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{{Short description|Mechanism with bendable rotation axis}}
{{Redirect|U-joint|the plumbing fixture|U-bend}}
[[Image:Universal joint.gif|right|frame|A universal joint]]

A '''universal joint''' (also called a '''universal coupling''' or '''U-joint''') is a [[joint (mechanics)|joint]] or [[coupling]] connecting rigid [[shaft (mechanical engineering)|shaft]]s whose [[wikt:axis#Noun|axes]] are inclined to each other. It is commonly used in shafts that transmit [[rotation around a fixed axis|rotary motion]]. It consists of a pair of [[hinge]]s located close together, oriented at 90° to each other, connected by a cross shaft. The universal joint is not a [[constant-velocity joint]].<ref>{{cite web|url=https://www.youtube.com/watch?v=VD1PuGdVB5U|title=Kinematics with MicroStation - Ch02 J Hookes Joint|last=UjjwalRane|date=8 July 2010|access-date=4 May 2018|via=YouTube|url-status=live|archive-url=https://web.archive.org/web/20160311050830/https://www.youtube.com/watch?v=VD1PuGdVB5U|archive-date=11 March 2016}}</ref>

U-joints are also sometimes called by various [[eponym]]ous names, as follows: 
* '''Cardan joint''', after [[Gerolamo Cardano]], a polymath of the 16th century who contributed to knowledge of various clever mechanisms, including [[gimbal]]s 
* '''Hooke joint''' or '''Hooke's joint''', after [[Robert Hooke]], a polymath of the 17th century who contributed to knowledge of various clever mechanisms 
* '''Spicer joint''', after [[Clarence W. Spicer]] and the [[Spicer Manufacturing Company]], who manufactured U joints
* '''Hardy Spicer joint''', after the [[Hardy Spicer]] brand, a successor to the Spicer brand 

== History ==
[[File:Universal Shaft.ogv|thumbnail|This video shows different parts and operation of the universal shaft.]]
[[File:Spicer u-joint advert in Automobile Trade Journal 1916.png|thumb|Spicer universal joints for motor cars, 1916.]]

