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A universal joint (also called a universal coupling or U-joint) is a joint or coupling connecting rigid
shaft Shaft may refer to: Rotating machine elements * Shaft (mechanical engineering), a rotating machine element used to transmit power * Line shaft, a power transmission system * Drive shaft, a shaft for transferring torque * Axle, a shaft around whi ...
s whose axes are inclined to each other. It is commonly used in shafts that transmit
rotary motion Rotation around a fixed axis is a special case of rotational motion. The fixed- axis hypothesis excludes the possibility of an axis changing its orientation and cannot describe such phenomena as wobbling or precession. According to Euler's ...
. It consists of a pair of hinges located close together, oriented at 90° to each other, connected by a cross shaft. The universal joint is not a
constant-velocity joint Constant-velocity joints (also known as homokinetic or CV joints) are mechanical joints which allow a drive shaft to transmit power through a variable angle, at constant rotational speed, without an appreciable increase in friction or play. The ...
. U-joints are also sometimes called by various
eponym An eponym is a person, a place, or a thing after whom or which someone or something is, or is believed to be, named. The adjectives which are derived from the word eponym include ''eponymous'' and ''eponymic''. Usage of the word The term ''epon ...
ous names, as follows: * Cardan joint, after
Gerolamo Cardano Gerolamo Cardano (; also Girolamo or Geronimo; french: link=no, Jérôme Cardan; la, Hieronymus Cardanus; 24 September 1501– 21 September 1576) was an Italian polymath, whose interests and proficiencies ranged through those of mathematician, ...
, a polymath of the 16th century who contributed to knowledge of various clever mechanisms, including gimbals * 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

The main concept of the universal joint is based on the design of gimbals, which have been in use since antiquity. One anticipation of the universal joint was its use by the ancient Greeks on ballistae. In Europe the universal joint is often called the Cardano joint (and a
drive shaft A drive shaft, driveshaft, driving shaft, tailshaft ( Australian English), propeller shaft (prop shaft), or Cardan shaft (after Girolamo Cardano) is a component for transmitting mechanical power and torque and rotation, usually used to conne ...
that uses the joints, a Cardan shaft), after the Italian mathematician
Gerolamo Cardano Gerolamo Cardano (; also Girolamo or Geronimo; french: link=no, Jérôme Cardan; la, Hieronymus Cardanus; 24 September 1501– 21 September 1576) was an Italian polymath, whose interests and proficiencies ranged through those of mathematician, ...
, who was an early writer on gimbals, although his writings mentioned only gimbal mountings, not universal joints. 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 Constant-velocity joints (also known as homokinetic or CV joints) are mechanical joints which allow a drive shaft to transmit power through a variable angle, at constant rotational speed, without an appreciable increase in friction or play. The ...
.Mills, Allan, "Robert Hooke's 'universal joint' and its application to sundials and the sundial-clock", ''Notes & Records of the Royal Society'', 2007, accesse
online
2010-06-16
Shortly afterward, between 1667 and 1675, Robert Hooke analysed the joint and found that its speed of rotation was nonuniform, but that this property could be used to track the motion of the shadow on the face of a sundial. 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''. He published a description in 1678,Review of Ferdinand Berthoud's Treatise on Marine Clocks, Appendix Art. VIII
The Monthly Review or Literary Journal
Vol. L, 1774; see footnote, page 565.
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 Constant-velocity joints (also known as homokinetic or CV joints) are mechanical joints which allow a drive shaft to transmit power through a variable angle, at constant rotational speed, without an appreciable increase in friction or play. The ...
. 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 analyzed the motion of the universal joint. By 1845, the French engineer and mathematician
Jean-Victor Poncelet Jean-Victor Poncelet (; 1 July 1788 – 22 December 1867) was a French engineer and mathematician who served most notably as the Commanding General of the École Polytechnique. He is considered a reviver of projective geometry, and his work '' ...
had analyzed the movement of the universal joint using spherical trigonometry. The term ''universal joint'' was used in the 18th century 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. Ephriam Shay's
locomotive A locomotive or engine is a rail transport vehicle that provides the motive power for a train. If a locomotive is capable of carrying a payload, it is usually rather referred to as a multiple unit, motor coach, railcar or power car; the ...
patent of 1881, for example, used double universal joints in the locomotive's
drive shaft A drive shaft, driveshaft, driving shaft, tailshaft ( Australian English), propeller shaft (prop shaft), or Cardan shaft (after Girolamo Cardano) is a component for transmitting mechanical power and torque and rotation, usually used to conne ...
. Charles Amidon used a much smaller universal joint in his bit-brace patented 1884. Beauchamp Tower's spherical, rotary, high speed steam engine used an adaptation of the universal joint circa 1885. 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 French (french: français(e), link=no) may refer to: * Something of, from, or related to France ** French language, which originated in France, and its various dialects and accents ** French people, a nation and ethnic group identified with Franc ...
or are strongly influenced by French usage. Examples include an 1868 report on the ''Exposition Universelle'' of 1867 and an article on the dynamometer translated from French in 1881. 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, farm equipment,
heavy equipment Heavy equipment or heavy machinery refers to heavy-duty vehicles specially designed to execute construction tasks, most frequently involving earthwork operations or other large construction tasks. ''Heavy equipment'' usually comprises five e ...
, and industrial machinery industries.


