Reformulation for the Solution of the Dynamic Response of Co-axial Segmented Beams

Stepped beams constitute an important class of engineering structures whose vibration response has been widely studied. Many of the existing methods for studying stepped beams manifest serious numerical difficulties as the number of segments or the frequency of excitation increase. In this paper, we focus on the Transfer Matrix Method (TMM), which provides a simple and elegant formulation for multi-step beams. The main idea in the TMM is to model each step in the beam as a uniform element whose vibration configurations are spanned by the segment's local eigenfunctions. Utilizing these local expressions, the boundary conditions at the ends of the multi-step beam as well as the continuity and compatibility conditions across each step are used to obtain the nonlinear eigenvalue problem. Also, and perhaps more importantly, we provide a reformulation for multi-step Euler-Bernoulli beams that avoids much of the numerical singularity problems that have plagued most of the earlier efforts. When this reformulation is extended to multi-segment Timoshenko beams, the numerical difficulties appear to be mitigated, but not solved.