Four-Bar Linkage Simulator

Switch on the wipers and a motor spins one way, round and round. The blades do something else. They sweep, stop and sweep back. Between them sits one of the plainest devices in machine design: four bars joined by four pins.

One of the four bars does not move. It is the frame, or ground link, the fixed distance between two pivots. Two side links hinge on it, and the fourth bar, the coupler, floats between their free ends. It is the simplest closed loop of bars that can still move, and a count of freedoms shows why it is so useful. Three moving bodies in a plane have nine ways to move. Four pin joints take away eight. Exactly one is left, so turning a single side link fixes the position of every other part.

That one freedom turns up everywhere. An oil-field pumpjack uses it to turn a motor’s rotation into the nodding of its beam. A cyclist’s thigh, shin, frame and pedal crank form a linkage on each side of the bicycle, turning a push into rotation. Door closers, treadle sewing machines, oscillating fans, folding chairs and step-on trash cans are on the usual lists of applications. A thirteenth-century sketch of a sawmill hints at one, and by the sixteenth century such linkages were in wide use.

Bodies use it too. A University of Illinois dynamics reference models the human knee as a four-bar linkage of femur, tibia and the two cruciate ligaments, and shows a parrotfish gaining biting force from one in its jaw. The knee model is only approximate, it warns, because ligaments can pull but not push.

One inequality sorts every linkage

What a linkage does depends on its four lengths and nothing else. More exactly, it depends on their ratios. Scale all four by the same factor and the angles the links turn through do not change.

The sorting rule is Grashof’s theorem. Call the shortest link s, the longest l, and the other two p and q. If s + l < p + q, at least one link can turn through a full circle, and which link is shortest settles the type. Make the shortest link the frame and both side links revolve: a double-crank, also called a drag link. Make it a side link and that link is a crank while the opposite side rocks, the crank-rocker behind the wiper. Make it the coupler and both side links rock, a double-rocker. If instead s + l is greater than p + q, nothing revolves and all three moving links rock.

Equality is the awkward case. When s + l equals p + q the four bars can fold flat along one line, a position called the change point. There the geometry allows two ways on, and the driven link may leave in either direction. A parallelogram, with opposite links equal, is the familiar example. A deltoid, two equal short links joined to two equal long ones, is another, also listed under the name Galloway. Set one up in the simulator with a short link as its frame, and the short crank needs two turns to bring the linkage back to where it started.

A worked example: 100, 40, 80 and 70

Take a ground link of 100 mm, a crank of 40, a coupler of 80 and a rocker of 70. The proportions come from a worked example in Ohio University’s kinematics notes, and the simulator above opens with them. Shortest plus longest is 40 + 100 = 140, less than 80 + 70 = 150, and the short link is a side link, so this is a crank-rocker. The rocker swings between two limits, reached when crank and coupler fall into one line, stretched out or folded back. Measured from the ground line the rocker stands at 92.0 degrees at one limit and 161.8 at the other, a swing of 69.8 degrees.

Now ask how well the coupler pushes the rocker. The angle between them at their shared pin is the transmission angle, and the law of cosines gives it from the distance between the crank pin and the rocker pivot. That distance is shortest, 60 mm, when the crank points at the rocker pivot, and longest, 140 mm, when it points straight away. Those two positions give the extremes, so through the whole turn the angle stays within 46.6 to 137.8 degrees.

Lengthen the crank to 60 and the sums become 160 against 150. Grashof’s test fails. The same notes work this case: the crank can no longer go round and is held within 137.9 degrees either side of the ground line.

Why 90 degrees is the number to watch

The transmission angle matters because of how a coupler carries force. In slow motion it can only push or pull along its own length, and that push turns the output link best when it arrives square to it. The turning effect on the output link goes as the sine of that angle, so it is greatest at 90 degrees. As the angle falls toward zero or climbs toward 180 the same push turns the link less and less, until friction in the pins can absorb it.

How far from 90 is too far? There is no law, only practice. A paper on the linkages that steer a solar tracker reports that several authors suggest keeping the angle above 30 degrees, which is no more than 60 away from square, and its own two designs were held at 39 degrees or better. By that measure the worked example is comfortable. Its worst case is 47.8 degrees from square.

