The definitions of “simply supported” that I found were that the bar was supported at the ends.
I never said that they are supported at the ends. I said they are supported at the nodes of the fundamental mode. I guess I could have been more clear and said the fundamental mode of the freely vibrating bar. But I did say the bars are supported some distance in from the ends, so it couldn’t be a simply supported bar, at least by the definitions of simply supported that I can find. If you went to the link, it would make clear that the support is indeed at the nodes of the fundamental mode of the freely vibrating bar.
If you look at a glockenspiel, the bars clearly aren’t supported at the ends. What definition of “simply supported” are you using?
OK, I see in your reply to MikeS that you seem to be using the standard definiton of simply supported. Xylophone and glockenspiel bars are not simply supported.
I’ve already said how I think they are different. Simply suported means the bar is supported at the very ends of the bar. Xylophone and glockenspiel bars are not supported at the ends. They are supported at the nodes of the fundamental mode of the freely vibrating bar. Look at the chart on the bottom right of page 2 of the link I gave. The bar is supported 0.224L in from the ends, where L is the bar length. That is not the same as simply supported.
In a practical system this difference is negligible – no one puts the supports in a xylophone at the very ends of the bar, because it would fall off. The position of the supports defines where the nodes are. The fundamental frequency and the modes depend upon the distance between the supports, rather than the actual bar length, but that’s a trivial point. There are some corrections that would need to bge added, in a strict mathematical sense, but the Euler-Bernoulli beam equatioin itself is a simplification and requires corrections for strict accuracy (it doesn’t include rotation of the elements during vibration, for instance).
To say a xylophone bar isn’t “simply supported” seems to me simply perverse. Simplly supported beam theory gives you the vibration modes and the frequencies you require. To say that the beam has to be supported at the very ends is absurdly pedantic.
In most real xylophones and the like the difference in result is negliugible. And the mechanical engineers I’ve interacted with use “simply supported” in the same sense I do,
By the way – your number of 0.224 L is irrelevant to me. Why do I care what the distance between the supports in some system you have is? My statement was for the general case of a simply supported beam with small overhang. Although an orchestral instrument will have shaped bars for perfect tuning, there are examples of bars of uniform cross-section and simply different lengths (and only slight overhang) providing a full scale. The Boston Museum of Science has one such example, made of rock cores.
In any event, the paper I link to above shows that the overtones are multiples of the fundamental.
As for a bar with non-negligible but significant overhang (not the example I was giving), this paper seems to have the details, but only the first page is free: