Modal properties of mechanical systems under geometric variations by perturbation theory

The availability of analytical expressions to obtain the system matrices of mechanical components subject to small geometric variations (Bouras and Carassale (2024)) enables efficient solutions of several technical problems, including uncertainty propagation and shape optimization. An important step to tackle these problems is the calculation of the modal properties of the modified system. This can be done by relying on system matrices that are expressed through approximations such as power series. This paper starts from a motivational example showing some unexpected results and then studies the mathematical problem using a perturbation approach. This formulation provides analytic expressions for the corrections up to the second order of the eigenpairs of systems whose matrices depend on a small parameter. The results obtained can be related to known expressions for the derivatives of eigenvalues and eigenvectors both for the case of isolated modes, as well as for repeated eigenvalues. Besides, the technically relevant case of closely-spaced modes is considered. On the other hand, the perturbation analysis enables the discussion of the relative importance of the terms contained in the asymptotic expressions and explains some previously obtained numerical results.