Engineering interpretation of eigenvector sensitivities using normalization

Design sensitivities are used in various engineering methods to arrive at optimal or advanced designs. The design sensitivity of a system, also known as the derivative to a design variable, represents the change of the system response due to an adjustment of a design variable. Scientists and engineers often use modal design sensitivities to predict the change of a transient response. These sensitivities are the derivatives of the eigenvalues and eigenvectors of the system. Many researchers have successfully computed modal sensitivities using several methods. Nelson's method [1] is widely used even though computationally expensive. Fox's method [2] uses a modal superposition to represent the eigenvector sensitivities by the system eigenvectors, but is inaccurate when using only a few eigenvectors. Hence, several iterative methods were proposed to increase the accuracy of Fox's method (e.g., [3] or [4]). The primary focus of the previous research was on the computational algorithms rather than the interpretation of the eigenvector sensitivities. This interpretation depends highly on the chosen normalization. An approach for incorporating these normalizations was presented by Smith [5]. This paper shows the consequence and potential use for interpretation of the chosen normalization. This is illustrated on an example using a generic two-degrees-of-freedom system, as shown in FIGURE 2.