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The equations described free longitudinal and lateral vibrations of the rectilinear elastic beams with the rectangular cross section are derived based on the theory of linear elasticity and the method of integrodifferential relations. The original system of partial differential equations is reduced to the system of ordinary differential equations with constant coefficients. The influence of geometrical and elastic properties of the beam on eigenvalues and eigenfunctions are investigated. The existence of the different displacement and internal stress types of the longitudinal motions are shown. The presence of two frequency zones corresponded to different kinds of the characteristic solutions for the lateral vibrations is detected. [ABSTRACT FROM AUTHOR]

The article presents the analysis of the stress-strain state of elastic bodies. It states that it is based on the linear theory of elasticity as well as the method of integro-differential relations. It notes that employing the integral formulation of Robert Hooke's law and polynomial representations of the obscure functions of stresses and displacements wherein it is reduced to an integrable system of ordinary differential equations with constant coefficients.