Specific Features of the Dynamics of the Rectilinear Motion of the Darboux Mechanism

Abstract

The Darboux mechanism is considered. It is proved that this hinge mechanism allows the rotational movement of one link to be converted into (strictly) straight linear movement of its top H. The links of the Darboux mechanism can form geometric shapes such as triangles and squares (with diagonals drawn). In the “square”-shaped configuration of the mechanism, geometrically, branching may occur when the vertex H can move both along a straight line L and along a curve γ. In this case, the rank of the holonomic constraints of the system diminishes by one. For direct linear motion of the vertex H, the Lagrange equation of the second kind in terms of the point H coordinates is derived. The coefficients of this equation can be smoothly continued through a branching point. The “limiting” behavior of the reaction forces in the rods is studied when the mechanism moves to the branching point. An external force that does not do work on point H leads to unlimited reactions in the rods. The kinematics at the branching point is also studied. The inverse problem of dynamics at the point where the rank of the holonomic constraints is not a maximum is solvable. The Lagrange multipliers Λ_i at the branching point are not defined in a unique way, but the corresponding forces acting on the mechanism vertices are uniquely defined.