Universal joint

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:

1. ${\displaystyle {\gamma }_1}$ the angle of rotation for axle 1
2. ${\displaystyle {\gamma }_2}$ the angle of rotation for axle 2
3. ${\displaystyle \beta }$ 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 with unit vectors ${\displaystyle \hat{\mathbf{x}}}$ and ${\displaystyle \hat{\mathbf{y}}}$ 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 (${\displaystyle {\hat{\mathbf{x}}}_1}$ and ${\displaystyle {\hat{\mathbf{x}}}_2}$) pointing from the origin towards one of the connection points. As shown in the diagram, ${\displaystyle {\hat{\mathbf{x}}}_1}$ is at angle ${\displaystyle {\gamma }_1}$ with respect to its beginning position along the x axis and ${\displaystyle {\hat{\mathbf{x}}}_2}$ is at angle ${\displaystyle {\gamma }_2}$ with respect to its beginning position along the y axis.

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

$${\displaystyle {\hat{\mathbf{x}}}_1=\left[\cos {\gamma }_1,\sin {\gamma }_1,0\right]}$$

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

$${\displaystyle {\hat{\mathbf{x}}}_2=\left[-\cos \beta \sin {\gamma }_2,\cos {\gamma }_2,\sin \beta \sin {\gamma }_2\right]}$$

A constraint on the ${\displaystyle {\hat{\mathbf{x}}}_1}$ and ${\displaystyle {\hat{\mathbf{x}}}_2}$ 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 equals zero:

$${\displaystyle {\hat{\mathbf{x}}}_1\cdot {\hat{\mathbf{x}}}_2=0}$$

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

$${\displaystyle \tan {\gamma }_1=\cos \beta \tan {\gamma }_2}$$

with a formal solution for ${\displaystyle {\gamma }_2}$:

$${\displaystyle {\gamma }_2=\arctan \left[\tan {\gamma }_1\sec \beta \right]}$$

The solution for ${\displaystyle {\gamma }_2}$ is not unique since the arctangent function is multivalued, however it is required that the solution for ${\displaystyle {\gamma }_2}$ be continuous over the angles of interest. For example, the following explicit solution using the atan2(y,x) function will be valid for ${\displaystyle -\pi <{\gamma }_1<\pi }$:

$${\displaystyle {\gamma }_2=\mathrm{atan2}\left(\sin {\gamma }_1,\cos \beta \cos {\gamma }_1\right)}$$

The angles ${\displaystyle {\gamma }_1}$ and ${\displaystyle {\gamma }_2}$ 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 ${\displaystyle {\omega }_1=\frac{d{\gamma }_1}{dt}}$ and ${\displaystyle {\omega }_2=\frac{d{\gamma }_2}{dt}}$:

$${\displaystyle {\omega }_2={\omega }_1\left(\frac{\cos \beta }{1-{\sin }^2\beta {\cos }^2{\gamma }_1}\right)}$$

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 ${\displaystyle a_1}$ and ${\displaystyle a_2}$:

$${\displaystyle a_2=\frac{a_1\cos \beta }{1-{\sin }^2\beta {\cos }^2{\gamma }_1}-\frac{{\omega }_1^2\cos \beta {\sin }^2\beta \sin 2{\gamma }_1}{{\left(1-{\sin }^2\beta {\cos }^2{\gamma }_1\right)}^2}}$$

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 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. The intermediate shaft will also have a sinusoidal component to its angular velocity, which contributes to vibration and stresses.

Mathematically, this can be shown as follows: If ${\displaystyle {\gamma }_1}$ and ${\displaystyle {\gamma }_2}$ are the angles for the input and output of the universal joint connecting the drive and the intermediate shafts respectively, and ${\displaystyle {\gamma }_3}$ and ${\displaystyle {\gamma }_4}$ 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 ${\displaystyle \beta }$ with respect to each other, then:

$${\displaystyle \tan {\gamma }_2=\cos \beta \tan {\gamma }_1\tan {\gamma }_4=\cos \beta \tan {\gamma }_3}$$

If the second universal joint is rotated 90 degrees with respect to the first, then ${\displaystyle {\gamma }_3={\gamma }_2+\frac{\pi }{2}}$. Using the fact that ${\displaystyle \tan \left(\gamma +\frac{\pi }{2}\right)=\frac{1}{\tan \gamma }}$ yields:

$${\displaystyle \tan {\gamma }_4=\frac{\cos \beta }{\tan {\gamma }_2}=\frac{1}{\tan {\gamma }_1}=\tan \left({\gamma }_1+\frac{\pi }{2}\right)}$$

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.