A NEW FOUR-BAR LINKAGE COMPLETING DELASSUS' FINDINGS

In 1922, E. Delassus found out four-bar linkages including parallel screw joints (or helical pairs), which are movable with one degree of freedom. The Delassus linkages can move only when geometric conditions based on the Euclidean metric are achieved; their mobility is said to be paradoxical. Delassus stated that there is no paradoxical four-bar linkage with three helical H pairs and one prismatic P pair. In fact, Delassus did one error of logical deduction. After the error detection, the paper introduces a new HHHP linkage with a paradoxical mobility. Its three H axes are parallel and intersect a plane that is orthogonal to the H axes in three points forming an isosceles triangle. The P pair is parallel to the triangle side, which has a variable length. The three pitches have to satisfy one scalar relation. Through the consideration of helical trajectories in relative motions, the new mechanism is related to a geometric property of the right helicoid. Helices with a pitch that is half the helicoid pitch can be drawn on the right helicoid.