Mechanism Design
Four-bar coupler-point curves Four-bar coupler-point curves
A coupler is the most interesting link in any linkage. It is in complex motion, and thus points on the coupler can have path motions of high degree. Four-bar coupler-point curves
The figure shows a four-bar linkage with its coupler extended to include a large number of points, each of which describes a different coupler curve. Note that these points may be anywhere on the coupler, including along line AB. There is an infinity of points on the coupler, each of which generates a different curve. Four-bar coupler-point curves
Fourbar coupler curves come in a variety of shapes which can be crudely categorized. There is an infinite range of variation between these generalized shapes. Some features of interest are the curve's double points, ones that have two tangents. They occur in two types, the cusp and the crunode. Four-bar coupler-point curves
Fourbar coupler curves come in a variety of shapes which can be crudely categorized. In general, a fourbar coupler curve can have up to three real double points which may be a combination of cusps and crunodes as can be seen in Figure. Four-bar coupler-point curves
A double point is a point on a curve at which the curve has two tangents. A double point may be of two types: - crunode – at which the tangents are distinct, the curve crossing itself - cusp – at which the tangents are coincident, the curve being tangent to itself Four-bar coupler-point curves
Cusps
The most familiar example of the cusp is derived from the curve traced by a point on the periphery of a rolling wheel. The curve is the common cycloid, one of the special cases of the trochoid. Point P is point on the centrode
We note that a cusp is a curve property associated with a point on a moving centrode and with the relative motion of centrodes.
Centrode in kinematics is the path traced by the instantaneous center of rotation of a rigid plane figure moving in a plane. Centrode in kinematics is the path traced by the instantaneous Four-bar center of rotation of a rigid plane figure moving in a plane. coupler-point curves
Cusps
The most familiar example of the cusp is derived from the curve traced by a point on the periphery of a rolling wheel. The curve is the common cycloid, one of the special cases of the trochoid.
We note that a cusp is a curve property associated with a point on a moving centrode and with the relative motion of centrodes.
We may see the action in a fourbar from figure (the coupler link is AB). Figure shows the fourbar linkage with the four coupler points (C, D, E, F) located on the moving centrode. The curves that these points trace on the fixed plane are shown in the figure. Each coupler curve shows a cusp. Four-bar coupler-point curves
Crunode
The crunode is a more obvious form of double point than the cusp. The curve crosses itself and therefore has two distinct tangents. Coupler curve with double points Q and Q´ A simple example again derives from a special case of the trochoid, specifically the prolate cycloid.
A coupler curve with two crunodes is shown in figure. For the crunode Q, there must be two positions of the coupler
AB such as A1B1 and A2B2 for which the coupler point M assumes the same position Q on the plane. Four-bar coupler-point curves
The atlas of fourbar coupler curves is a useful reference which can provide the designer with a starting point for further design and analysis.
Figure (b) shows a "fleshed out" linkage drawn on top of the atlas page to illustrate its relationship to the atlas information. The double circles in Figure (a) define the fixed pivots. The crank is always of unit length. The ratios of the other link lengths to the crank are given on top of the page. The actual link lengths can be scaled up or down to suit your package constraints and this will affect the size but not the shape of the coupler curve. Anyone of the ten coupler points shown can be used by incorporating it into a triangular coupler link. The location of the chosen coupler point can be scaled from the atlas and is defined within the coupler by the position vector R whose constant angle is measured with respect to the line of centers of the coupler. 1st step
For any position of the mechanism, we can find an infinite number of positions of the double point - Kd curve 2nd step
3rd step 4th step
Repeating the steps 2-4 with a different angle For any position of the mechanism, we can find an infinite number of positions of the double point - Kd curve Cognates
It sometimes happens that a good solution to a linkage synthesis problem will be found that satisfies path generation constraints but which has the fixed pivots in inappropriate locations for attachment to the available ground plane or frame. In such cases, the use of a cognate to the linkage may be helpful. Cognates
The term cognate was used by Hartenberg and Denavit to describe a linkage, of different geometry, which generates the same coupler curve. Samuel Roberts (1875) and Chebyschev (1878) independently discovered the theorem which now bears their names: Roberts-Chebyschev Theorem Three different planar, pin-jointed fourbar linkages will trace identical coupler curves.
Hartenberg and Denavit presented extensions of this theorem to the slider-crank and the six-bar linkages: Two different planar slider-crank linkages will trace identical coupler curves. The coupler-point curve of a planar fourbar linkage is also described by the joint of a dyad of an appropriate six bar linkage. Roberts-Chebyschev Theorem Three different planar, pin-jointed fourbar linkages will trace identical coupler curves.
Linkage OA A B OB have a common coupler point M, which traces the same coupler curves. Linkage OA A1 C1 OC The three linkages are called cognate mechanisms Linkage OB B2 C2 OC Roberts-Chebyschev Theorem
A fourbar linkage and its coupler curve Cognates of the fourbar linkage (Chebyshev straight line linkages) (Hoekens straight line linkages)