The main concept of the universal joint is based on the design of [[gimbal]]s, which have been in use since antiquity. One anticipation of the universal joint was its use by the ancient Greeks on [[ballistae]].<ref>see: "Universal Joint - Invented by Gerolamo Cardano" {{cite web |url=http://www.edubilla.com/invention/universal-joint/ |title=Universal Joint,Universal Joint inventors |access-date=2017-04-21 |url-status=live |archive-url=https://web.archive.org/web/20170422130444/http://www.edubilla.com/invention/universal-joint/ |archive-date=2017-04-22 }}</ref> In Europe the universal joint is often called the Cardano joint (and a [[drive shaft]] that uses the joints, a Cardan shaft), after the 16th century Italian mathematician, [[Gerolamo Cardano]], who was an early writer on gimbals, although his writings mentioned only gimbal mountings, not universal joints.<ref>See:
* Tony Rothman (2013) "Cardano v. Tartaglia: The Great Feud Goes Supernatural", p. 25. Available on-line at: [https://arxiv.org/ftp/arxiv/papers/1308/1308.2181.pdf Arxiv.org]. (Note that Rothman mentions Wikipedia's error regarding Cardano's supposed invention of the universal joint.)
* Hans-Christoph Seherr-Thoss, Friedrich Schmelz, Erich Aucktor, ''Universal Joints and Driveshafts: Analysis, Design, Applications'' (Berlin, Germany: Springer Verlag, 1992), [https://books.google.com/books?id=-iLyCAAAQBAJ&pg=PA1 p. 1.]
* Marie Boas, ''The Scientific Renaissance: 1450-1630'' (New York, New York: Harper Brothers, 1962), [https://archive.org/stream/scientificrenais007153mbp#page/n209/mode/2up p. 186] {{webarchive|url=https://web.archive.org/web/20160411103028/https://archive.org/stream/scientificrenais007153mbp |date=2016-04-11 }}.
* James Eckman, ''Jerome Cardan'' (Baltimore, Maryland: The Johns Hopkins Press, 1946.), p. 77.
* Hieronymi Cardanime (Gerolamo Cardano), ''De Subtilitate Libri XXI.'' (On subtle things in 21 books) (Basel, Switzerland: Sebastian Henric Petri, 1553), ''Liber XVII. De Artibus, Artificiosisque; rebus.'' (Book 17. On crafts and ingenious devices), p. 817. (Note: (1) This book is a reprint of the 1500 original. (2) In the margin of p. 817 is printed: ''Sedes mira'' (miraculous chair).) [https://archive.org/stream/immagineDE295MiscellaneaOpal#page/n895/mode/2up From p. 817:] {{webarchive|url=https://web.archive.org/web/20171011035211/https://archive.org/stream/immagineDE295MiscellaneaOpal |date=2017-10-11 }} ''"Simili ratione inventũ est, ut Cæsaris sedes ita disponeretur, ut quocumque situ constituatur, ille immobilis, ac commodè dum vehitur sedeat. Hoc tractum ex armillarum ratione: cum enim circuli tres chalybei constituentur, polis sursum, deorsum, antè, retro, dextra ac sinistra mobilibus, cum plures non possint esse situs, necesse est ipsum in essedo quomodocumque agatur quiescere perpetuò."'' (By similar reasoning, [it] has been found that the Emperor's chair might be so arranged that he [remain] fixed in whatever orientation be decided and he sit comfortably while he is transported. This is based on the logic of the gimbal mounting: the three steel rings are arranged by the movable poles [i.e., ends of the axes] upwards, downwards, forwards, backwards, right and left, when more [motions] cannot be allowed, [because it] is necessary [that] he in the carriage somehow be made to remain still constantly.)
* Hieronymi Cardani (Gerolamo Cardano), ''Mediolanensis Philosophi ac Medici Celeberrimi Operum'' [Of the very famous works of the Milanese philosopher and physician] (Lyon (Lugdunum), France: Jean Antoine Huguetan and Marc Antoine Ravaud, 1663), vol. 10: ''Opuscula miscellanea'' (Miscellaneous works), ''Paralipomenon'' (Supplement), ''Liber V. De rebus factis raris & artificiis'' (Book 5. On rare and ingeniously made things), ''Caput VII. De Armillarum instrumento'' (Chapter 7. On the armillary), [https://books.google.com/books?id=kgxTAAAAcAAJ&pg=PA488 pp. 488-489.]</ref>

