A Virtual Joints Based Transfer Matrix Method for Dynamic Performance Analysis of Periodic Beams

Abstract

An updated Transfer Matrix (TM) method involving spatial discretization through the introduction of virtual joints is proposed for the dynamic analysis of periodic structures with large spacing between discontinuity sections. In the traditional TM method, long transfer distances cause rapid exponential growth of hyperbolic terms, which can lead to significant numerical errors. This issue is effectively addressed by discretizing the original beam segments with virtual joints. The Joint Coupling Matrix (JCM) method is used to universally describe the restraints and interactions at joints, leading to the closed-form solutions for the dynamic responses of periodic beams. To validate the updated TM method, the natural frequencies of a simply supported periodic beam obtained from the Finite Element Method (FEM) are compared with those from the updated TM method, showing good agreement between the two methods. Then, the updated TM method is used to investigate the vibration mitigation performance of periodic structures. Results from both free and forced vibration analyses indicate that periodic structures can effectively reduce vibrations when designed with appropriate configurations of structural periodic parameters.

Keywords: Periodic beam, Vibration mitigation, transfer matrix method, Joint coupling matrix, State space method