Speaking from 30,000 feet, power transmission couplings are devices used to connect two shafts together, transmitting system torque from one shaft to the other. (When one shaft spins, the coupling's job is to make the other shaft spin.) Within the huge array of coupling solutions, couplings can be broadly broken down into two primary types: rigid (which we will briefly touch on) and flexible (which we will then dive into).
Rigid couplings (an example pictured left) are exactly that... rigid. They firmly connect the two shafts together without any additional features or capability of accommodating/handling system misalignment. Rigid couplings are generally simple and cost-effective for applications where misalignment is not a concern.In contrast to rigid couplings, flexible couplings (the focus of this article) have an integrated flexing element or design component that allows for some degree of misalignment handling & management. Within the vast flexible coupling world, couplings can further be broken into two major sub-groups: elastomeric couplings and metallic couplings.
Elastomeric (flexible) couplings - Elastomeric couplings are couplings that include a flexing rubber or plastic element to both accommodate misalignment and dampen system vibrations. Within the elastomeric coupling subset, there are many style and designs. This post will focus in on and provide a quick cliff notes overview of the 3 major elastomeric flexible coupling product types: compression loaded, shear loaded, and torsional.
(Note: Lovejoy's "The Coupling Handbook" takes a much deeper dive into everything this blog post covers, and the Mechanical Power Transmission Association also has an excellent 7 page PDF document, title Elastomeric Coupling Primer, that would make for great follow-on reading as well.)
Compression Loaded: Let's start the elastomeric compression loaded discussion with the bread and butter all-purpose industrial coupling: Lovejoy's very own straight jaw coupling. This coupling comprises two hubs with straight jaws that are interlocked with an elastomeric spider (we call it a spider because, yes, it looks like one) in between the two hubs (with the spider serving as the flexing element) transmitting torque in compression. One feature of this coupling is that it is considered "fail-safe". If/when the spider eventually wears out (which, after a full service life it will, as any/every rubber based product will eventually break down), the metallic jaw hubs can then continue to carry the load (though not as smooth, and with a bit more noise). (Note: It is certainly preferable to replace a spider before it fails, generally when it has been compressed to 75% its original size... as this is much cheaper and quicker than replacing both hubs and the spider, which will be required if the hubs start to wear on each other if the coupling is run in the absence of a functional spider.)
The basic jaw coupling (known as an L-line, and pictured above next to "Compression Loaded" title), has many variations for specific applications. These include aluminum (lightweight) and stainless steel (for food and pharmaceutical applications) hubs, special spiders made for high temperatures, high torques, chemical and oil resistance (and even a bronze spider for high torque, low speed applications), quick change out radial spiders, drop out spacers, and a variation (known as a jaw in-shear & pictured left) that turns a "fail safe" jaw coupling (where the elastomer is in compression) into a "non-fail safe" coupling (where the elastomer is a combination of both in-shear and in compression, preferable for applications where torque transmission should cease should the coupling not be operating at full capacity).
In Europe, a curved jaw coupling variation has become the standard industrial coupling, and Lovejoy sells a large number of curved jaw couplings into Europe through Lovejoy's German affiliate. Lovejoy also replaces a significant number of curved jaw coupling components on equipment in the United States that has been imported from overseas. (Note: Just like Lovejoy's L line, curved jaw couplings are manufactured, finished, and readily available from Lovejoy's Downers Grove, IL manufacturing facility.) While curved jaw couplings cannot be turned into jaw in-shear like straight jaw (due to their tooth profile), one nice feature of curved jaw couplings is that, by tightening their tolerance and using a very stiff elastomer, the curved jaw couplings can be turned into a very affordable backlash free coupling. (Backlash refers to the looseness of fit of a coupling, which is generally undesirable in very precise motion control applications.) 
Shear Loaded: In addition to the Jaw In-Shear couplings, there are several other popular elastomer in-shear designs. These popular designs include the tire (or tyre) coupling, and the sleeve coupling (or S-Flex coupling, pictured left). A common trait among all of these shear loaded designs is that system torque transmission will cease if/when the elastomer fails. In this capacity, the coupling is acting similar to a non-calibrated fuse. (Elastomers generally are not rated or designed to fail under specific load conditions, and should not be used or trusted to be a fuse. However, they are often designed as the weak point in a power transmission system... and will fail if there is a major system lockup. Solutions to include adding rated shear pins and/or a torque limiter to the coupling design are available if a user is looking/requiring their coupling to act explicitly as a fuse.)
Torsional: The third and final elastomeric coupling grouping to cover are torsional couplings. Mechanical power transmission systems can have devastating natural frequencies (think Tacoma Narrows Bridge Collapse). Diesel engine applications are one of the most common applications where natural frequencies need to be managed. The goal of a torsional coupling is to tune the system above or below its natural frequencies... and both torsionally soft (incorporating soft rubber) and torsionally hard (often incorporating hard plastics) are available to tune the system. Selecting the proper torsional coupling is not a trivial task (generally involves a formal torsional vibration analysis), and it is highly recommended that you consult with a coupling manufacturer's staff prior to making your own product selection. (Lovejoy offers multiple types of torsional couplings, and can be interchangeable with specific other manufacturers.)
Metallic (flexible) couplings - Metallic couplings are different from elastomeric couplings in that they do not employee elastomeric (soft) materials to provide coupling flexibility & dampening. The breadth of metallic coupling offerings is massive (covered in depth in The Coupling Handbook), and this post will focus in on and provide a quick cliff notes overview of the 2 major metallic flexible coupling product types: lubricated and non-lubricated.
Lubricated metallic couplings achieve flexibility through loose fitting parts rolling or sliding against one another, while non-lubricated metallic couplings achieve flexibility through a flexing or bending of a metal component itself. Lubricated couplings are generally less expensive, but do require periodic maintenance/more lubrication, and will eventually "wear out". Non-lubricated are generally more expensive, require minimal maintenance, and categorizes as having theoretical "infinite life" (no metal on metal wearing parts).
(Note: While the Lovejoy brand is near synonymous with the elastomeric coupling market, the company has been a major player in the metallic coupling industry for several decades. Many people are surprised to learn that Lovejoy's metallic coupling sales are on par with Lovejoy's elastomeric coupling sales, and that their knowledge of the products and applications is so great.)
Lubricated Couplings: The three major types of lubricated metallic couplings are: gear, grid, and chain. The primary form of failure for these type couplings is wear (metal on metal contact), meaning torque peaks/overloads as well as poor or improper lubrication/grease maintenance will shorten the coupling's life.
Of the three major lubricated metallic coupling types, gear couplings (where misalignment is achieved through crowning on the gear tooth surfaces) are historically the big boy on the block. Gear couplings have a very high power density (can carry huge torque loads in a small footprint), have many available custom application options (i.e. - flanged or continuous sleeves, spacers, floating shafts, limited end float, sliders, insulation), can be balanced to operate at high RPMs, and are generally lower cost than other equivalent high torque coupling alternatives. Inclusive of Lovejoy, a number of manufacturers' flanged gear couplings are half coupling for half coupling interchangeable through size 9, given they adhere to the common AGMA standard (AGMA 9008-B00: Flexible Couplings -- Gear Type -- Flange Dimensions, Inch Series). Please remember, interchangeability does not mean coupling quality, reliability, ratings, or performance characteristics are equivalent... so please proceed cautiously when selecting a vendor. 
Grid couplings (where misalignment is achieved between a single spring steel serpentine grid wrapped around two flanges) are also a very well respected lubricated metallic coupling. One advantage of this coupling is that the flexibility of the grid provides it an ability to spread out impact energy over time... allowing the coupling an opportunity to reduce the magnitude of peak loads. Grid couplings can be used in both horizontal and vertical axis applications, and also have may additional feature upgrades (floating shafts, break discs, spacers, etc.). Grid couplings generally compete with large elastomeric couplings (which may be too large to fit inside the system's space constraints), as well as with gear couplings (given the grid coupling's enviable all-metal ability to modestly dampen vibration). Most major grid coupling manufacturers (inclusive of Lovejoy) have kept the majority of their grid coupling components directly interchangeable... with the exception being unique seals and gaskets. (If interchanging manufacturers, corresponding seals and gaskets should be procured from the same manufacturer to avoid potential sealing issues). In contrast to gear and grid, chain couplings are somewhat of a dirty step child. In full disclosure, Lovejoy does not manufacture chain couplings... so we may be a bit biased... but, generically speaking, chain couplings are found and used on unsophisticated applications (i.e. - makeshift farming equipment). Chain couplings are known for being relatively rugged and very low cost. A chain coupling consists of two sprocket hubs with a single double roller chain connecting the two hubs. These couplings are relatively easy to install, maintain, and rough align.
Non-Lubricated Couplings: Popular non-lubricated metallic flexible couplings include the disc (or disk), diaphragm, link, spiral wound, bellows, and beam coupling types. All six have a theoretical infinite life (meaning they have no metal on metal wearing parts), assuming the flexing or load carrying methods stay within the mechanical endurance limits of the flexing metal material. (Overload on these type couplings, be it continuous torque or cyclic misalignment forces, will result in fatigue failure.)
