Vibrating beam

[Boustrophedon]

Consider a xylophone made of uniform beams: Where is the best place to support the beams, that is, where is the point of least vibration?
Now for the first fundamental frequency my, quite well known and highly regarded, university text book shows the nodes right at the end, which is ok for calculating the frequency, but does not reflect reality. When the middle goes up the ends go down.
So where are those bl..dy nodes? Unsubstantiated statements on t'internet suggest 22.4% of the length of the beams in from the end. This is very close to the Bessel points for supporting a beam (or a bookshelf) for minimum sag at 22.05%.

[Boustrophedon]

OK so it is clock related, think westminster chimes, tubular bells and scaffolding poles...

[shpalman]

The nodes are only at the end if the ends are fixed. The ends and the middle are obviously antinodes if it would be somehow floating in space, although I wouldn't know how to quantify the amplitudes at the ends as compared to the middle. I expect the way to do it is to consider half a beam, forcing oscillation at one end at its resonant frequency, and then consider a free beam to be two of those connected together.

[dyqik]

The nodes are only at the end if the ends are fixed. The ends and the middle are obviously antinodes if it would be somehow floating in space, although I wouldn't know how to quantify the amplitudes at the ends as compared to the middle. I expect the way to do it is to consider half a beam, forcing oscillation at one end at its resonant frequency, and then consider a free beam to be two of those connected together.

One solution is to find the center of mass as a function of time in the end supported case, transform to the (oscillating) center of mass frame, and then find the stationary points.

[Boustrophedon]

After a lot of thought I decided that the beam as a whole cannot be moving up or down. So assuming a sine wave and calculating an average height of said wave gives an answer of 2/pi this crosses the sine wave at 39.54 deg. which is 22.2% of the length. The sine wave assumption may not reflect reality, but is the approximation used in my undergrad textbook.

[Boustrophedon]

Of course having wasted half an afternoon trying to google the answer, I find the answer looking for Airy points.
https://mechanicsandmachines.com/?p=330

Thanks for the help.