The mechanism was later described in ''Technica curiosa sive mirabilia artis'' (1664) by [[Gaspar Schott]], who mistakenly claimed that it was a [[constant-velocity joint]].<ref name="Mills2007">Mills, Allan, "Robert Hooke's 'universal joint' and its application to sundials and the sundial-clock", ''Notes & Records of the Royal Society'', 2007, accessed [http://rsnr.royalsocietypublishing.org/content/61/2/219.full.pdf+html online] {{webarchive|url=https://web.archive.org/web/20150925130716/http://rsnr.royalsocietypublishing.org/content/61/2/219.full.pdf+html |date=2015-09-25 }} 2010-06-16</ref><ref>Gasparis Schotti, ''Technica Curiosa, sive Mirabilia Artis, Libris XII. … '' [Curious works of skill, or marvelous works of craftsmanship] (Nuremberg (Norimberga), (Germany): Johannes Andreas Endter & Wolfgang Endter, 1664), ''Liber IX. Mirabilia Chronometrica, … '' (Book 9. Marvelous Clocks, … ), ''Caput V. Signa chronometrica optica, seu indices.'' (Chapter 5. Marvelous visual clocks, or clocks with hands), [https://books.google.com/books?id=dhRTAAAAcAAJ&pg=PA664 pp. 664-665:] ''Propositio XX. Indicem sinuosum & obliquatum per anfractus quosvis, sine Rotis dentatis quocumque lubet educere.'' (Proposition 20. [How], without any gears, to lead the twisting, turning pointer [i.e., the shaft that drives the clock's hands] through any bend one pleases.) In the margin is printed: ''Vide Iconism. VII. Fig. 32.'' (See [https://books.google.com/books?id=DKPSEzEWz8EC&pg=PA662-IA2 Plate 7, Figure 32.]), which depicts Schott's universal joint. Schott first notes that there may be occasions when a clock's gear works and its face can't be conveniently aligned; e.g., public clocks installed in towers. He then mentions, in the description of its construction (''Technasma'', the Greek word for "artifice"), that the universal joint resembles a gimbal that is used to hold an oil lamp so that it won't spill oil. Schott's joint consists of two forks (''fuscinula''), each of which consists of a shaft to which a metal strip, bent into a semicircle, is attached to one end. Near each end of the semicircle, a hole is drilled. A cross with four perpendicular arms (''crux sive 4 brachia'') is also made. The holes in each semicircle fit over the ends of an opposing pair of arms. The angle between the shafts must be greater than a right angle. In discussing the joint's motion (''Motus''), Schott claims that the two shafts move at the same speed (i.e., they form a constant-velocity joint): "'' … horum autem ductum necesse est sequatur & altera fuscinula, parique cum priore illa feratur velocitate: unde si fuerit unius fuscinulae motus regularis circularis, erit similis & alterius … ''" ( … but this driven [fork] must follow the other [driving] fork, and it be born at a speed equal to the former: whence if one fork's motion were regularly circular, it will be similarly with the other … ).</ref><ref>For a (partial) history of universal joints, see: Robert Willis, ''Principles of Mechanism'' … , 2nd ed. (London, England: Longmans, Green, and Co., 1870), Part the Fifth: On Universal Joints, [https://books.google.com/books?id=KFjHSfh561MC&pg=PA437 pp. 437-457.]</ref> Shortly afterward, between 1667 and 1675, [[Robert Hooke]] analysed the joint and found that its speed of rotation was nonuniform, but that property could be used to track the motion of the shadow on the face of a sundial.<ref name="Mills2007"/> In fact, the component of the [[equation of time]] which accounts for the tilt of the equatorial plane relative to the ecliptic is entirely analogous to the mathematical description of the universal joint. The first recorded use of the term 'universal joint' for this device was by Hooke in 1676, in his book ''Helioscopes''.<ref>"universal, ''a. (adv.)'' and ''n.''", para.13, Oxford English Dictionary Online, accessed 2010-06-16</ref><ref>Hooke first described a universal joint in [[Johannes Hevelius|Hevelius']] instrument in: Robert Hooke, ''Animadversions on the first part of the Machina Coelestis'' … (London, England: John Martyn, 1674), [https://books.google.com/books?id=KAtPAAAAcAAJ&pg=PA73 p. 73.] Here he calls the joint a "universal Instrument". From page 73: I shall show " … what use I have made of this Joynt, for a universal Instrument for Dialling, for equalling of Time, for making the Hand of a Clock move in the Shadow of a Style, and for performing a multitude of other Mechanical Operations." The joint is depicted on Plate X, Fig.s 22 and 23, which are available at: [http://posner.library.cmu.edu/Posner/books/pages.cgi?call=530_H78AA&layout=vol0/part0/copy0&file=0093 Posner Memorial Collection - Carnegie Mellon University] {{webarchive |url=https://web.archive.org/web/20151117025019/http://posner.library.cmu.edu/Posner/books/pages.cgi?call=530_H78AA&layout=vol0%2Fpart0%2Fcopy0&file=0093 |date=2015-11-17 }}</ref><ref>Robert Hooke, ''A Description of Helioscopes, and Some Other Instruments'' (London, England: John Martyn, 1676), p. 14. [https://books.google.com/books?id=KQtPAAAAcAAJ&pg=PA14 From p. 14:] "The ''Universal Joynt'' for all these manner of Operations, having not had time to describe the last Exercise, I shall now more particularly explain." Illustrations of Hooke's universal joint appear on p. 40, Fig.s 9 and 10; available at: [http://www.e-rara.ch/zut/content/pageview/731094 ETU Library ; Zurich, Switzerland] {{webarchive|url=https://web.archive.org/web/20150923225412/http://www.e-rara.ch/zut/content/pageview/731094 |date=2015-09-23 }}.</ref> He published a description in 1678,<ref name=Berthoud>Review of Ferdinand Berthoud's Treatise on Marine Clocks, Appendix Art. VIII, [https://books.google.com/books?id=JYMCAAAAYAAJ&pg=PA565 The Monthly Review or Literary Journal], Vol. L, 1774; see footnote, page 565.</ref> resulting in the use of the term ''Hooke's joint'' in the English-speaking world. In 1683, Hooke proposed a solution to the nonuniform rotary speed of the universal joint: a pair of Hooke's joints 90° out of phase at either end of an intermediate shaft, an arrangement that is now known as a type of constant-velocity joint.<ref name="Mills2007"/><ref>Gunther, Robert Theodore, ''Early Science in Oxford'', vol. 7: "Life and work of Robert Hooke, Part II" (Oxford, England:  Dawsons of Pall Mall, 1930), pp. 621–622.</ref> [[Christopher Polhem]] of Sweden later re-invented the universal joint, giving rise to the name ''Polhemsknut'' ("Polhem knot") in Swedish.

