Derivation and significance of second-order modal design sensitivities

If the system is considered to be unsatisfactory in some way, e.g., a natural frequency to the system coinciding with a constant shaft speed, then it is of interest to find the design change which will give maximum benefit with minimum disturbance to the system. This may be quantified in terms of design sensitivities of the structure. Fox and Kapoor derived an expression for the sensitivity of an eigenvalue for self-adjoint systems (i.e., where the left and right eigenvectors are simple transposes, typical of conservative systems and Duncan forms). Garg extended this result to apply to non-self-adjoint systems (e.g., where dissipative and gyroscopic effects are present). The analysis for eigenvectors was presented by Plaut and Huseyin for the general case. Most recently, Nelson has provided an alternative approach to eigenvector sensitivity. The sensitivities derived by all of the above methods are expressed in the form of (first) derivatives and, hence are properties local to the original solution of the system. In a recent paper by Vanhonacker, there is a clear implication that these first derivatives may be extrapolated for gross system changes. Clearly this assumption can be valid only if the value of the second derivative is negligible.