On the geometry of the planar 4-bar mechanism

Abstract

The paper seeks to elucidate the geometry of a simple engineering mechanism, comprising four bars smoothly jointed together to form a movable quadrilateral with one fixed side. The configurations of this mechanism correspond to the points of an elliptic curve, to which is associated interesting geometry and Morse theory. By appropriate projection, this curve yields the 2-parameter family of plane curves described by points rigidly attached to the side of the quadrilateral opposite the fixed side: the geometry of the general projection is related to the configuration of lines on a Segre quartic surface.

Real inflections of hinged planar four-bar coupler curves

“…In fact for any coupler point (except the ends of the coupler bar) the complex coupler curve is a projection of the complex residual curve in P C 6 described in section 2 and so has degree six (Gibson and Newstead [5]). The coupler curve has the same geometric genus as the corresponding residual curve, that is to say it is elliptic (genus 1) for generic four-bars (as we have already said) and rational (genus 0) for Grashof four-bars.…”

“…Since there are altogether only three finite double points on such a coupler curve, at most one can be an acnode. The result now follows from equation (5). However no direct method presents itself of ascertaining whether coupler curves exist which exhibit 12 inflections.…”

“…Following Darboux, the configuration space of the mechanism can be described in terms of unit vectors (x i , y i ), i = 1, 2, 3 along the corresponding bars (see, for example Gibson and Newstead [5]). The six variables are connected by five equations, expressing the fact that each vector has unit length and that the four-bar forms a closed quadrilateral.…”

“…Four-bars for which (2) does not hold shall be called generic. Those that satisfy (2) are classified in [5] according to their singularities or equivalently by the geometry of the quadrilateral formed by the four-bar. The classification can be visualised in terms of the equations (2).…”

“…Let E = e 1 + e 4 − e 2 − e 3 denote the sum of the shortest and longest sides minus the lengths of the other two. Then it is a standard fact (again, derived in [5]) that for E < 0 the configuration space is a one-dimensional manifold consisting of two connected components (topologically circles), while if E > 0 the manifold has only one component. An invaluable refinement is the classification of generic four-bars, described by Hain [6], in which they are described by means of the rocking (R)…”