In 1841, the English scientist [[Robert Willis (engineer)|Robert Willis]] analyzed the motion of the universal joint.<ref>Willis, Robert, ''Principles of Mechanisms'', … (London, England: John W. Parker, 1841), [https://books.google.com/books?id=1CCEKSqQaqcC&pg=PA272 pp. 272-284.]</ref> By 1845, the French engineer and mathematician [[Jean-Victor Poncelet]] had analyzed the movement of the universal joint using spherical trigonometry.<ref>J. V. Poncelet, ''Traité de mécanique appliquée aux machines'', Part 1 (Liége, France: Librairie scientifique et industrielle, 1845), [https://books.google.com/books?id=Qqw-AAAAYAAJ&pg=PA121 pp. 121-124.]</ref>

The term ''universal joint'' was used in the 18th century<ref name=Berthoud /> and was in common use in the 19th century. Edmund Morewood's 1844 patent for a metal coating machine called for a universal joint, by that name, to accommodate small alignment errors between the engine and rolling mill shafts.<ref>Edmund P. Morewood, Improvement in Coating Iron and Copper, [https://patents.google.com/patent/US3746 U.S. Patent 3,746], Sept. 17, 1844.</ref> Ephriam Shay's [[Shay locomotive|locomotive]] patent of 1881, for example, used double universal joints in the locomotive's [[drive shaft]].<ref>Ephraim Shay, Locomotive-Engine, [https://patents.google.com/patent/US242992 U.S. Patent 242,992], June 14, 1881.</ref> Charles Amidon used a much smaller universal joint in his [[Brace (tool)|bit-brace]] patented 1884.<ref>Charles H. Amidon, Bit-Brace, [https://patents.google.com/patent/US298542 U.S. Patent 298,542], May 13, 1884.</ref> [[Beauchamp Tower]]'s spherical, rotary, high speed steam engine used an adaptation of the universal joint {{circa|1885}}.<ref>{{cite web |author=Douglas Self |author-link=Douglas Self |url=http://www.douglas-self.com/MUSEUM/POWER/tower/tower.htm |title=The Tower Spherical Engine
}}</ref>

The term 'Cardan joint' appears to be a latecomer to the English language. Many early uses in the 19th century appear in translations from [[French language|French]] or are strongly influenced by French usage. Examples include an 1868 report on the [[Exposition Universelle (1867)|''Exposition Universelle'' of 1867]]<ref>William P. Blake, Report of the Commissioner to the Paris Exposition, 1867, Chapter 1, [https://books.google.com/books?id=peENdsQu6cgC&pg=PA257 Transactions of the California State Agricultural Society, During the Years 1866 and 1867], Vol X, Gelwicks, Sacramento, 1868.</ref> and an article on the [[dynamometer]] translated from French in 1881.<ref>The Dynamometer Balance, [Van Nostrand's Engineering Magazine], Vol. XXV, No. CLVI (Dec. 1881); page 471.</ref>

In the 20th century, [[Clarence W. Spicer]] and the [[Spicer Manufacturing Company]], as well as the [[Hardy Spicer]] successor brand, helped further popularize universal joints in the [[automotive industry|automotive]], [[agricultural machinery|farm equipment]], [[heavy equipment]], and [[outline of industrial machinery|industrial machinery]] industries.

