Applications of Analytical Sensitivities of Principal Components in Structural Dynamics Analysis

Principal Components Analysis (PCA), which is more recently referred to as Proper Orthogonal Decomposition (POD) in the literature, is a popular technique in many fields of engineering, science, and mathematics for analysis of time series data. The benefit of PCA for dynamical systems comes from its ability to detect and rank the dominant coherent spatial structures of dynamic response, such as operating deflection shapes or mode shapes. Structural dynamics is one area in which it is useful because PCA results in modal-like dynamic properties for linear and nonlinear dynamical systems. Most notable is the use of PCA to generate efficient basis sets for developing reduced order models for fluid dynamics and structural dynamics problems. This is but one application of PCA in the science and engineering literature. The objective of this work is to investigate the applications of principal component sensitivities in structural dynamics analysis to evaluate where opportunities may exist for linear systems and potential new nonlinear analyses. Sensitivity analysis is standard tool used by analysts and has the potential to impact many applications of PCA. In this paper, an analytical approach for sensitivity analysis of PCA, developed and verified in previous work by the authors, is applied to structural dynamics analyses. Analytical sensitivities provide the necessary derivatives for gradient-based algorithms. The proposed applications include an analytical framework for evaluating structural property changes, calibration of linear and nonlinear structural dynamics models, and sensitivity with respect to forcing inputs. Examples applications are considered.