# sounds of polygonal ring chimes of metal

Suppose we have a number of rings, each of them is constructed from some mass of wrought iron or steel: each ring is a polygon, which I would like to suppose is constructed without welding (so they are cast and then machined to tolerances). The side length is $$a$$ and the radius of each side (assume they are locally cylindrical away from the joins) is $$r$$. They are suspended at a point (like the musical instrument, the triangle)

• If the rings are of constant internal area, what frequency sound do they make when struck?
• If the rings are of constant side length, what frequency sound do they make when struck
• What have you done so far in looking at this? Also, the sounds you hear when playing the triangle instrument greatly depends on how and where you strike it. I don't think a general "strike" is enough to specify a solution – BioPhysicist Mar 27 '19 at 19:58
• I do not know where to look: this is not a homework problem, it's something that's been nagging at me for years. – graveolensa Mar 27 '19 at 20:16
• I think that there's a symmetry argument which reduces the number of strike-places one needs to think about. A strike is likely /impact of some force of $n$ Newtons at a point on the surface at some angle. But I don't know where to go next. The Laplacian I can probably do numerically, but I'm unsure about where to go analytically here. – graveolensa Mar 27 '19 at 20:24
• The homework-and-exercises tag isn't just for assigned homework problems – BioPhysicist Mar 27 '19 at 20:52
• This is tricky! I assume your chimes are solid, not hollow, and they don't have a gap (like an orchestral triangle has), so they should be fairly tonal (whereas a triangle is fairly atonal, due to the gap). I can't find much relevant info, but some info relating to bells & chimes ought to apply to your polygons, eg en.wikipedia.org/wiki/Strike_tone I assume the polygonal geometry could have some influence on the partial tones ("harmonics") of these instruments, compared to a straight chime bar, but I have no idea how much. – PM 2Ring Mar 28 '19 at 6:38