In this thread, we are talking about two kinds of vibrating objects – free reed tongues (also called bars) and gongs (also called plates) - composed of two kinds of materials – metals and concrete.
I believe your main interest concerns the vibration of concrete gongs, and perhaps your interest in metallic bars is in order to help you get an understanding of concrete gongs.
I point out these differences because each of these special cases requires different theoretical treatment, and one can error by confusing each of them.
My friends asks me if the modulus density ratio is the same , the fundemental frequency is the same. They asks me what will be the harmonics and their sustain for each different.
I read your post on concertina reeds and as fas as I understand , you defend every characteristics would be the same.
Is it true ?
Yes, in the case of vibrating bars that are homogeneous (have the same material and cross sectional area throughout their lengths), as described by the Euler-Bernoulli governing equation (which works well for the vibration of tongues of the Free Reed), if the ratios E/Rho and the geometries are the same.
As a matter of fact, in these cases, the Euler-Bernoulli ( E-B ) equation allows us to be even more specific; i.e., for two vibrating bars of different materials, if the ratio E*k^2/Rho is the same and if the bar lengths are the same, the vibrational response will also be the same. Here, k^2 is the “radius of gyration of the cross section,” assumed to be constant over the length of the bar, and for a rectangular cross section, equals (a^2)/12, where a is the (constant) thickness of the bar.
As an aside, this result suggests a very interesting experiment that concertina reed makers might want to try. Make two different tongues of different materials (say steel and brass), with constant cross sectional area and having the same length and the same parameter E*(a^2)/Rho. According to these theoretical considerations, both tongues should produce the same acoustic sound: fundamental plus overtones. My feeling is that, if this conclusion can be experimentally verified, our understanding of the free reed would be significantly increased.
Is total mass of the gong would be the same also ? Or is it not important ?
For gongs, we are having another kind of discussion, for at least two reasons: 1) materials, and 2) governing equation of motion.
For gongs, metals can behave much differently than non metals. For instance, concrete is supposed to have many micro-cracks throughout its interior, and this will affect the “resonant” or decay-time properties of the gong.
Concerning the governing equation of motion for such gongs, since they are relatively thick, rotational aspects of small sections of the gong can be important. The Euler-Bernoulli governing equation neglects these rotational aspects, because the thickness of the bar it describes is presumed to be small enough, which is pretty much true. But for gongs, it’s often best to use a more precise formulation, called the Timoshenko governing equation.
One main criterion deciding which equation to use is to compare the thickness of the gong to the wavelengths of the acoustic frequencies one is interested in. As usual with acoustic phenomenon, certain “lengths” of the physical system become important, and the major criterion when such lengths become important is a comparison of such lengths to the wavelengths making up the acoustic result of interest. For a vibrating gong, I’d guess that one would be interested in frequencies up to around 10,000 Hz, which is an upper limit to the overtones average adult hearing responds to. Corresponding wavelengths here are about 3 cm, which is probably getting down to the thickness of practical gongs. Thus, one would expect that the Timoshenko governing equation should be used for gongs.
An aside here is to notice the difference between a vibrating bar, or the tongue of the free reed, and a gong. For the tongue, the frequency response required need not go above a couple thousand Hz, because it responds to only the fundamental of the music tone. The overtones are produced by the dynamical behavior of air flow. For a gong, the musical tone in the vibration must be able to respond to all the overtones, which occur at many times the frequency of the fundamental.
Thus, for a gong, it’s more appropriate to use the Timoshenko governing equation, which is more complicated than the E-B version. I was not able to find a clear way to write (display) these equations here, but anyone interested can refer to Wikipedia. The Timoshenko equation contains not only Young’s Modulus and density, but also the shear modulus, Poisson’s ratio, along with geometry, and all these parameters occur in complex ways. There is thus no simple way to relate gongs of different materials using the more accurate Timoshenko beam theory.
However, there is perhaps an intermediate approach considered by Phillip Morse, in Vibration and Sound, a classic text. In this approach, Morse considers a circular “plate,” which is the two dimensional analog of the vibrating bar, which as we have seen, can be represented well enough by the E-B governing equation. This restriction considers only a circular plate (gong) that is fixed at its periphery, although I don’t think this is the usual way to mount a gong. Regardless, this circumferential boundary condition, produces a governing equation much simpler than the Timoshenko formulation. With this “plate” equation, a comparison of different materials can be made in a way similar to that done with the E-B equation: as long as the parameter E*t^2/Rho/(1-p^2) and the gong diameter are the same, the response of two different materials should be the same. Here, p is Poisson’s ratio and t is the thickness of the gong. All these parameters are considered constant throughout the gong disk.
What makes bronze does not dissipate energy ?
I learned carbon fiber or more exactly epoxy does not waste energy when hit ? Is it true for e glass epoxy laminates ?
Although dissipation is a complicated issue, it must have something to do with the internal molecular structure of the material, and it is very dependent on the kind of material. I think Poisson’s ratio tells us something about dissipation because this parameter is a measure of the relative motion between microscopic elements. An incompressible material will have a maximum Poisson’s ratio of 0.5, and many metals have around 0.2 – 0.3. These values suggest that microscopic elements change volume when stressed, and this suggests dissipation. With metals, different kinds of dissipation are considered, depending how the stress tensor is modeled in relation to it. There are viscous, Coulomb, and hysteretic models, and one can find much description of these on the web.
For concrete, a common model for dissipation considers the relative motion between the sides of micro cracks in the material, which interestingly doesn’t degrade the material. This, I believe, is entirely different than the models used for metals, and thus, understanding why bronze is less dissipative than other metals may not give clues as to the dissipation in concrete. It’s my understanding that Bell bronze (approximately 70% Copper and 30% Tin) has been regarded as a desirable material for gongs and bells for over a thousand years or so, and this is based on empirical evidence. I don’t know how much scientific understanding is behind this practice, but the link below should provide you with some useful information (“A micro-structural model for dissipation phenomena in concrete”). I believe additives are introduced into concrete sometimes in order to increase its internal dissipation, as a way to make the material more resistant to earthquakes.