Leonardo da Vinci's Perpetual Motion Wheels Part I

A great inventor

Many people regard Leonardo da Vinci as the greatest inventor of all time. Yet, while he is praised for being far ahead of his time for his many inventions of technologies that now exist overtly in the 21st century, some commentators seem a bit embarrassed by his inventions in the field of perpetual motion, which does not yet exist overtly. They assure us that he didn't spend much time on it; that he quickly saw how impossible it was, that he labelled his wheels "for studies on the impossibility of perpetual motion" and so on. 

This view tends to overlook what an inventor really does — to visualise something that is not actually realised (i.e. built) yet, and which won't be built until some future time. Usually the time interval between visualisation and realisation is only a few years at most, but for a few exceptionally far-seeing inventors like Leonardo, it can extend to centuries. Perhaps we just have to wait a bit longer for mechanical perpetual motion to appear?

However, other commentators (and I'm one of them) would say that Leonardo's basic and admittedly undeveloped idea of a weight-driven perpetual motion machine has already been realised, if not by the Marquis of Worcester in around 1639, then, to a very high degree of probability, by Johann Bessler in 1712.

Some of Leonardo da Vinci's perpetual motion drawings

I found the above image at http://www.leonardodavincisinventions.com, and decided to make silux models of two of Leonardo's machines. The two devices I chose are those drawn again by Leonardo side by side in the document now known as "Codex Forster II", in the section he labelled in clear Latin as Mechanica Potissimum, i.e. "Highest-Power Engineering". See http://www.vam.ac.uk/content/articles/e/leonardo-da-vinci-explore-the-forster-codices/ Volume 2, pages 90 verso and 91 recto. Leonardo wrote fourteen lines of script in his Italian shorthand mirror-writing (which I cannot read) under each of these drawings. Although some of his writing in Codex Forster II has been translated at the vam.ac.uk website, unfortunately these two pages have not been, so far.

On the page shown above, Leonardo obviously wasn't concerned about having gravity oriented in the same direction for all his drawings. The uppermost drawing has gravity oriented along the centerline he has drawn at 60 degrees from the vertical. The drawing furthest to the left has gravity oriented along the horizontal centerline.

It's pretty obvious that neither of these machines will deliver any net energy, but let's see what happens anyway.

Leonardo's "ratchet wheel" with pawl, and internal arms and weights

This drawing, from Codex Forster II, clearly shows the ratchet pawl, which is almost invisible in Leonardo's first drawing of this machine, at the top of this page. So the wheel must be intended to rotate anti-clockwise.

When the weights fall over-center on the descending (left) side, the first drawing shows them falling against the curved teeth of the wheel's rim, where they are held at a radial position. The geometry doesn't really permit that, and even at the lowest position where a weight could just have fallen against a wheel tooth, the impact will be an undesirable "glancing" one, for the round weights shown.

In my model, to get good perpendicular impacts at radial positions for the weights, I used separate wheel-rests (although they are incorporated into the wheel, all as one object). More wheel-rests are used for the arms, to keep the weights positioned as drawn on the ascending side. (A "wheel-rest" is a component of the wheel which is interacted with, i.e. "hit" by other objects).

Silux model of Leonardo's "ratchet wheel" machine, at start of simulation

Data: Wheel: diameter approx 1 meter; mass 20kg
           Weights: 4kg each
           Arms: 0.1kg each
           Gravity active, friction negligible.
           Weight to wheel impacts: 10% elastic/90% plastic.

There are two ways of running the model: 1) with the pawl always able to engage the ratchet teeth, and 2) with the pawl always disengaged.

Results

1) With pawl. For this version, I didn't bother to model the pawl. I just held the wheel stationary (i.e. "Simulation Member" was unchecked in the Edit Object box) until just before the falling weight on the left side had fallen far enough to hit its wheel-rest. The wheel was then activated, and the results recorded.

Wheel angle vs time with pawl active

The weight in unstable equilibrium on the left side was initially given a very light push of 0.01N for 0.1s, to start its fall. It fell down onto its wheel-rest at 1.436 seconds, bouncing three or four times. This, and the slight bouncing of other weights, accounts for the somewhat noisy wheel angle graph at first. The wheel reaches a maximum angle of only 0.0812 radians at 1.83 seconds, not enough to turn the wheel by even one ratchet tooth, let alone the four teeth needed to cause another weight to start its fall. The wheel angle returns to zero at 2.22 seconds, where the pawl would once again engage the original wheel tooth.

2) Pawl disengaged. For this version, the wheel is always active.

Wheel angle vs time with pawl disengaged

Because there is initially excess weight on the right-hand side of the wheel, it accelerates clockwise at first. This causes the weight in unstable equilibrium on the left side to fall down onto its wheel-rest, which occurs at 0.699 seconds. It bounces (noticeably) only twice. The wheel then decelerates to zero rotational speed at 1.840 seconds; then it starts to turn anticlockwise, but never enough to cause another weight to fall. It just oscillates at a slowly diminishing amplitude, as shown.