Understanding Universal Joints

Gears require axles geometrically fixed to each other to work at all. If driving and driven axle however exhibit an angular movement relative to each other, special elements are needed to compensate this - called joints.
The most popular type, called Universal Joint, consists of a little cross. The ends of one cross member are held by the forked end of the driving shaft, the other cross member is connected to the driven shaft likewise. The center of the cross stays at all times at the exact point where driving and driven shaft would intersect were they long enough.
Now let's turn the driven shaft by a certain angle (ß). The picture below left illustrates this configuration. For convenience lets adjust a three-dimensionial orthogonal coordinate system in such a way that the origin is in the center of the cross (blue), the X axis runs in the core of the driving shaft (red). The driven shaft (green) finally is kept in the plane defined by axis X and Z.

Turning the driving shaft, the ends of the cross define two planes:
The coordinates of point A can easily be given as function of angle w (Let the drawn point A define w = 90° and the distance AB be 2):

(1) A(w) = {xa; ya; za} = {0; sin(w); cos(w)}

With a little more effort the coordinates of  point D can be expressed depending from  angles ß and ø (Hint: Look top down on the drawn configuration; planes E and F then look like lines with included angle ß):

(2) D(ß,ø) = {xd; yd; zd} = {sin(ß)*sin(ø); cos(ø); -cos(ß)*sin(ø)}

For the net step consider that the distance AD is fixed by the cross itself and therefore constant sqrt(2) when distance AB is 2. On the other hand the distance AD can be calculated from the coordinates. So trigoniometric wizardry results in

(3) tan(w) = tan(ø) * cos(ß)

Note: If you are ambitious, you can obtain exactly the same equation using spherical geometry. Key is the observation that diameters AB and CD each define any number of great circles. Just pick the ones intersecting the driving resp. driven shaft. There is only one great circle however containing all four points.
These three great circles now intersect, producing interesting triangles...

Ok, so we know by now that the tangent functions of the driving and driven shafts' rotation are proportional for a given misalignment angle. And exactly there we have a problem! Equation (3) shows quite clearly, that driving and driven shaft do not rotate with the same angular velocity all the time. The diagram above right shows the lag/advance of the driven shaft over half a revolution of the driving shaft for various values of misalignment angle ß.

As we see, the error is not constant and quite considerable sometimes..

Double Joints

But let's go back to equation (3). The misalignment ß could be represented as well by two equal-sized but smaller angles ß1 and ß2. With an intermediate shaft turning an angle ð we get:

(4) tan(w) = tan(ð) * cos(ß1)

(5) tan(ð) = tan(ø) * cos(ß2)

Simply eliminating ð we get:

(6) tan(w) = tan(ø) * cos2(ß/2)

Not much help - oh wait, we missed one important thing: The initial orientation! In the graph above driving shaft angle (w) zero means fork ends A and B on the Z-axis. C and D are therefore sitting on the Y axis in the beginning. If the second joint now would start with its A' and B' on a parallel to C and D, its equation would be different. As we all know (don't we?):

(7) tan(ð+90°) = -1 / tan(ð)

Rotating equation (5) gives

(8) tan(ø) = tan(ð) * cos(ß2)

Now (8) matches (4), which happens to prove - as long as cos(ß1) = cos(ß2)

(9) ø = w

In words: The rotation of the driving and driven shafts is identical at all times when the misalignment angle is ß
a) compensated by two joints,
b) each misaligned by the same angle ß/2 and
c) the intermediate shaft has parallel forks.

Each configuration obeying these rules is called a Constant Velocity Joint (CV).

Applications

The best known application is the drive shaft in any car. Here rule b) is interpreted a little more flexible: As driving and driven shaft are parallel, the two misalignment angles are equal, but of opposite sign. The cos function in equation (8) mends this conveniently.


© odts 2004