A poor design is easy to make. Give the simulator a frame of 100, a crank of 50, a coupler of 60 and a rocker of 100. It passes Grashof’s test, 150 against 160, and turns happily on paper, with a rocker swing of 61.0 degrees. Its transmission angle sinks to 22.3 degrees once every revolution, 67.7 degrees away from square, and the readout turns amber.

Push the angle all the way to 0 or 180 and the linkage reaches a toggle, or dead point: two links in one line. Seen from the driven end it is a wall. No torque on the far link can start the aligned pair moving, which is why crank-driven machines rely on the energy stored in a flywheel, or on a second crank set out of step with the first, to carry them past. Seen from the other end it is a lock, and toggle mechanisms that fold just beyond such a position are used as clamps.

A toggle also marks the edge of the possible. In a linkage that fails Grashof’s test, the input link stops where the coupler and the output link line up.

The point that draws

The side links can only swing on circles. The coupler is different. Pick any point on it, on the bar itself or on a plate fixed to the bar, and that point traces a closed path that is generally neither a circle nor anything else with a common name. Six lengths shape it, as Alfred Kempe counted in an 1877 lecture: the four links and the two distances that place the tracing point.

Designers treat these curves as a catalog of motions to choose from. In 1951 Hrones and Nelson published an atlas of thousands of such paths, because, as a Dutch scholar remarked, the complicated analytical theory of the curve had kept engineers from using it. And the Roberts-Chebyshev theorem says that three different four-bar linkages trace exactly the same coupler curve.

The most famous coupler curve was drawn out of necessity. James Watt’s double-acting steam engine drove its piston both up and down, and the chains that had hung earlier pistons from the beam could only pull. He needed the top of the piston rod to travel in a straight line, guided by nothing but pinned bars. In 1784 he found that the middle of a short link joining two swinging arms would very nearly do it, and put the linkage into a patent that year. “I have started a new hare,” he wrote to his partner, calling the idea “one of the most ingenious simple pieces of mechanism I have contrived.”

It is not straight. Let the tracing point run its whole course, Kempe told his audience, and it describes a figure of eight, the shape now called Watt’s curve, whose crossing portions are only nearly straight. He added that the exact problem cannot be solved with three moving links at all. Every pin-jointed four-bar straight line is an approximation.

That limit fascinated Pafnuty Chebyshev. He toured France, London and Germany in 1852 and studied steam engines at work. His paper on these mechanisms, published in 1854, is where the polynomials that carry his name first appear, and a Smithsonian history of kinematics says his interest in straight-line linkages amounted to an obsession. His own design has proportions of 5 : 4 : 2, two swinging arms of 5, pivots 4 apart and a coupler of 2 with the tracer at its middle. Kempe showed one with five-inch arms and noted that it meets the true line at exactly three points.

Other four-bar designs keep a crank that turns fully. In the linkage known as Hoecken’s, or the Chebyshev lambda, the crank is 1, the pivots are 2 apart, and the coupler and rocker are 2.5 each, with the tracer at twice the coupler’s length. A table in Norton’s Design of Machinery lists proportions of this family whose path holds straight for half of every crank revolution with a listed error of 0.096 percent.

The exact straight line arrived in 1864, from a French engineer officer named Peaucellier, and it took more than four bars. By then it hardly mattered to industry. Machines that could plane a flat guide had made the quest, in the Smithsonian historian’s words, largely academic. What survived was the mathematics, and the habit of asking how good an approximation has to be.

What a kinematic model leaves out

Everything above is geometry, and geometry is all the simulator computes. It assumes rigid bars, pins with no play and motion in a single plane, and it knows nothing about force. The transmission angle is a stand-in for force, not a measurement of it. It is read from the shape alone, so it says nothing of the loads that come from the mass and speed of the moving bars, and those take a dynamic analysis to find. Bars drawn as lines never collide; real ones must be stacked in layers to pass each other.

Nor does the model choose at a change point or a toggle, where two motions are equally possible on paper and something outside the geometry, such as the inertia of the links, decides in the machine. It also covers only the planar case, though spherical and spatial four-bar linkages exist and are used. Within those limits the old lesson holds: four lengths, one inequality, one angle to watch and one point to follow will tell you most of what a pin-jointed machine is going to do before any metal is cut.