Parametric Eigensolution Sensitivity Analysis Using the Transfer Matrix Method

The transfer matrix method has long been recognized as a computationally efficient approach for the eigenanalysis of discrete and continuous dynamic systems, particularly in structural and rotor dynamics. In this study, the transfer matrix method framework is extended to incorporate parametric sensitivity analysis, providing the foundations required for continuation-based investigations. Sensitivity analysis is performed by evaluating the parametric derivatives of eigenvalues and eigenvectors directly from the system’s characteristic equation, clarifying the influence of parameter perturbations on system dynamics. To reduce computational effort, the intermediate terms required for the generic sensitivity matrix are accumulated and stored, enabling efficient evaluation of the overall sensitivity of the transfer matrix to frequency. Eigenvector normalization within the sensitivity analysis resolves the intrinsic non-uniqueness of eigenvectors and ensures consistent and stable computation of their sensitivities. The availability of consistent eigenvalue and eigenvector sensitivities naturally supports continuation procedures, enabling the systematic tracking of eigensolutions as system parameters vary and facilitating the identification of characteristic behaviors such as eigenvalue crossings and veering. Analytical sensitivity formulas inside the transfer matrix method framework and eigenvector normalization are used to address numerical challenges commonly encountered in transfer-matrix-based calculations. The methodology is verified through benchmark examples of increasing complexity, demonstrating its ability to capture critical system trends while preserving the compact structure of the transfer matrix formulation. Overall, the study demonstrates that integrating sensitivity analysis into the transfer matrix method significantly enhances its effectiveness for parametric investigations and early-stage design evaluations of complex dynamical systems, naturally supporting continuation-based analyses.