Critical re-examination and numerical verification of Sorokin and Arkhipov's approach to free transversal vibrations of beams using plane elasticity theory

The paper written by Sorokin and Arkhipov, which appeared in 1966 in a commemorative volume, and is still largely unknown to the non-Russian speaking scientific community, is here thoroughly analyzed and commented. For the convenience of potential readers, a complete, unabridged translation into English is also provided. The transverse free vibrations of a beam are analyzed in this work: the problem is formulated within the framework of plane elasticity theory, following the original approach, where the beam is considered as a two-dimensional elastic body, in plane stress conditions. The equations describing its dynamic behavior are deduced, along with the relevant boundary conditions, from the general theory by a truncated power series method. All details of the original formulation have been re-examined and verified, and a critical check of the numerical results, which are relevant to the free vibration of a simply supported beam, confirmed their correctness. New results, which extend the cases analyzed in the original paper, are presented and discussed. In particular, a promising field of investigation stems from the obtained results, which outline the need to validate such higher-order beam models both analytically and through experimental tests. Scholars interested in the dynamics of beams treated as a 2-D problem of elasticity will find a useful reference and interesting insights for their current and future work.