Control of a System with Two Degrees of Freedom by Means of Potential Forces

Abstract

This work determines the restoring forces of the elastic elements of a system with two degrees of freedom by its predetermined movement. Two bodies of a certain mass are considered. The first body is connected by the first elastic element with a fixed base; and the second body is connected with the first body by the second elastic element. The movement occurs in a straight line in a horizontal plane. Gravity and friction forces are not taken into account. Then, the trigonometric functions of the law of motion of each body and the restoring forces of both elastic elements are analytically determined. These restoring forces are the control functions by which the desired movement is realized. Technically, such controls can be implemented in various ways, in particular, through passive elastic elements, the design and calculation method of which is proposed in the article. A variant of creating elastic elements with the restoring forces based on springs moving between two guides of the calculated form perpendicular to their axis of symmetry is given.

APPENDIX

To verify the results, analytical dependences (2.6) and (2.7) were also obtained in a table. From (2.2) for the moments of time selected with a certain step t_i values were found ${{x}_{1}}\left( {{{t}_{i}}} \right)$, ${{x}_{2}}\left( {{{t}_{i}}} \right)$, and from (2.4) were found $F_{1}^{/}\left( {{{t}_{i}}} \right)$, $F_{2}^{/}\left( {{{t}_{i}}} \right)$. After which, excluding time points t_i, points with coordinates $\{ {{x}_{{1i}}}$, ${{F}_{{1i}}}\}$ were obtained superimposed on the dependence determined by formula (2.6) (Fig. 7a), and points with coordinates $\{ ({{x}_{{2i}}} - {{x}_{{1i}}})$, ${{F}_{{2i}}}\}$, which are superimposed on the dependences obtained by formula (2.7) (Fig. 7b), where i is a natural number corresponding to the number of points. As can be seen from this figure, the coincidence is almost complete, which confirms the correctness of the previous calculations.

Fig. 7.