Transfer matrix method for free and forced vibrations of multi-level functionally graded material stepped beams with different boundary conditions

Functionally graded materials (FGMs) are a novel class of composite materials that have attracted significant attention in the field of engineering due to their unique mechanical properties. This study aims to explore the dynamic behaviors of an FGM stepped beam with different boundary conditions based on an efficient solving method. Under the assumptions of the Euler-Bernoulli beam theory, the governing differential equations of an individual FGM beam are derived with Hamilton's principle and decoupled via the separation-of-variable approach. Then, the free and forced vibrations of the FGM stepped beam are solved with the transfer matrix method (TMM). Two models, i.e., a three-level FGM stepped beam and a five-level FGM stepped beam, are considered, and their natural frequencies and mode shapes are presented. To demonstrate the validity of the method in this paper, the simulation results by ABAQUS are also given. On this basis, the detailed parametric analyses on the frequencies and dynamic responses of the three-level FGM stepped beam are carried out. The results show the accuracy and efficiency of the TMM.