Four-bar coupler-point curves

Mechanism Design

Four-bar coupler-point curves

A coupler is the most interesting link in anylinkage. It is in complex motion, and thus

points on the coupler can have path motionsof high degree.

The figure shows a four-bar linkagewith 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, includingalong line AB. There is an infinity ofpoints 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 betweenthese 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 twotangents. A double point may beof two types: - crunode – at which the

tangents are distinct, the curvecrossing itself

- cusp – at which the tangentsare coincident, the curve beingtangent to itself

Four-bar coupler-point curves

Cusps

The most familiar example of the cusp is derived from thecurve traced by a point on the periphery of a rolling wheel. The curve is the common cycloid, one of the special casesof the trochoid.

We note that a cusp is a curve property associated with a point on a moving centrode and with the relative motionof centrodes.

Four-bar coupler-point curves

Centrode in kinematics is the path traced by the instantaneous center of rotation of a rigid plane figure moving in a plane.

Point P is point on the centrode

Cusps

The most familiar example of the cusp is derived from thecurve traced by a point on the periphery of a rolling wheel. The curve is the common cycloid, one of the special casesof the trochoid.

We note that a cusp is a curve property associated with a point on a moving centrode and with the relative motionof centrodes.

Four-bar coupler-point curves

We may see the action in a fourbar from figure (thecoupler link is AB). Figure shows the fourbar linkage withthe four coupler points (C, D, E, F) located on the movingcentrode. The curves that these points trace on the fixedplane are shown in the figure. Each coupler curve shows a cusp.

Centrode in kinematics is the path traced by the instantaneous center of rotation of a rigid plane figure moving in a plane.

A coupler curve with two crunodes is shown in figure. Forthe crunode Q, there must be two positions of the couplerAB such as A1B1 and A2B2 for which the coupler point M assumes the same position Q on the plane.

Four-bar coupler-point curves

Coupler curve with double points Q and Q´

Crunode

The crunode is a more obviousform of double point than the cusp.

The curve crosses itself and therefore has two distinct tangents.

A simple example again derivesfrom a special case of the trochoid,

specifically the prolate cycloid.

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 bescaled 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.

Four-bar coupler-point curves

CognatesIt sometimes happens that a good solution to a linkage

synthesis problem will be found that satisfies pathgeneration 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 cognateto the linkage may be helpful.

CognatesThe 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:

Three different planar, pin-jointed fourbar linkages will trace identical coupler curves.

Roberts-Chebyschev Theorem

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.

Three different planar, pin-jointed fourbar linkages will trace identical coupler curves.

Roberts-Chebyschev Theorem

Linkage OA A B OB

Linkage OA A1 C1 OC

Linkage OB B2 C2 OC

have a common coupler point M, which traces the same coupler curves.

The three linkages are called cognate mechanisms

A fourbar linkage and its coupler curve Cognates of the fourbar linkage

(Chebyshev straight line linkages) (Hoekens straight line linkages)

Roberts-Chebyschev Theorem