Specificity of the Darboux mechanism rectilinear motion

Abstract

Mechanisms of a parallel structure may have singularities in configuration space in which system controllability may be lost or additional instantaneous degrees of freedom may appear. These features have kinematic base. But interest is also represented by geometric singularities, when the mechanism in some configuration could change the type of motion. The example of a mechanism with branch points, the Darboux mechanism, is considered. It is proved that this hinged mechanism can transform the rotational motion of one rod into a (strictly) rectilinear motion of its vertex H. The rods of Darboux’s mechanism can form geometric figures, such as triangles and a square (with diagonals drawn). In the “square” configuration of the mechanism the branch point arises: the vertex H can move both along the straight line L and along the curve γ. The rank of the holonomic constraints of the system in singular point falls by one. For a rectilinear motion of the vertex H, the Lagrange equation of the second kind is written in terms of the coordinate of H. The coefficients of this equation smoothly extend through the branch point. The “limiting” behavior of reaction forces in rods is analyzed when the mechanism moves to the branch point. An external force that does not work on the point H leads to unlimited reactions in the rods. Kinematics at the branch point are studied. The inverse problem of dynamics at a point where the rank of holonomic constraints is not maximal is solvable. The Lagrange multipliers of Λi at the branch point are not uniquely determined, but the forces corresponding to them acting on the vertices of the mechanism are uniquely determined.