The universal joint

…No, it isn’t something everyone smokes…

But it is common in machine mechanisms. The universal joint is a neat way of turning rotation in one plane into rotation in another. A common use is on driveshafts where you want the direction of the shaft to bend. There’s a neat animation on Wikipedia of how the thing moves. Despite seeing them in action (including in our teaching lab) I’m always amazed that it works. Let’s think about it. There are four pins on the joint – two for one shaft, and two for the other. They have to stay in the same geometry (namely at the four corners of a square) as both shafts rotate. That doesn’t seem possible. Three points are what defines a plane – put in a fourth and surely it’s not, in general, going to sit on that plane, let alone stay at an angle of 90 degrees in the plane from its two neighbours. The problem is ‘solved’ when you realize that the two shafts do not rotate at exactly the same rates. What I mean by that is that if you rotate the first shaft at constant rotation speed (angular speed) the second does not respond with a constant rotation. At some parts of its cycle it speeds up slightly, and at some parts it slows down. The extent of this speed-up and slow-down depends on the angle through which you bend your drive-shaft. A large angle of bend will cause a considerable fluctuation in rotation rate of the driven shaft as it goes through a cycle.  However, for small angles, this fluctuation is pretty small.

These fluctuations can be important and problematic, since a fluctuating rotation rate causes a fluctuating torque on the equipment.

Now, here’s the really neat bit about the universal joint as far as I’m concerned. I don’t care much for mechanical mechanisms (Hmm – maybe my third year mechanical engineering class shouldn’t hear that…) but I do like astronomy. The maths governing the fluctuation in rotation rate of the driven driveshaft is exactly the same as the maths determining one of the two contributions to the Equation of Time. This equation is what determines how ‘fast’ or ‘slow’ solar time (that is, what a sundial would measure) is compared to clock time.

A day is only 24 hours long on average through the year. Sometimes it is about half a minute shorter, sometimes about half a minute longer. We can see the effects of this at the moment – as we come out of summer (sigh…) the long days are ending. The mornings are now considerably darker (sunrise is much later) than it used to be.  But the evenings are still pretty light. Sunset hasn’t shifted a lot. This is because our days have actually been longer than 24 hours for a few weeks. Clock time has gradually got ahead of solar time. (But, fear not, it will reverse itself quickly.) There are two reasons for this effect, which is pretty noticeable around November (light mornings compared to evenings) and February (light evenings compared to mornings). First, there’s a contribution due to the earth’s orbit around the sun being elliptical, not circular. The earth moves quicker in December/January than it does in June/July. Secondly, there’s an effect due to the fact that the plane of rotation of the earth on its axis (and its the earth’s rotation that obviously controls clock time) is tilted compared to the plane of orbit of the earth around the sun. The two planes are not the same. The maths for this second contribution is pretty horrendous but it was done long ago and if you are interested you can go and look it up.

Note that we are talking about two different planes of rotation, slightly tilted with respect to each other. That’s exactly the same as for the universal joint. The same mathematics applies to both. Which makes the universal joint, after all, something worth looking at.





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