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.