Free Reed Scaling

[ttonon]

Greetings,

 

I started this topic in another forum, and there’s a bit more to say on it, though I’ve moved it here to where its subject matter is more appropriate.

 

As with Pistachio Dreamer, along with the image that moniker stirs up, there’s sometimes the issue of matching the properties of reeds having different tongue materials. As I mentioned in the other post, the problem of the vibrating cantilever has been solved in the simplest case; i.e., freely, with no air resistance, and wherein the beam has constant cross section. It may be useful to extract from this formulation as much as we can, as a way to make practical comparisons between reeds with different materials and geometries.

 

The attached pdf file gives four equations that may be of some interest. The first numbered equation gives the vibrational frequency of the first mode of vibration. The next three allow the calculation of a parameter for reed 2, given the same parameter from reed 1, which differs from 2 in its length, thickness, and material composition.

 

I should describe a bit where these four equations came from. The first falls out of the solution of the governing partial differential equation of motion. The next three originate not by solving this equation, but by simply nondimensionalizing it. Nondimensionalization of equations is a common trick among engineers and mathematicians as a way to understand these equations, and in particular, the method allows a comparison of the relative magnitudes of the various terms. In some applications, some terms are very much larger than other terms, in which case, the smaller terms can be simply thrown away, at least to some specified level of accuracy. Such an approach is fundamental to acoustics and other forms of “small perturbations.”

 

Thus, I nondimensionalized the basic equation, using the reed’s length as characteristic length, and the period of oscillation as characteristic time. As usual, the process spits out a nondimensional group, or combination of parameters, and characteristically, such a term can teach a lot about the behavior of the solution – without even performing the solution. I call this group, for want of a better name, the “Reed Number,” RN, as shown in the attached file to be a grouping of frequency, Young's Modulus, material density, tongue length, and tongue thickness.

 

Okay, perhaps I’m getting into too much abstruse material here, but before I stop, I should really present when such results are expected to be accurate. By nondimensionalizing with respect to the tongue length, as opposed to, say, the wavelength of oscillation, the convenient result is that the equation describes the position of the tongue, all along its length, as a fraction of its length. Thus, if two different reeds – different frequency, material, length, and (constant) thickness – have the same RN, their vibrational shape will be identical. Yes, one reed will be longer than the other, but the difference is only scale - no more than looking at a given reed under different magnifications (we are not concerned in the simplest case here the width of the tongue).

 

As mentioned, the above reasoning applies strictly to the simplest case; i.e., no air resistance and constant cross section. Real free reeds do experience air resistance, taper (change of width along the axis), profile (varying thickness), and slots beveled in different ways. However, what are the magnitudes of these complicating features? In general, concertina reeds have little taper, and I don’t expect too many problems there. It’s also my guess that the air resistance, even as influenced by the different slot angles, is relatively small. Yes, I understand that these things affect sound, but our first concern here is frequency and its sensitivity to material properties and geometry. I think the most serious deviation from the simplest case arises in tongues that have a large amount of profiling: appreciable modification in thickness along their axis.

 

How well the scaling parameter RN can account for real effects can only be determined from experiment. In some cases, the tongues in question might conform very closely to the simplest geometry, and in other cases, perhaps the profiling of two different reeds might be purposely made to be the same, and in still other cases, this approach might only provide an initial guess, and hopefully, subsequent necessary modification of profile might be minimized.

 

The simplest problems, and ones for which the most (relative) accuracy should be expected, involve calculations of frequency. Solutions that seek to produce the same sound between two different reeds are more difficult and require more accuracy, perhaps in some cases beyond what this approach can provide. The proper approach would thus appear to be: use the equations as a way to work among the parameters f, a, L, Y, and rho. These equations, however, can give no guarantee about the sound, a property that is subject to the subtlest of parameter variation, and for that, we need experimental confirmation.

 

I would be interested in hearing feedback from those experienced in making reeds, especially reeds of different tongue materials. There are also perhaps already rules of thumb that are used in order to accomplish what I’m suggesting here. If so, I’d be curious to know how well they agree/disagree with this proposed approach.

 

Best regards,

Tom

www.bluesbox.biz

Free Reed Scaling.pdf

[Dana Johnson]

This seems a very handy approach. Very straightforward . Despite the use of the simplest case, I don't see why it won't apply to other profiles since the same factors of bending and mass distribution as they are controlled by Young's modulus apply equally I'd think. The only issue I think that has to be included in the decision making process that isn't scaleable by the same method is the apparent volume produced by the reeds. I don't honestly know how the factors of operating pressure, aspect ratio and reed area combine to provide a certain volume level at a given deflection, just that all else being in correct balance ( optimized for the reed design, not necessesary equal ) that larger/ longer reeds are more powerful than smaller / shorter ones. Even when using the same material, you need to adjust your reed scale to balance the power of the long low reeds with the shorter mid range, and the increased "apparent loudness" of the higher range notes those of us who still retain their normal hearing experience. I expect the actual power output of a given reed size is not complicated to calculate and if you had a chart from an audiologist for average ear sensitivity at each pitch, you could find an appropriate place to plug that in.

Then you just have to make the rest of your concertina acoustically match the new reed! Have fun with that!

Best wishes,

Dana

[JimLucas]

Still one of my favorite quotes:

"In theory, there's no difference between theory and practice... but not in practice."

