Unified Spectro-Geometric Framework for Multi-Span Beam Vibrations under Arbitrary Boundary and Coupling Conditions

A unified spectro-geometric framework is presented for the transverse, longitudinal and torsional vibration analysis of elastic multi-span beams under arbitrary boundary and coupling conditions. Improved Fourier series with auxiliary functions are employed to satisfy derivative continuity in both second and fourth order governing equations, while translational and rotational spring groups are introduced to parametrize general end and inter-span constraints. Applying Hamilton’s principle and the Ritz method establishes a unified eigenvalue formulation applicable to all vibration types. Comparisons with finite-element and reference solutions demonstrate excellent accuracy and fast convergence, achieving frequency deviations below 4.43% with only 12 retained terms. The results reveal distinct stiffness-sensitive intervals for boundary and coupling springs, inherent coupling among axial, bending, and torsional modes, and consistent monotonic effects of geometric and material non-uniformity on modal frequencies. Forced vibration analyses further confirm the correctness of the model. The proposed method provides an efficient and general approach for vibration prediction, stiffness tuning, and dynamic optimization of multi-span beam structures in engineering applications.