Marvelous Mechanisms: The Ubiquitous Four Bar Linkage

The four bar linkage is a type of mechanical linkage that is used in many different devices. A few examples are: locking pliers, bicycles, oil well pumps, loaders, internal combustion engines, compressors, and pantographs. In biology we can also find examples of this linkage, as in the human knee joint, where the mechanism allows rotation and keeps the two legs bones attached to each other. It is also present in some fish jaws that evolved to take advantage of the force multiplication that the four bar mechanism can provide.

How It Works

Deployable mirror with scissor linkages. By [Catsquisher] via Wikimedia Commons
The study of linkages started with Archimedes who applied geometry to the study of the lever, but a full mathematical description had to wait until the late 1800’s, however, due to the complexity of the resulting equations, the study and design of complex linkages was greatly simplified with the advent of the digital computer.

Mechanical linkages in general are a group of bodies connected to each other to manage forces and movement. The bodies, or links, that form the linkage, are connected to each other at points called joints. Perhaps the simplest example is the lever, that consists of a rigid bar that is allowed to pivot about a fulcrum, used to obtain a mechanical advantage: you can raise an object using less force than the weight of the object.

Two levers can be connected to each other to form the four bar linkage. In the figure, the levers are represented by the links a (A-D) and b (B-C).  The points A and B are the fulcrum points.  A third link f (C-D) connects the levers, and the fourth link is the ground or frame g (A-B) where the mechanism is mounted. In the animation below, the input link a (the crank) performs a rotational motion driving the rocker rod b and resulting in a reciprocating motion of the link b (the rocker).

This slider-crank arrangement is the heart of the internal combustion engine, where the expansion of gases against a sliding piston in the cylinder drives the rotation of the crank. In a compressor the opposite happens, the rotation of the crank pushes the piston to compress the gas in the cylinder. Depending on how the mechanism is arranged, it can perform the following tasks:

  • convert rotational motion to reciprocating motion, as we just discussed above.
  • convert reciprocating motion to rotational motion, as in the bicycle.
  • constrain motion, e.g. knee joint and car suspension.
  • magnify force, as in the parrotfish jaw.
Locking pliers mechanism. Image from [Engineering made easy]

Some Applications

One interesting application of the four bar linkage is found in locking pliers. The B-C and C-D links are set at an angle close to 180 degrees. When force is applied to the handle, the angle between the links is less than 180 (measured from inside the linkage), and the resulting force in the jaws tries to keep the handle open. When the pliers snap into the locked position that angle becomes less than 180, and the force in the jaws keeps the handle in the locked position.

In a bicycle, the reciprocating motion of the rider´s legs is converted to rotational motion via a four bar mechanism that is formed by the two leg segments, the bicycle frame, and the crank.

An example from nature, the Moray eel. Image from [Matthew West]
As with many other inventions of humankind, we often find that nature has already come up with the same idea via evolution. The parrotfish lives on coral reefs, from which it feeds, and has to grind the coral to get to the polyps inside. For that job, they need a very powerful bite. The parrotfish obtains a mechanical advantage to the muscle force by using a four bar linkage in their jaws! Other species also use the same mechanism, one is the Moray eel, shown in the image, which has the very particular ability to launch its jaws up in the mouth to capture its prey, much like the alien from the film series.

The joints connecting the links in the linkage can be of two types. A hinged joint is called a revolute, and a sliding joint is called a prismatic. Depending on the number of revolute and prismatic joints, the four bar linkage can be of three types:

  • Planar quadrilateral linkage formed by four links and four revolute points. This is shown in the animation above.
  • Slider-crank linkage, formed by three revolute joints and a prismatic joint.
  • Double slider formed by two revolute joints and two prismatic joints. The Scotch yoke and the trammel of Archimedes are examples.

There are a great number of variations for the four bar linkage, and as you can guess, the design process to obtain the forces and movements that we need is not an easy task. An excellent resource for the interested reader is KMODDL (Kinematic Models for Design Digital Library) from Cornell University. Other interesting sites are the 507 mechanical movements, where you can find nice animations, and [thang010146]’s YouTube channel.

We hope to have piqued your curiosity in mechanical things. In these times of ultra fast developments in electronics, looking at the working of mechanisms that were developed centuries ago, but are still present and needed in our everyday lives can be a rewarding experience. We plan to work on more articles featuring interesting mechanisms so please let us know your favorites in the comments below.

Don’t Fear the Filter: Cascading Sallen-Keys

In the last edition of Don’t Fear the Filter, we built up two examples of the simplest and most-used active filter of all time: the two-pole Sallen-Key lowpass. This time, we’re going to put two of these basic filter blocks in a row, and end up with a much sharper lowpass filter as well as a bandpass filter. For the bandpass, we’ll need to build up a quick highpass filter as well. Bonus!

I claimed last time that the Sallen-Key lowpass would cover something like 80% of your filtering needs. (And 72.4% of all statistics are totally made up!) These two will probably get you through another 10% or so. Honestly, I’ve never built a standalone active highpass, for reasons we’ll see below, but the active bandpass filter that we’re building it for is a great tool to have in your belt, especially for anything audio.

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Getting Sparks from Water with Lord Kelvin’s Thunderstorm

In the comments to our recent article about Wimshurst machines, we saw that some hackers had never heard of them, reminding us that we all have different backgrounds and much to share. Well here’s one I’m guessing even fewer will have heard of. It’s never even shown up in a single Hackaday article, something that was also pointed out in a comment to that Wimshurst article. It is the Lord Kelvin’s Water Dropper aka Lord Kelvin’s Thunderstorm, invented in the 1860s by William Thomson, 1st Baron Kelvin, the same fellow for whom the Kelvin temperature scale is named.  It’s a device that produces a high voltage and sparks from falling drops of water.

