The D-shaped Locus

The Schwiers/Richard device

In 1790 Conradus Schwiers of Hoxton, Middlesex, England obtained Patent No 1745 for "A machine on a self-moving principle, or perpetual motion". In 1858 Pierre Richard of Paris, France obtained Patent No 1870 for "Improvements in apparatus for obtaining motive power".

Both of these devices work on essentially the same principle. Richard's version is slightly simpler and easier to understand. 

This is Richard's device, as drawn on p482 of Perpetuum Mobile... by Henry Dircks, 1861, where a full description of the operating principle can be found. Briefly, a number of weights are equally spaced around an endless chain so that they rise along the vertical centerline, and are then guided into cups on the descending side of the wheel. So the weights follow a D-shaped path. If N weights are rising, about π/2 × N ≈ 1.57N weights must be falling.

Equal weights in a D-shaped tube. The highest weights are not at the same level.

A question

Needless to say, the device as invented by Schwiers and Richard does not work. But I spent some time investigating developments of it. Here is a question: is the configuration above at equilibrium? There are eighteen 4kg weights of 10cm diameter stacked into a fixed D-shaped tube. Gravity is active. Friction is negligibly small.

The answer is —  yes, it is at equilibrium, even though the highest weight on the curved side is almost 3cm higher than that on the straight side. So, could we replace these 18 weights with more weights of smaller diameter, and exploit the height difference by allowing them to roll down from the top of the curved stack to the top of the vertical stack?

Such a simple approach won't work, because the smaller the weights are made, the smaller the height difference becomes. (In the limit, this would be similar to filling the D tube with liquid, and getting zero height difference). However, I leave as an open question whether it is completely impossible to exploit the height difference that occurs with large-diameter weights. Oscillation of the weights within the stack, or vertical oscillation of the stack as a whole, perhaps combined with composite weights, might be worth further investigation.