These couplings have been historically complex to understand and evaluate... as significant stress analysis (finite element analysis) must be conducted to flush out performance characteristics (inclusive of taking torque load, misalignment, temperature, and varying system speeds into consideration). Non-lubricated couplings generally have a higher upfront cost, relative to traditional lubricated couplings, but can offer long term "total cost of ownership" savings opportunities.
Of all non-lubricated coupling types, disc couplings (with multiple layers of flexing discs) are the most popular... and they continue to pick up steam both in new designs (inclusive of boiler feed pumps, gas and steam turbines,
compressors, high-speed test stands, marine propulsion systems, and wind energy) and in replacing traditional installed gear coupling applications (where either maintenance, reliability, or environmental concerns arise). Disc couplings can excel at high speeds (where balancing and lubrication concerns put gear couplings at a disadvantage), and, furthermore, have the built in advantage of no backlash which lubricated metallic couplings cannot claim). One drawback of disc couplings is that they are generally less tolerant of misalignment (queue smiles from the shaft laser alignment product sales folks). Without diving too deep into disc couplings, disc packs can be circular (call it version 1.0), flat sided (version 2.0), or scalloped (version 3.0) on the outer dimension... with each revision offering improved performance. (Circular disc packs acts as a beam... stressing the extreme edges, flat sided packs avoid the curved disc pack drawbacks, and scallop disc packs both avoid the curved disc pack drawbacks and provide more flexibility/misalignment handling capability. The additional capability can be attributed to the disc pack's reduced cross-section... which requires less force to flex, translating to lower reactionary loads on the system's adjacent bearings.) All three disc pack styles are readily available on the market (though Lovejoy only sells the scalloped version).
Diaphragm couplings were originally introduced to service very high speed, high horsepower applications in the petrochemical industry... and has since progressed to other extreme applications such as helicopter drives. Diaphragm couplings handle misalignment through use of a flexing metal plate (or series of flexing metal plates in parallel). The metal plate(s) is loaded in shear, with torque being introduced at the outside diameter of the coupling and then transferred on to the inside diameter. (The process reversed at the opposite flex point.)
Diaphragm couplings are known for their large outside diameters, and, generally, very high cost. Diaphragm couplings are generally sold as custom solutions, and there are a wide variety of options to consider... so those seriously in the market for this coupling should spend a considerable amount of time speaking with manufacturers' application engineering staffs. (Note: Lovejoy does not sell diaphragm couplings, though Lovejoy does sell API 610 or 671 disc couplings that can sometime compete for the same application.)
Taking a huge step back from size and cost of diaphragm couplings, triple wound spring couplings are generally small and much more an off the shelf mainstream product (translation: Lovejoy has lots of these in stock). These couplings handle torques generally only up to ~1800 in-lbs, 213 Nm), with bores up to 1.5 inches (38 mm), and speeds up to 30,000 RPM. They operate by having three tension springs mounted (one inside the other) to the two shaft hubs. The middle spring runs in the opposite direction of the inner and outer spring to allow the coupling to transmit torque in either direction. As each spring can flex independently, both angular and parallel misalignment can be addressed with this coupling. This coupling does have both backlash and windup, but their values are known... so these couplings can be found in use on index positioning and robotic applications. 
The three link couplings is variation/cousin to the traditional disc pack coupling, where two sets of three flat strip springs are attached to a triangular plate at the inner diameter and a circular flange at the outer. Links function similar to disc legs, only they are thick enough to be loaded in compression on one side and in tension on the other. With two flex planes, this coupling was designed for lightly loaded, high misalignment (up to ~5° angular) applications with smooth non-cyclic torque loads. 
In addition to disc couplings, bellows couplings (pictured left) and beam couplings (pictured below left) are considered well suited all-metal motion control couplings (Motion control applications include shaft encoders, resolvers, all forms
of servo devices, linear and ball screw actuators, robots, step motors,
light duty pumps and metering devices, plotters, medical equipment,
positioning tables, computers and radar.) Key features/requirements for motion control couplings are their torsional rigidity, low inertia, constant velocity, low radial
stiffness, zero backlash, corrosion resistance and the capability of
cyclic (repeated start/stop/reverse) activity.
Bellows couplings use thin tubular metal bellows (either one or two layers) formed with annual corrugations as the flexing element, while beam couplings (pictured left) use a single piece of aluminum or stainless steel (generally bar stock) cut with single or double flex planes. While there are many similarities between bellows and beam couplings that make them both well suited for many common applications, one notable performance variance is that bellows couplings are generally more torsionally stiff than beam couplings (though beam couplings are also fairly stiff).
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