Equation of motion

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: # \gamma_1 the angle of rotation for axle 1 # \gamma_2 the angle of rotation for axle 2 # \beta 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 A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in ...
with unit vectors \hat and \hat and the 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 (\hat_1 and \hat_2) pointing from the origin towards one of the connection points. As shown in the diagram, \hat_1 is at angle \gamma_1 with respect to its beginning position along the ''x'' axis and \hat_2 is at angle \gamma_2 with respect to its beginning position along the ''y'' axis. \hat_1 is confined to the "red plane" in the diagram and is related to \gamma_1 by: \hat_1 = \left cos\gamma_1\,,\, \sin\gamma_1\,,\,0\right/math> \hat_2 is confined to the "blue plane" in the diagram and is the result of the unit vector on the ''x'' axis \hat = , 0, 0/math> being rotated through Euler angles pi\!/2\,,\, \beta\,,\, \gamma_2 \hat_2 = \left \cos\beta\sin\gamma_2\,,\, \cos\gamma_2\,,\, \sin\beta\sin\gamma_2\right A constraint on the \hat_1 and \hat_2 vectors is that since they are fixed in the gimbal, they must remain at right angles to each other. This is so when their
dot product In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an alg ...
equals zero: \hat_1 \cdot \hat_2 = 0 Thus the equation of motion relating the two angular positions is given by: \tan\gamma_1 = \cos\beta\tan\gamma_2\, with a formal solution for \gamma_2 = \tan^\left tan\gamma_1 \sec\beta\right, The solution for \gamma_2 is not unique since the arctangent function is multivalued, however it is required that the solution for \gamma_2 be continuous over the angles of interest. For example, the following explicit solution using the
atan2 In computing and mathematics, the function atan2 is the 2-argument arctangent. By definition, \theta = \operatorname(y, x) is the angle measure (in radians, with -\pi < \theta \leq \pi) between the positive
(y, x) function will be valid for -\pi < \gamma_1 < \pi: \gamma_2 = \operatorname\left(\sin\gamma_1, \cos\beta\, \cos\gamma_1\right) The angles \gamma_1 and \gamma_2 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 \omega_1 = d\gamma_1/dt and \omega_2 = \omega_1\left(\frac\right) 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 a_1 and a_2 = \frac - \frac


Double Cardan shaft

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 A drive shaft, driveshaft, driving shaft, tailshaft ( Australian English), propeller shaft (prop shaft), or Cardan shaft (after Girolamo Cardano) is a component for transmitting mechanical power and torque and rotation, usually used to conne ...
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.Electronically-controlled adjustable height bearing support bracket - US Patent 6345680
The intermediate shaft will also have a
sinusoidal A sine wave, sinusoidal wave, or just sinusoid is a mathematical curve defined in terms of the '' sine'' trigonometric function, of which it is the graph. It is a type of continuous wave and also a smooth periodic function. It occurs often i ...
component to its angular velocity, which contributes to vibration and stresses. Mathematically, this can be shown as follows: If \gamma_1\, and \gamma_2\, are the angles for the input and output of the universal joint connecting the drive and the intermediate shafts respectively, and \gamma_3\, and \gamma_4\, 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 \beta\, with respect to each other, then: \tan\gamma_2 = \cos\beta\,\tan\gamma_1\qquad \tan\gamma_4 = \cos\beta\,\tan\gamma_3 If the second universal joint is rotated 90 degrees with respect to the first, then Using the fact that \tan(\gamma + \pi/2) = 1/\tan\gamma yields: \tan\gamma_4 = \frac = \frac = \tan\left(\gamma_1 + \frac\right)\, 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

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

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

* Canfield joint *
Constant-velocity joint Constant-velocity joints (also known as homokinetic or CV joints) are mechanical joints which allow a drive shaft to transmit power through a variable angle, at constant rotational speed, without an appreciable increase in friction or play. The ...
* Elastic coupling * Gear coupling * Hotchkiss drive * Rag joint * Twin Spring Coupling joint


Notes


References


''Theory of Machines 3''
from National University of Ireland


External links

*

' by Sándor Kabai, Wolfram Demonstrations Project. *
DIY: Replacing Universal Joints
' at About.com.
Thompson Couplings Limited
explanation of the Thompson coupling.
Universal Joint Failure - Custom Solutions Address Common Problems
*Universal Joint Phasing
The Concept and Importance of Drive Shaft Phasing and Alignment


by ABC Television (''The New Inventors'', broadcast Feb 2007). * (constant-velocity coupling).
About universal joints
at McMaster Carr.
Cardan Shaft
at McMaster Carr. {{DEFAULTSORT:Universal Joint Rotating shaft couplings Mechanisms (engineering) Automotive transmission technologies Articles containing video clips