Method of Reverberation-Ray Matrix for the Free Vibration of Euler-Bernoulli Beam with Variable Cross-Section

The method of reverberation-ray matrix （MRRM） is an efficient approach for analyzing the dynamic behavior of truss and frame structures. Due to its advantages of clearer physical interpretation and better numerical stability, this method has been extensively studied and employed by scholars. However, most existing studies of MRRM remain focused on the uniform elements, with limited consideration in the elements with variable cross-section. The main reason lies in the fact that, in the elements with variable cross-section, the phase relation in MRRM becomes difficult to be well established. In view of this, this paper derives the backward ray matrix method formulation for the free vibration problem of a specific variable-section Euler–Bernoulli beam, and uses this to calculate the natural frequencies and corresponding modes of the structure; the results obtained have been effectively validated by finite element simulation.