Second‐order sensitivity analysis in vibration and buckling problems

A perturbation approach for the first‐ and second‐order sensitivity analysis of eigenvalue problems, as they arise in buckling and vibration of structural systems, is presented. A particular feature introduced in the formulation is the dependence of all matrices that model the problem not only on the design parameter, but also on a pre‐critical state. Thus, the sensitivity of the response of the fundamental path in buckling problems, or a pre‐stressed state, in vibration problems, are introduced in the analysis. Two techniques are considered in this work: the direct and the adjoint methods. First‐ and second‐order sensitivity equations for eigenvalues and eigenvectors are obtained, and it is shown that second‐order sensitivity of an eigenvalue can be accomplished by using only one adjoint problem.