Eigenvalue sensitivity analysis based on the transfer matrix method

For linear mechanical systems, the transfer matrix method is one of the most efficient modeling and analysis methods. However, in contrast to classical modeling strategies, the final eigenvalue problem is based on a matrix which is a highly nonlinear function of the eigenvalues. Therefore, classical strategies for sensitivity analysis of eigenvalues w.r.t. system parameters cannot be applied. The paper develops two specific strategies for this situation, a direct differentiation strategy and an adjoint variable method, where especially the latter is easy to use and applicable to arbitrarily complex chain or branched multibody systems. Like the system analysis itself, it is able to break down the sensitivity analysis of the overall system to analytically determinable derivatives of element transfer matrices and recursive formula which can be applied along the transfer path of the topology figure. Several examples of different complexity validate the proposed approach by comparing results to analytical calculations and numerical differentiation. The obtained procedure may support gradient-based optimization and robust design by delivering exact sensitivities.