Vibration Analysis of a Uniform Beam Fixed at One End and Restrained Against Translation and Rotation at the Second One

The problem of analyzing the transverse vibrations of uniform beams was largely reported in the scientific literature. The purpose of this work is to study the particular case of a uniform beam fixed at its left end and elastically restrained at the right end by two springs K_T and K_r. The latter are relative to the vertical translation and the rotation. The investigation consists of analyzing the influence of K_T and K_r on the uniform beam vibration behavior. The situations of extreme values of the two springs representing free, pinned, guided and fixed conditions at the right end served for the validation for which a good agreement was noted. Thereafter, it was extended to take into account the effect of the two springs acting simultaneously. The results are summarized under the form of plotted curves representing the variations of the angular vibration frequency ω₁ as a function of both the K_T and K_r values.