Spectral Analysis of Large-Amplitude Nonlinear Free Vibrations of Beams: Influence of Boundary Conditions on Displacement Localization and Frequency–Amplitude Relationships

This paper presents a comparative spectral analysis of large-amplitude geometrically nonlinear free vibrations of beams under different boundary conditions, namely clamped–clamped (CC) and clamped–simply supported (CSS) configurations. Based on Hamilton’s principle, a nonlinear formulation accounting for bending and axial strain energies induced by large transverse displacements is developed. The nonlinear eigenvalue problem is solved using an iterative numerical procedure, allowing the evaluation of frequency–amplitude relationships and associated transverse displacement fields. Particular attention is paid to the spatial localization of maximum displacement and its dependence on boundary conditions and vibration symmetry. The results demonstrate that boundary conditions significantly affect both the nonlinear frequency shift and the position of peak transverse displacement along the beam span. For CC beams, symmetric and antisymmetric modes exhibit distinct displacement localizations, whereas CSS beams show a pronounced shift of maximum displacement toward intermediate span locations. Quantitative comparisons reveal notable differences in frequency ratios and displacement amplitudes for identical vibration levels. These findings provide physical insight into the role of boundary conditions in nonlinear beam dynamics and offer useful guidance for the design and optimization of slender structural components operating under large vibration amplitudes.