The article is devoted to the numerical analysis of the calculation of rods in linear and geometrically nonlinear formulations. The article presents graphs of the results of longitudinal displacement along the length of the rod at different time points. The transverse vibration propagation along the length of the rod is shown for boundary conditions of rigid fixing at two ends in a linear formulation at different time points. The sections of the rod where the largest excitation is observed are determined, and it is shown that the results differ at different times at different points along the length of the rod. Several practical problems can be solved setting specific values of parameters (Young’s modulus, Poisson’s ratio (for steel), rod length, cross-sections under consideration, surface loads). Considering the theoretical foundations, a software package was created using finite difference methods, the Bubnov–Galerkin method, the V. L. Rvachev R-functions method (RFM), and the method of successive approximations. The method of successive approximations is used in the article, by virtue of which linear and nonlinear problems are solved at each iteration step. The results obtained are presented in the form of graphs. The article highlights the importance of numerical methods in the analysis of bar structures and provides valuable information for engineers and researchers involved in this field.