Optimal gyrodine rotor shape in the class of conical bodies

A. E. Biryuk ; M. I. Drobotenko ; I. Riadchikov ; A. A. Svidlov

Matematičeskaâ fizika i kompʹûternoe modelirovanie, Tome 23 (2020) no. 4, pp. 69-81.

Résumé

The problem is to find the optimal shape of the gyrodine rotor and its angular rotation speed that maximizes the angular momentum relative to the axis of rotation at a fixed radius, mass and material of the rotor, taking into account the final strength of the material. The gyrodine rotor is a body of revolution, the thickness of which depends only on the distance $r$ to the axis of rotation, $ r \in [0, R] $, where $R$ is the radius of the rotor. The rotor surface is defined by rotating the curves $ z = \pm z (r) $ around the axis. When the rotor is spinning, it undergoes deformation due to centrifugal forces. Normal stress fields appear: radial and annular. Assuming the rotor to be thin, deformations can be described by the functions of the radial displacement of the rotor points $u(r)$. Stress fields can be expressed in terms of this function. The functions $ u (r) $ and $ z (r) $ are related by the equation of the elastically deformed state. This equation is supplied with the boundary conditions for the absence of radial stresses at $r = R$ and the condition for the absence of displacement on the axis of rotation $u (0) = 0$. Using the numerical solution of the equation, the problem is solved for the class of conical rotors $ z (r) = a + br $ with two parameters $a$ and $b$. The numerical method is used due to the fact that even in this relatively simple case the problem cannot be solved analytically. Several integrable cases are used to analyze the calculation error in the numerical solution of the problem. The dependence of the problem on the Poisson ratio $\mu\in (-1.1)$ is investigated, with the remaining parameters fixed. The gain in the angular momentum relative to a rotor of constant thickness is compared. The optimal steel conical rotor ($\mu=0.3$) is $2.068$ times thicker at the center than at the edge. Its advantage in the angular momentum over the rotor of constant thickness is $3.2~\%$.

Détail

Mots-clés : Poisson's ratio
Keywords: material strength, rapidly rotating disk, boundary value problem, extremal problem, finite-difference method, dichotomy method.