Locomotives with Non-Circular Wheels
There's no point in pretending that this is going to be a large gallery in the Museum. Locomotives with non-circular wheels have always been something of a minority taste.
In 1887, in the USA, the Swinerton Locomotive Driving Wheel Company took delivery of a unique locomotive confidently called "Onward" built by the Hinkley Locomotive Works of Boston. Its unique feature was that its wheels were not round.
C E Swinterton specifed that each driving wheel should actually be a polygon with 118 sides, each segment being about 2 inches long. The theory behind this quite daft arrangement was that the line contact at the intersection of each segment would have better rail adhesion than a circular wheel, which was supposed to have only a point contact; this is not of course actually the case as even steel deforms under enough pressure, so there is on fact a contact area even with a circular wheel.
Left: The Swinerton Locomotive "Onward"
The Swinerton Locomotive had a single driving axle, an arrangement long abandoned elsewhere because of its poor adhesion, presumably to emphasise the effectiveness of the invention. The "Onward" was tried on several north-eastern lines, but worked about as well as you would think it would, and failed to convince practical railway men that it had any special advantages. It was sold to the Portland and Rochester Railroad- after the polygonal driving wheels had been replaced with conventional ones. After a few years the P & R sent it to the Manchester Locomotive Works for reconstruction as an ordinary eight-wheeler. "Onward" was cut up for scrap in 1905.
No other hard information on the performance of the polygon wheels has so far been unearthed, but it seems very probable that the non-constant radius would have caused very severe vibration. It also seems likely to destroy brake-blocks very quickly- there is clearly a brake-block acting on the single driver in the photograph above.