Why do Brass Tongues Break?

[ttonon]

Hi Chris,

 

I find your questions stimulating, and I’ll try my best to respond, but not necessarily in the order you asked in your two posts.

 

The profiles were the same to within a thou or so and the maximum lift above the frame pretty much the same.

I would find this most interesting if the pitches and tongue lengths of the brass and steel reeds are the same. Are they?

 

They can only be created by the tip of the reed passing through the frame, and presumably the only variable in this is speed,,.

 

…I take it the pure sinusoidal swing you mentioned earlier would be a theoretical movement with absolutely no decel and accel, just an instant change of direction?

 

Let me try to answer all your related questions here by giving a more detailed picture on how I understand it. Again, this is my own intuitive feel, and I don’t mean to declare it as the truth. I'll try to be clear where I think I'm guessing.

 

The governing partial differential equation for the motion of a vibrating beam is typical in that it has a “free vibration” solution, as well as a “forced vibration” solution. By free is meant that no external forces act on it, and in practice, it would closely assume this motion sometime after given a pluck. The pluck would excite many modes of vibration, but all the higher modes would die out faster than the first mode, which assumes the shape of a swinging cantilever. Friction (air resistance) would of course continually reduce the amplitude of this first mode vibration to zero. Ideally, without friction, this first mode would persist indefinitely with time-dependent sinusoidal motion. This free vibration has been solved exactly, and out of that solution pops the geometric shape of the bending tongue, which is made up of sine, cosine, hyperbolic sine, and hyperbolic cosine terms. Keep in mind here there are two independent variables: distance along the beam, x, and time. The geometric dependence is with x, and the sinusoidal dependence is with time.

 

The forced vibration case is the case that happens when the beam is operating as a tongue in a concertina reed, and in principle, its vibration can go on forever, with energy from the air stream maintaining periodic motion against friction, emitting a musical tone. The solution in this case – the free reed case – has been obtained only approximately and partially, and it is much more complicated. A key difference between the tongue motion in forced vibration from its motion in free vibration depends upon how much the aerodynamic forces of the air modify the geometrical curve (with x) of the tongue during its periodic bending, which is sinusoidal with time.

 

You might imagine how the air can impart different kinds of forces to the tongue, along the length of the tongue, as it vibrates through the slot. From experimental measurements performed by Cottingham at Coe College (I can send you the paper if you’d like), the bending of the tongue in forced vibration, however, is very close to the bending that occurs in free vibration. This is understandable because the stiffness of the steel is quite large, compared to the differences in air pressure forces that would cause departure from this shape. But – and I think this is the most important part – there are measured differences, and this is where I suspect different tongue materials can have an influence, producing subtle differences in shape that induce their own characteristic sound.

 

If I might say it another way, the first order solution to the forced vibration problem produces the free vibration shape to the bending tongue. But there are second order effects that cause small departures from this first order solution, and these second order effects are still large enough to cause differences in sound among different materials. As I mentioned previously, the ratio Y/rho is very important in this regard. Perhaps a better term than “specific stiffness” can be found by looking at the units of this ratio, being force-length/mass; i.e. work, or energy, per unit mass. In this case, it’s elastic energy per unit mass. Thus, can you imagine that the same air forces will bend a tongue of one material slightly different than they bend another tongue of a different material? Further, the higher this ratio, the greater the effect of stiffness, for a given mass. Thus, perhaps steel yields less to these air forces than brass, causing a more sudden clap as its tongue goes in and out of the slot. I don’t know if this is true, because the thickness of the tongue also contributes to the stiffness. However, at least some of the thickness effect is accounted for by the "per unit mass" part of Y/rho, and if we impose that the two tongues we are comparing have the same length and vibration frequency, then it may be that material properties then dominate in this proposed mechanism. This is my guess.

 

Being ever motivated by the practical, the reason for my particular interest here is, should it be possible to define a continuum of metallurgical characteristics which would range between a "fast" reed and a "slow" reed of the same pitch, such that one could easily select characteristics you want in the final sound without having to compromise on clearances or restrictive woodwork to remove higher partials, it would be easier to build concertinas with a wider range of tones without losing speed.

 

Perhaps you’re a mind reader. I’m just finishing another post that I hope deals with some of your curiosities. So stay tuned. I kindly mention that your distinction, “fast” and “slow,” I think is misplaced. If brass and steel tongues with the same length vibrate at the same frequency and at the same amplitude, their velocity amplitudes must be the same.

 

Best regards,

Tom

[ttonon]

Quite right, I knew the shape but was suffering from brain fade. I have often wondered what the true shape would be. All of the down accelerations and speeds would be faster than the up ones, and there would be a substantial speed increase when the reed broaches the frame on the way down and a slowing when it does so on the way up.

 

Chris, I think you make a valid point here. If I may elaborate, the down motion is with the air stream, and the up motion is against the air stream. Thus, you might expect an asymmetry in the time dependent motion. It would look something like a sine wave with the first (+) hump smaller (less time interval) than the second (-) hump. Such asymmetry would indeed cause higher harmonics in the Fourier series representing the resulting air pressure pulses. I can also believe that the ratio Y/rho (elastic energy per unit mass), would influence the amount of this asymmetry, providing a feasible mechanism to explain why steel and brass sound so differently.

 

Best regards,

Tom