[ttonon]

This seems a very handy approach. Very straightforward . Despite the use of the simplest case, I don't see why it won't apply to other profiles since the same factors of bending and mass distribution as they are controlled by Young's modulus apply equally I'd think.

Hi Dana,

 

It’s very nice to hear from you again.

 

Good intuition, and I have similar suspicions. When the cross section varies, additional terms are added to the governing equation, and the effects of these extra terms can be broken down into two parts: 1) the effect of the material, and 2) the effect of a changing cross section. The material effect is the easier one, simply because the material is the same throughout the tongue length, and putting your intuition in mathematical terms, it enables you to factor out the ratio Y/rho from all those additional terms. Thus, the effect of material, is basically the same, and it’s confined to the simple ratio, Y/rho. The additional terms involve S, the cross sectional area, and k, its moment of inertia. I myself am not familiar how much k and S change in typical reed tongues, but if the changes are not too great, the problem is amenable to the technique of small perturbations, and indeed, this problem has been partly solved in some instances and completely solved in others, as appearing Morse, Vibration and Sound. Thus, there’s a lot more future work that can be done with this problem, and when I get time, I’ll pursue it and report back. But at this point, I think it’s safe to say that an x percent total change in either S or k^2 will produce about an x percent change in the results, provided that x is confined to about 10 or so, and less.

The only issue I think that has to be included in the decision making process that isn't scaleable by the same method is the apparent volume produced by the reeds.

Very good observation. As I mentioned, predicting the sound of the reed is most difficult, and theory so far is incomplete in this regard. But we can certainly speculate, based on our intuition. Let’s take the simplest case. We then are concerned mainly with the effect of air on tongue motion, and its reverse, tongue on air motion. Two tongues of different lengths, but with the same Reed Number (RN), will vibrate with the same frequency and shape, as scaled to its length. This means that the larger tongue will have to experience higher velocities, because its tip experiences larger amplitude. It’s accelerations must likewise be larger, which leads us to conclude that the longer tongue will impart more force in moving the air out of the way. Hence the larger tongue should experience more volume for two reasons: larger pressure pulses and more area to generate sound with. Thus, as you suggest, it would be helpful to be able to quantify the relationship between tongue length and overall sound power.

I expect the actual power output of a given reed size is not complicated to calculate and if you had a chart from an audiologist for average ear sensitivity at each pitch, you could find an appropriate place to plug that in.

Although it might not be very easy to calculate this from theory, there’s hope that empirical relationships can be found. The scaling relationships I presented can be perhaps a part of the decision process involved with balancing reeds. For instance, if changing the tongue length is desirable because of frequency considerations, but such change will probably produce an imbalance in volume, then perhaps a change in thickness would be a better way to address the frequency issue.

 

Best regards,

Tom

[jrintaha]

Solving for thickness, a = (f * l^2) / (sqrt(Y / rho) * 3.18), one gets to the unintuitive conclusion that as the desired frequency is lowered, while the reed length is kept constant, the thickness of the reed decreases. (As we all know, in real instruments the reeds get progressively thicker, not thinner, when we go down towards the deepest notes.) However, none of the instruments I own have reed tongues of uniform thickness (well, maybe the very highest pitched reeds might be); they're indeed always tapered so that the root of the tongue is much thinner than its tip.

 

Assuming uniform thickness, a couple of notes (which I picked because I happened to have a Lachenal 20-key left reedpan at hand) and their respective thicknesses:

 

G4 (the one above middle C): ~392 Hz, ~22mm tongue length, thickness would be ~0.0115mm

C3 (the lowest sound on a C/G 20-key anglo): ~131 Hz, ~32mm tongue length, thickness would be ~0.0081mm

 

A reed that thin would be impossible to file, I imagine, and extremely difficult to manufacture. (I also find it hard to fathom it could perform well in reality either, it's just so incredibly thin.)

 

How would one factor in the relative thickness of the tip compared to the root? Some sort of empirical correction factor (or a table of correction factors for different operating points and their respective neighborhoods) applied to the formula?

 

(Wish I had the time to further discuss this, but I have to go to sleep...)

 

Cheers,

Jori

[Theo]

You have made some assumptions that are not in agreement with real reeds. Thickness of many low pitched reeds - it is usual for these to be thicker near the tip, and to have a minimum thickness somewhere in the middle third of the reed.

[ttonon]

Hi Jrintaha, I just noticed your response, and sorry for the delay. The formula for frequency that I gave, as you realize, is valid for a reed tongue of constant cross section. The formula is correct. If the desired frequency is to be increased, for the same length, the tongue thickness must increase, because increasing the thickness increases the stiffness of the cantilever, resulting in higher vibration frequency. It’s true that increasing the thickness also increases the mass of the vibrating portion of the reed – an effect that would tend to lower the frequency. However, this latter effect is more than compensated for by the increase in thickness near the root, where the spring effect is most experienced. Beam rigidity is dependent upon beam thickness to the fourth power, while mass is directly proportional to beam thickness.

 

The reason why reed tongues get thicker for the lower notes is because of the increase in their lengths. If you notice, the length dependence in the formula is squared, which is a large effect. Increasing the tongue length requires an increase in thickness as partial compensation, otherwise the frequencies would be too low.

 

Hi Theo, I believe I acknowledge your point in my text: “I think the most serious deviation from the simplest case arises in tongues that have a large amount of profiling: appreciable modification in thickness along their axis,” and in other comments.

 

Best regards,

Tom