Editing Universal joint (section)

==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.)