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A 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 hinges located close together, oriented at 90° to each other, connected by a cross shaft.

[edit] 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.

[edit] Equation of motion

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 β with respect to each other. The angular displacement (rotational position) of each axle is given by

γ 1

and

γ 2

respectively, which are the angles of the unit vectors $\hat{x}_1$ and $\hat{x}_2$ with respect to their initial positions along the x and y axes. The $\hat{x}_1$ and $\hat{x}_2$ vectors are fixed by the gimbal connecting the two axles and so are constrained to remain perpendicular to each other at all times.

The Cardan joint suffers from one major problem: even when the drive shaft axle rotates at a constant speed, the driven 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:

$γ 1$ The angle of rotation for axle 1
$γ 2$ The angle of rotation for axle 2
$β$ 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 $\hat{\mathbf{x}}$ and $\hat{\mathbf{y}}$ 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 ( $\hat{\mathbf{x}}_1$ 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 $γ 1$ with respect to its beginning position along the x axis and $\hat{\mathbf{x}}_2$ is at angle $γ 2$ 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 $γ 1$ by:

$\hat{\mathbf{x}}_1=[\cos\gamma_1\,,\,\sin\gamma_1\,,\,0]$

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

$\hat{\mathbf{x}}_2 = [-\cos\beta\sin\gamma_2\,,\,\cos\gamma_2\,,\,\sin\beta\sin\gamma_2]$

A constraint on the $\hat{\mathbf{x}}_1$ and $\hat{\mathbf{x}}_2$ vectors is that since they are fixed in the gimbal, 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:

$\sin\gamma_1\cos\gamma_2=\cos\beta\cos\gamma_1\sin\gamma_2\,$

The angles $γ 1$ and $γ 2$ 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 $ω 1 = d γ 1 / d t$ and $ω 2 = d γ 2 / d t$ :


Angular (rotational) output shaft speed $\omega_2\,$ versus rotation angle $\gamma_1\,$ for different bend angles $\beta\,$ of the joint	Output shaft rotation angle, $\gamma_2\,$ , versus input shaft rotation angle, $\gamma_1\,$ , for different bend angles, $\beta\,$ , of the joint

$\omega_2=\frac{\omega_1\cos\beta}{1-\sin^2\beta\cos^2\gamma_1}$

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 $a 2$ :

$a_2 = \frac{a_1 \cos\beta }{1-\sin^2\beta\,\cos^2\gamma_1}-\frac{\omega_1^2\cos\beta\sin^2\beta\sin 2\gamma_1}{(1-\sin^2\beta\cos^2\gamma_1)^2}$

[edit] Double Cardan Shaft

Universal joints in a driveshaft

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 $\beta\,\le$ 90°. 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 parallel, if $\beta\,>$ 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.^[1] The intermediate shaft will also maintain a sinusoidal angular velocity, which contributes to vibration and stresses.

[edit] Double Cardan Joint

Main article: Constant-velocity joint#Double Cardan

A double cardan joint consists of two Hookes joints mounted back to back, with no intermediate shaft. The second UJ cancels the velocity errors introduced by the single Hookes joint, and so they act as a CV joint.

[edit] Thompson Coupling

Main article: Constant-velocity joint#Thompson coupling

A Thompson Coupling is a refined version of the Double Cardan Joint. It offers slightly increased efficiency with the penalty of some increase in complexity.

[edit] See also

[edit] References

Theory of Machines 3 from National University of Ireland

^ Electronically-controlled adjustable height bearing support bracket - US Patent 6345680

[edit] External links

[1] by Sándor Kabai, Wolfram Demonstrations Project.
DIY: Replacing Universal Joints, About.com.
Thompson Couplings Limited - Explanation of the Thompson coupling
The Thompson Coupling - invented by Glenn Thompson ABC Television - The New Inventors - broadcast Feb 2007
U.S. Patent 7,144,326 - Constant velocity coupling
Video on Youtube: http://www.youtube.com/watch?v=xgQgm3GwaFs
Video on Youtube: http://www.youtube.com/watch?v=Dh5C4e4exhM

[0] Electronically-controlled adjustable height bearing support bracket - US Patent 6345680

[1]

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