Mathematical Modeling of Transverse Vibrations for Cantilever Rod of Constant Cross Section with Longitudinal Force

This article is devoted to obtaining an exact solution to the problem of transverse bending vibrations of a straight cantilever rod with a constant cross section loaded with a constant longitudinal force along the length of the rod. The problem based on dimensionless coordinates allows to reveal in a generalized form the influence of the longitudinal force, rod dimensions and its material on the natural frequencies for bending vibrations. The exact values of the natural frequencies of a rod with specific dimensions and a specific longitudinal force have been determined based on the generalized solution results. The obtained exact solutions are further considered as a standard for estimating the accuracy of the numerical method of fifth degree algebraic polynomials (M5DAP) in calculate the eigenfrequencies of transverse vibrations for cantilever rod of constant cross section with longitudinal force on different mesh. The paper shows the M5DAP possibility to obtain numerical results with an accuracy of 4–6 significant digits for the first ten modes on a mesh of 1000 nodes. Withal, the monotonic convergence with approximately second-order is observed, that makes possible to obtain even more accurate results with the appropriate software implementation.