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Shut Up and Say Something: Amateur Radio Digital Modes

In a recent article, I lamented my distaste for carrying on the classic amateur radio conversation — calling CQ, having someone from far away or around the block call back, exchange call signs and signal reports and perhaps a few pleasantries. I think the idle chit-chat is a big turn-off to a lot of folks who would otherwise be interested in the World’s Greatest Hobby™, but thankfully there are plenty of ways for the mic-shy to get on the air. So as a public service I’d like to go over some of the many digital modes amateur radio offers as a way to avoid talking while still communicating.

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Say It With Me: Aliasing

Suppose you take a few measurements of a time-varying signal. Let’s say for concreteness that you have a microcontroller that reads some voltage 100 times per second. Collecting a bunch of data points together, you plot them out — this must surely have come from a sine wave at 35 Hz, you say. Just connect up the dots with a sine wave! It’s as plain as the nose on your face.

And then some spoil-sport comes along and draws in a version of your sine wave at -65 Hz, and then another at 135 Hz. And then more at -165 Hz and 235 Hz or -265 Hz and 335 Hz. And then an arbitrary number of potential sine waves that fit the very same data, all spaced apart at positive and negative integer multiples of your 100 Hz sampling frequency. Soon, your very pretty picture is looking a bit more complicated than you’d bargained for, and you have no idea which of these frequencies generated your data. It seems hopeless! You go home in tears.

But then you realize that this phenomenon gives you super powers — the power to resolve frequencies that are significantly higher than your sampling frequency. Just as the 235 Hz wave leaves an apparent 35 Hz waveform in the data when sampled at 100 Hz, a 237 Hz signal will look like 37 Hz. You can tell them apart even though they’re well beyond your ability to sample that fast. You’re pulling in information from beyond the Nyquist limit!

This essential ambiguity in sampling — that all frequencies offset by an integer multiple of the sampling frequency produce the same data — is called “aliasing”. And understanding aliasing is the first step toward really understanding sampling, and that’s the first step into the big wide world of digital signal processing.

Whether aliasing corrupts your pristine data or provides you with super powers hinges on your understanding of the effect, and maybe some judicious pre-sampling filtering, so let’s get some knowledge.

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How Does a Voltage Multiplier Work?

If you need a high voltage, a voltage multiplier is one of the easiest ways to obtain it. A voltage multiplier is a specialized type of rectifier circuit that converts an AC voltage to a higher DC voltage. Invented by Heinrich Greinacher in 1919, they were used in the design of a particle accelerator that performed the first artificial nuclear disintegration, so you know they mean business.

Theoretically the output of the multiplier is an integer times the AC peak input voltage, and while they can work with any input voltage, the principal use for voltage multipliers is when very high voltages, in the order of tens of thousands or even millions of volts, are needed. They have the advantage of being relatively easy to build, and are cheaper than an equivalent high voltage transformer of the same output rating. If you need sparks for your mad science, perhaps a voltage multiplier can provide them for you.

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Negative Resistance: It Shouldn’t Make Sense!

When you leaf through a basic electronics textbook, you’ll find chapters describing in detail the operation of the various components. Resistors, capacitors, inductors, and semiconductors. The latter chapter will talk about P and N type regions, introduce us to the diode, and then deal with the transistor: its basic operation, how to bias it, and the like.

A tunnel diode amplifier circuit. Chetvorno [CC0]
Particularly if your textbook is a little older, you may find a short section talking about the tunnel diode. There will be an odd-looking circuit that seems to make no sense at all, an amplifier formed from just a forward-biased diode and a couple of resistors. This logic-defying circuit you are told works due to the tunnel diode being of a class of devices having a negative resistance, though in the absence of readily available devices for experimentation it can be difficult to wrap your head around.

We’re all used to conventional resistors, devices that follow Ohm’s Law. When you apply a voltage to a resistor, a current flows through it, and when the voltage is increased, so does the current. Thus if you use a positive resistance device, say a normal resistor, in both the top and the bottom halves of a potential divider, varying the voltage fed into the top of the divider results in the resistor behaving as you’d expect, and the voltage across it increases.

In a negative resistance device the opposite is the case: increasing the voltage across it results in decreasing current flowing through it. When a large enough negative resistance device is used in the lower half of a resistive divider, it reduces the overall current flowing through the divider when the input voltage increases. With less current flowing across the top resistor, more voltage is present at the output. This makes the negative resistor divider into an amplifier.

The tunnel diodes we mentioned above are probably the best known devices that exhibit negative resistance, and there was a time in the early 1960s before transistors gained extra performance that they seemed to represent the future in electronics. But they aren’t the only devices with a negative resistance curve, indeed aside from other semiconductors such as Gunn diodes you can find negative resistance in some surprising places. Electrical arcs, for example, or fluorescent lighting tubes.

A typical negative resistance I-V curve. Chetvorno [CC0]
The negative resistance property of electric arcs in particular produced a fascinating device from the early twentieth century. The first radio transmitters used an electric arc to generate their RF, but were extremely inefficient and wideband, causing interference. A refinement treated the spark not as the source of the RF but as the negative resistance element alongside a tuned circuit in an oscillator, These devices could generate single frequencies at extremely high power, and thus became popular as high-powered transmitters alongside those using high-frequency alternators until the advent of higher powered tube-based transmitters around the First World War.

It’s unlikely that you will encounter a tunnel diode or other similar electronic component outside the realm of very specialist surplus parts suppliers. We’ve featured them only rarely, and then they are usually surplus devices from the 1960s. But understanding something of how they operate in a circuit should be part of the general knowledge of anyone with an interest in electronics, and is thus worth taking a moment to look at.

1N3716 tunnel diode header image: Caliston [Public domain].