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by David W. Henderson and Daina Taimina


The universal joint P01 can be considered to be a spherical 4bar linkage with , the output crank b variable in the range , and the fixed (grounded) link g being the angle between the input and output shafts which can be adjusted in the range . Note that the links a, h, and g are not actually links, however, the constraints of the mechanism operate as if they were spherical links. See 4bar linkages for a discussion of the eight main types of spherical 4bar linkages.
When this mechanism is used in the drive shaft of an automobile, then it is usual that and in this case the mechanism is called Hooke’s joint. See History of Hooke’s Joint. For the Hooke’s joint, using the notation from 4bar linkages, we have . Thus (noting that on the sphere, l_{S} = l, if l , and l_{S} =  l, if l )
From the chart in 4bar linkages we have that this is a double crank, which means that both the input and output shafts can rotate completely around, as can be seen in the movie above. {The reader can check that as long as b is within of and g is within of , then this universal joint will still be a double crank. In fact, this is important in its automotive use because as an automobile goes over bumps the angle g between the shafts will change.}
We now look at the relationship between the rotation of the input shaft to the rotation of out shaft. One of the problems with Hooke’s joint is that, though one rotation of the input shaft results in one rotation of the output shaft, the rotations are not in sync during the revolution. We look at the standard case of , which we depict in Figure 1. Since , the corresponding arcs are of great circles, thus A must the pole for the great circle (dashed in the figure) passing through O and B; and the great circle arcs h and a must intersect this great circle at right angles (since A is the pole). Likewise, B must be the pole for the great circle passing through A and C; and the great circle arcs h and b must also intersect this great circle at right angles. Since the arc h is perpendicular to both of the dashed great circles, then the arc must bisect lune determined by these two great circles. Since h is great circle in length, then angle of this lune (at P) must be , as marked. In addition, we use to label the angle at A and to label the angle at B. Now, since A is the pole for the great circle POB then the arc OB subtends the angle at the center of the sphere and so the radian measure of OB is . By the same reasoning the radian measure of the arc AC is .
The angle measures the rotation of the Input Shaft and the angle measures the rotation of the output shaft. Note that, when =0 then =0 also. As changes in the positive (CCW) direction, will be changing in the negative (CW) direction; thus, when is positive, will be negative and we will need to use when denoting the angle in the triangle OCA.
In order to find the relationship between g (in radian measure) and the angles and , we use properties of spherical triangles that are familiar to anyone who has studied spherical geometry but unfortunately few people study spherical geometry these days. You can find the formulas we need at the end of this document.
If we apply the “Spherical Pythagorean Theorem” to the right triangle OPC, we obtain:
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If we apply the Law of Sines to triangle OCA, then we obtain:
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From triangle OCB we obtain:
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Putting these together we get:
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Solving
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We put the negative as indicated because g in practice is near and thus is positive. Note that when then and thus the two shafts turn in unison.
Spherical Trigonometry Formulas
If we have a spherical triangle with sides r, s, t (in radian measure) whose opposite angles are labeled , then the Law of Sines states:
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The Law of Cosines states:
In the case that the angle opposite c is a right angle then we get the “Spherical Pythagorean Theorem”:
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For more information, see
Henderson, David W., and Taimina, Daina. Experiencing Geometry: Euclidean and NonEuclidean With History. Upper Saddle River, NJ: Prentice Hall, 2005.