Equilibrium state stabilisation of material point on three crank-and-rods supports

Abstract

Mechanical system consisting of a heavy-weight material point, which is based on the three weightless crank-connecting rod bearing is considered. Cranks are driven by servo motors. For this system with three degrees of freedom, has solved the kinematics task, which given the coordinates of a material point with help of the cranks rotation angles relative to the axis servo drives. To solve the dynamics problem the mechanical system with equivalent forces is considered. In the new equivalent system material point is based on three weightless rods with variable lengths. For this system kinematics problems is solved and theLagrangian equation of the secondkindisderived.Direct andinversedynamicsproblems are resolved. The state of equilibrium is considered. The linearized system of the second kind Lagrange equations is splitintothreeindependent equations with respecttoeach of allgeneralized coordinates.By theLyapunov theorem receivedfeedbackparameters,providing the asymptotic stability of the equilibriumposition.The calculation results are presented.

Keywords:

crank mechanism, the Lagrange equations, Lyapunov’s theorem on stability in the first approximation, equivalent system of forces