== Equation of motion ==
[[Image:UJoint.png|thumb|right|Diagram of variables for the universal joint. Axle 1 is perpendicular to the red plane and axle 2 is perpendicular to the blue plane at all times. These planes are at an angle &beta; with respect to each other. The angular displacement (rotational position) of each axle is given by <math>\gamma_1</math> and <math>\gamma_2</math> respectively, which are the angles of the unit vectors <math>\hat{x}_1</math> and <math>\hat{x}_2</math> with respect to their initial positions along the x and y axis. The <math>\hat{x}_1</math> and <math>\hat{x}_2</math> vectors are fixed by the gimbal connecting the two axles and so are constrained to remain perpendicular to each other at all times.]]

[[File:UJoint_3D.png|alt=A sample universal joint colour-coded to the diagrams about the equation of motion. The red and blue planes are visible.|right|thumb|A sample universal joint colour-coded to the diagrams about the equation of motion. The red and blue planes are visible.]]

{{multiple image
 | width = 300
 | direction = vertical
 | image1 = UJoint1.png
 | caption1 = Angular (rotational) output shaft speed <math>\omega_2\,</math> versus rotation angle <math>\gamma_1\,</math> for different bend angles <math>\beta\,</math> of the joint
 | image2 = UJoint2.png
 | caption2 = Output shaft rotation angle, <math>\gamma_2\,</math> versus input shaft rotation angle <math>\gamma_1\,</math> for different bend angles <math>\beta\,</math> of the joint
}}

The Cardan joint suffers from one major problem: even when the input drive shaft axle rotates at a constant speed, the output drive shaft axle rotates at a variable speed, thus causing vibration and wear. The variation in the speed of the driven shaft depends on the configuration of the joint, which is specified by three variables:
# <math>\gamma_1</math> the angle of rotation for axle 1
# <math>\gamma_2</math> the angle of rotation for axle 2
# <math>\beta</math> the bend angle of the joint, or angle of the axles with respect to each other, with zero being parallel or straight through.

These variables are illustrated in the diagram on the right. Also shown are a set of fixed [[coordinate axes]] with unit vectors <math>\hat{\mathbf{x}}</math> and <math>\hat{\mathbf{y}}</math> and the [[Plane of rotation|planes of rotation]] of each axle. These planes of rotation are perpendicular to the axes of rotation and do not move as the axles rotate. The two axles are joined by a gimbal which is not shown. However, axle 1 attaches to the gimbal at the red points on the red plane of rotation in the diagram, and axle 2 attaches at the blue points on the blue plane. Coordinate systems fixed with respect to the rotating axles are defined as having their x-axis unit vectors (<math>\hat{\mathbf{x}}_1</math> and <math>\hat{\mathbf{x}}_2</math>) pointing from the origin towards one of the connection points. As shown in the diagram, <math>\hat{\mathbf{x}}_1</math> is at angle <math>\gamma_1</math> with respect to its beginning position along the ''x'' axis and <math>\hat{\mathbf{x}}_2</math> is at angle <math>\gamma_2</math> with respect to its beginning position along the ''y'' axis.

<math>\hat{\mathbf{x}}_1</math> is confined to the "red plane" in the diagram and is related to <math>\gamma_1</math> by:

<math display=block>\hat{\mathbf{x}}_1 = \left[\cos\gamma_1\,,\, \sin\gamma_1\,,\,0\right]</math>

<math>\hat{\mathbf{x}}_2</math> is confined to the "blue plane" in the diagram and is the result of the unit vector on the ''x'' axis <math>\hat{x} = [1, 0, 0]</math> being rotated through [[Euler angles]] <math>\left[\tfrac{\pi}{2}\,,\, \beta\,,\, \gamma_2\right]</math>:

