(untitled) (Journal of Mathematics Mechanics and Computer Science, 2019)
To date, a huge number of studies have been carried out to bring a three-dimensional problem to a two-dimensional one by engineering and mathematical methods. But these studies do not exhaust the problem completely. The solution of this problem for bodies with different geometries continues today, as evidenced by the publications of domestic, Russian and foreign scientists. Adjacent to them is the problem of studying the dynamic behavior of circular rods interacting with the deformable medium on the basis of oscillation equations derived using a rigorous mathematical apparatus. Approximate equations of oscillation of rod systems above the second order with respect to the derivatives of the desired function and the theory of oscillations of a circular cylindrical shell, in particular torsional oscillations, taking into account the inertion of rotation and the strain of the transverse shear, are devoted to a relatively small number of scientific publications. The approximate equations of rod systems of variable thickness presented in this paper allow us to construct approximate theories of oscillation depending on the conditions at the ends of the rod, on the order of the derivatives sought in the approximate equations and initial conditions. The results obtained make it possible to formulate boundary value problems in solving particular problems of oscillations of a cylindrical shell under various conditions at the end of the shell