Universal joint

A universal joint, U joint, Cardan joint, Hardy-Spicer joint, or Hooke's joint is a joint in a rigid rod that allows the rod to 'bend' in any direction, and is commonly used in shafts that transmit rotary motion. It consists of a pair of ordinary hinges located close together, but oriented at 90° relative to each other.

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. The first person known to have suggested its use for transmitting motive power was Gerolamo Cardano, an Italian mathematician, in 1545, although it is unclear whether he produced a working model. Christopher Polhem later reinvented it and it was called "Polhem knot". In Europe, the device is often called the Cardan joint or "Cardan shaft". Robert Hooke produced a working universal joint in 1676, giving rise to an alternative name, the "Hooke's joint". Though the first use of the name universal joint is sometimes attributed to American car manufacturer Henry Ford, the term appeared in patent documents as early as 1884 when Charles H. Amidon was awarded United States Letters Patent No. 298,542 for a bit brace.

Equation of motion

The configuration of the universal joint can be specified by three variables:
* $gamma\_1$ The angle of rotation of axle 1
* $gamma\_2$ The angle of rotation of axle 2
* The angle of the axles with respect to each other, 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 $hat\{mathbf\{x$ and $hat\{mathbf\{y$ and the planes of rotation of each axle. These planes of rotation are perpedicular to the axes of rotation and do not move as the axles rotate. The two axles are joined by a gymbal which is not shown. However, axle 1 attaches to the gymbal 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 ($hat\{mathbf\{x\_1$ and $hat\{mathbf\{x\_2$) pointing from the origin towards one of the connection points. As shown in the diagram, $hat\{mathbf\{x\_1$ is at angle $gamma\_1$ with respect to its beginning position along the "x" axis and $hat\{mathbf\{x\_2$ is at angle $gamma\_2$ with respect to its beginning position along the "y" axis.

$hat\{mathbf\{x\_1$ is confined to the "red plane" in the diagram and is related to $gamma\_1$ by:

:$hat\{mathbf\{x\_1=\; [cosgamma\_1,,,singamma\_1,,,0]$

$hat\{mathbf\{x\_2$ is confined to the "blue plane" in the diagram and is the result of the unit vector on the "x" axis $hat\{x\}\_1=\; [1,0,0]$ being rotated through Euler angles ] :

:

A constraint on the $hat\{mathbf\{x\_1$ and $hat\{mathbf\{x\_2$ vectors is that since they are fixed in the gymbal, they must remain at right angles to each other:

:$hat\{mathbf\{x\_1\; cdot\; hat\{mathbf\{x\_2\; =\; 0$

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

:

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 $omega\_1=dgamma\_1/dt$ and $omega\_2=dgamma\_2/dt$:

:

As shown in the plots, the angular velocities are not linearly related, but rather are periodic with a period twice that of the rotating shafts. The angular velocity equation can again be differentiated to get the relation between the angular accelerations $a\_1$ and $a\_2$:

:

Double cardan

A configuration known as a double cardan joint drive shaft partially overcomes the problem of jerky rotation. In this configuration, two U-joints are utilised where the second U-joint is phased in relation to the first U-joint in order to cancel the changing angular velocity, and an intermediate shaft connects the two U-joints. In this configuration, the assembly will result in an almost constant velocity, provided both the driving and the driven shaft are parallel and the two universal joints are correctly aligned with each other - usually 45°. This assembly is commonly employed in rear wheel drive vehicles.

Even when the driving and driven shafts are parallel, if 0°, 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. [ [http://www.patentstorm.us/patents/6345680/description.html Electronically-controlled adjustable height bearing support bracket - US Patent 6345680 ] ] The intermediate shaft will also maintain a sinusoidal angular velocity, which contributes to vibration and stresses.

In practice, it is often impossible to maintain a strict geometric relationship between the driving and driven shafts, and the intermediate shaft, giving rise to greater vibrations and mechanical stresses. The stresses can be reduced by the use of a smaller and lighter intermediate shaft, ensuring the driven and driving shafts share as close to the same angle in relation to the intermediate shaft, and reducing the angle of the joints.

Joints have been developed utilizing a floating intermediate shaft and centering elements to maintain equal angles between the driven and driving shafts, and the intermediate shaft. This overcomes the problem of differential angles between the input and output shafts.

A recent innovation, the Thompson coupling is a further development of the double cardan joint, which does not rely on friction or sliding elements to maintain a strict geometric relationship within the joint, and which is capable of transmitting torque under axial and radial loads with low frictional losses.

ee also

* Cardan shaft
* Constant-velocity joint
* Elastic coupling
* Gear coupling
* Rag joint

References

* [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, The Wolfram Demonstrations Project.
* " [http://autorepair.about.com/cs/doityourself/a/aa102602a.htm DIY: Replacing Universal Joints] ", About.com.