Abstract
This article deals with a rather complex problem of the rod system stability, the solution of which is complicated by the inconstancy of the bending stiffness along the length of a cantilever rod compressed by a concentrated axial force. The original fourth-order differential equation is split into a system of two second-order differential equations that, under given boundary conditions at the ends of the rod, are solved by the numerical finite difference method. The transition from the fourth-order equation to the second-order equations made it possible to eliminate the need to record the "second" contour nodes outside the material part of the structure under study, which made it possible to obtain a more "correct" (a physically determined value) of critical forces. Reliability of the presented theoretical provisions and applied results is confirmed by comparison with the known exact solutions of the other authors. In the course of the study, along with the values of critical forces, four forms of loss of stability are presented due to the corresponding "standing" waves along the length of the considered cantilever rods of variable bending stiffness. The scientific and applied results presented in this article will be useful in developing methods of strength calculations in the mechanics of deformable solids, and will also serve as a methodological guide for practicing engineers and designers