<math display=block>
\hat{\mathbf{x}}_2 = \left[-\cos\beta\sin\gamma_2\,,\, \cos\gamma_2\,,\, \sin\beta\sin\gamma_2\right]
</math>

A constraint on the <math>\hat{\mathbf{x}}_1</math> and <math>\hat{\mathbf{x}}_2</math> vectors is that since they are fixed in the [[gimbal]], they must remain at [[Orthogonality|right angles]] to each other. This is so when their [[dot product]] equals zero:

<math display=block>
\hat{\mathbf{x}}_1 \cdot \hat{\mathbf{x}}_2 = 0
</math>

Thus the equation of motion relating the two angular positions is given by:

<math display=block>
\tan\gamma_1 = \cos\beta\tan\gamma_2\,
</math>

with a formal solution for {{nowrap|1=<math>\gamma_2</math>:}}

<math display=block>\gamma_2 = \arctan\left[\tan\gamma_1 \sec\beta\right]\,</math>

The solution for <math>\gamma_2</math> is not unique since the arctangent function is multivalued, however it is required that the solution for <math>\gamma_2</math> be continuous over the angles of interest. For example, the following explicit solution using the [[atan2]](''y'',''x'') function will be valid for <math>-\pi < \gamma_1 < \pi</math>:

<math display=block>\gamma_2 = \operatorname{atan2}\left(\sin\gamma_1, \cos\beta\, \cos\gamma_1\right)</math>

The angles <math>\gamma_1</math> and <math>\gamma_2</math> in a rotating joint will be functions of time. Differentiating the equation of motion with respect to time and using the equation of motion itself to eliminate a variable yields the relationship between the angular velocities <math>\omega_1 = \frac{d\gamma_1}{dt}</math> and {{nowrap|1=<math>\omega_2 = \frac{d\gamma_2}{dt}</math>:}}

<math display=block>
\omega_2 = \omega_1\left(\frac{\cos\beta}{1 - \sin^2\beta\,\cos^2\gamma_1}\right)
</math>

As shown in the plots, the angular velocities are not linearly related, but rather are periodic with a period half that of the rotating shafts. The angular velocity equation can again be differentiated to get the relation between the angular accelerations <math>a_1</math> and {{nowrap|1=<math>a_2</math>:}}

<math display=block>
a_2 = \frac{a_1\cos\beta}{1 - \sin^2\beta\,\cos^2\gamma_1} - \frac{\omega_1^2\cos\beta\,\sin^2\beta\,\sin 2\gamma_1}{\left(1 - \sin^2\beta\,\cos^2\gamma_1\right)^2}
</math>

==Double Cardan shaft==
[[Image:Cardan Shaft.jpg|thumb|right|Universal joints in a driveshaft]]
A configuration known as a double Cardan joint drive shaft partially overcomes the problem of jerky rotation. This configuration uses two U-joints joined by an intermediate shaft, with the second U-joint phased in relation to the first U-joint to cancel the changing angular velocity. In this configuration, the angular velocity of the driven shaft will match that of the driving shaft, provided that both the driving shaft and the driven shaft are at equal angles with respect to the intermediate shaft (but not necessarily in the same plane) and that the two universal joints are 90 degrees out of phase. This assembly is commonly employed in [[rear wheel drive]] vehicles, where it is known as a [[drive shaft]] or propeller (prop) shaft.

Even when the driving and driven shafts are at equal angles with respect to the intermediate shaft, if these angles are greater than zero, oscillating moments are applied to the three shafts as they rotate. These tend to bend them in a direction perpendicular to the common plane of the shafts. This applies forces to the support bearings and can cause "launch shudder" in rear wheel drive vehicles.<ref>[http://www.patentstorm.us/patents/6345680/description.html Electronically-controlled adjustable height bearing support bracket - US Patent 6345680<!-- Bot generated title -->] {{webarchive |url=https://web.archive.org/web/20090205182015/http://www.patentstorm.us/patents/6345680/description.html |date=February 5, 2009 }}</ref> The intermediate shaft will also have a [[sine wave|sinusoidal]] component to its angular velocity, which contributes to vibration and stresses.

