenclature K = stiffness matrix for spring -mass system M = mass matrix for spring -mass system � = eigenvalue for spring -mass system � = eigenvector for spring -mass system ( ) ' = derivative of ( ) w.r.t. design variable