Mathematically, this can be shown as follows: If <math>\gamma_1\,</math> and <math>\gamma_2\,</math> are the angles for the input and output of the universal joint connecting the drive and the intermediate shafts respectively, and <math>\gamma_3\,</math> and <math>\gamma_4\,</math> are the angles for the input and output of the universal joint connecting the intermediate and the output shafts respectively, and each pair are at angle <math>\beta\,</math> with respect to each other, then:

<math display=block>\tan\gamma_2 = \cos\beta\,\tan\gamma_1\qquad \tan\gamma_4 = \cos\beta\,\tan\gamma_3</math>

If the second universal joint is rotated 90 degrees with respect to the first, then {{nowrap|1=<math>\gamma_3 = \gamma_2 + \tfrac{\pi}{2}</math>.}} Using the fact that <math>\tan\left(\gamma + \tfrac{\pi}{2}\right) = \frac{1}{\tan\gamma}</math> yields:

<math display=block>\tan\gamma_4 = \frac{\cos\beta}{\tan\gamma_2} = \frac{1}{\tan\gamma_1} = \tan\left(\gamma_1 + \tfrac{\pi}{2}\right)\,</math>

and it is seen that the output drive is just 90 degrees out of phase with the input shaft, yielding a constant-velocity drive.

NOTE: The reference for measuring angles of input and output shafts of universal joint are mutually perpendicular axes. So, in absolute sense the forks of the intermediate shaft are parallel to each other. (Since, one fork is acting as input and the other fork is acting as output for shafts and above 90 degree phase difference is mentioned between the forks.)

==Double Cardan joint==
{{Main|Constant-velocity joint#Double Cardan}}

A double Cardan joint consists of two universal joints mounted back to back with a centre yoke; the centre yoke replaces the intermediate shaft. Provided that the angle between the input shaft and centre yoke is equal to the angle between the centre yoke and the output shaft, the second Cardan joint will cancel the velocity errors introduced by the first Cardan joint and the aligned double Cardan joint will act as a CV joint.

===Thompson coupling===
{{Main|Constant-velocity joint#Thompson joints}}

A Thompson coupling is a refined version of the double Cardan joint. It offers slightly increased efficiency with the penalty of great increase in complexity.

==See also ==
{{div col|colwidth=23em}}
* [[Canfield joint]]
* [[Elastic coupling]]
* [[Gear coupling]]
* [[Hotchkiss drive]]
* [[List of auto parts]]
* [[Rag joint]]
* [[Coupling#Twin Spring Coupling|Twin Spring Coupling joint]]
{{div col end}}

==Notes==
{{Reflist|2}}

== References ==
* [https://web.archive.org/web/20120207082441/http://www.nuigalway.ie/mechbio/downloads/FinThMachines3.doc ''Theory of Machines 3''] from National University of Ireland

== External links ==
* ''[http://demonstrations.wolfram.com/UniversalJoint/]'' by Sándor Kabai, [[Wolfram Demonstrations Project]].
* ''[http://autorepair.about.com/cs/doityourself/a/aa102602a.htm DIY: Replacing Universal Joints]'' at About.com.
* [http://cvcoupling.com/index.php?option=com_content&task=view&id=36&Itemid=58 Thompson Couplings Limited] explanation of the Thompson coupling.
*[https://www.beldenuniversal.com/universal-joint-problems Universal Joint Failure - Custom Solutions Address Common Problems]
* [https://www.beldenuniversal.com/installation-maintenance/u-joint-phasing Universal Joint Phasing - The Concept and Importance of Universal Joint Phasing and Drive Shaft Alignment]
* [http://www.abc.net.au/tv/newinventors/txt/s1849939.htm The Thompson Coupling - invented by Glenn Thompson] by ABC Television (''The New Inventors'', broadcast Feb 2007).
* {{US patent|7144326}} (constant-velocity coupling).
* [http://www.mcmaster.com/#about-u-joints/ About universal joints] at McMaster Carr.
*[https://www.aavadh.in/ Cardan Shaft] at McMaster Carr.

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