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  1. My thoughts have swung to free reeds again and it occurred to me that there may be an interesting way to theoretically determine an optimum static offset distance for the quiescent tongue. As most of us here know, the offset is the distance the tip of the motionless tongue stands away from the plane of entrance to the slot below. I’m sure some of the makers here ask, what’s the need for such a theory? There isn’t really; makers get along well without it. But for those of us fascinated by the operational details of the free reed, such a theory can illustrate more of the intricate physical principles by which our musical source works. My interest in this topic was piqued during a discussion on this forum when several of us were postulating just how the tongue starts vibrating. Someone posted slow motion videos of the starting tongue and what struck me was that the first sign of motion of the tongue tip was a minute vibration that slowly grew in amplitude, until the tip entered the plane of the slot opening. At that instant, the amplitude of vibration increased very rapidly. The western free reed has a notoriously slow start transient – the time it takes from the application of pressure difference to the moment in which the musical tone can be considered to be fully developed. In fact, there was a period during the 20th Century when free reed organ pipes were in disfavor, precisely because of that. Typically it takes many tens of milliseconds for the transient. I myself think this feature is a hindrance to the bellows driven version in some musical settings. Perhaps it can be improved by somehow linking the key to an arm that flips the tongue the moment the key is pressed. Such a mechanism might eliminate the sluggish start period of time in which the tip is building up the amplitude of its vibration outside the slot. Apart from the time delay, a short, sudden start transient usually adds color and character to the entire musical tone. When we hear the fully developed tones of many musical instruments, when the start transient has been digitally removed, we can’t distinguish them, for instance a violin from a free reed, or even a beating reed, such as a saxophone, or a string sound from a guitar or piano. A crisp start transient also helps distinguish a musical instrument from other instruments in an ensemble, and it helps distinguish one note from the instrument from another note from the same instrument. The piano is a marvelous instrument with a wonderful, percussive start transit. In my opinion, it’s why it works so well in Jazz, hammering out complicated chords in which individual notes can be well distinguished, much unlike the accordion, or English concertina. This is an interesting, though large topic, and let’s get back to nudging the tongue in order to start speaking. The fact is, when the tongue is in that entrance plane, the static pressure force on the tongue is maximum. Assuming that we have a perfectly made reed with a tongue that perfectly fits its slot, this maximum pressure force is equal to the (static) bellows pressure difference times the footprint area of the tongue. There are no other static pressure forces in the system that can be larger than this, because before that moment, with the tongue still vibrating outside the slot, the static pressure in the air flowing around the sides of the tongue, under the tongue and into the slot is everywhere a little above the static pressure on the underside of the slot. It’s when the static pressure under the tongue equals the static pressure below the plate that the maximum bellows static pressure force is experienced by the tongue, and that occurs when the tongue completely covers the slot, blocking all air flow. This explains the observed very rapid increase in vibration amplitude. At that moment, the mechanism for nudging the tongue changes from eddy induced vibration (explained below) to a more direct and much larger pressure force that acts uniformly over the total top area of the tongue. At that moment of tongue coverage, we can write, during push of the bellows and a rectangular tongue, J = (Pb – Pa)*L*W, where J is total pressure force distributed uniformly over the tongue, Pb is bellows pressure, Pa is atmospheric pressure, L is tongue length, and W is tongue width. The above static pressure description is really only secondary to the offset theory I’d like to present here. However, it prepares ground for good visualization of the physics. My other key observation of the slow motion video start of tongue vibration was that the initial, small amplitude vibration of the tongue – before being forced into the slot – was obviously (to me) the result of vortex induced vibration (VIV), or at least periodic eddy induced vibration. VIV is an extremely well studied phenomenon, being of interest to architects and aeronautical engineers concerned for the potential damage such a process can cause to large, expensive manmade structures such as bridges, buildings, transmission towers and lines, etc. and aircraft, rockets, and the like. Most of us know about the collapse of the Tacoma Narrows bridge, only four months after it was built over Puget Sound, Washington, in which self-induced vibrations shook the bridge to pieces in only a 40 mph wind, causing the death of a pet dog in a car, both of which were never recovered in the 200 foot deep water below. For many such structures, VIV is bad; for free reeds, it’s good, because your concertina can’t speak without it (I think). Two most notable individuals associated with understanding vortex formation are Vincenc Strouhal, a Czech, and Theodore Von Karman, a Hungarian-American. Strouhal studied the inherently unsteady process of vortex formation in the wake of bluff bodies, arriving at well-known correlations between Strouhal Number and Reynolds Number. Von Karman shed (no pun) much light on what are called “vortex streets,” which are periodic formations of vortices in the wake of a blunt body in a fluid flow stream. Depending upon the Reynold’s Number (Re), or for a given geometry and fluid, the fluid velocity, turbulent eddys in the wake of the body form various patterns in space and time. A large regime for Re produces von Karman vortex streets, wherein vortices appear alternatively on both sides of the object, becoming regularly spaced and periodic in time. These periodic structures in turn cause periodic forces on the structures, and when these periodic forces couple to the natural vibration modes of the structures, large amplitudes of periodic structural motions can occur, called “galloping,” or “flutter.” Such vortex streets encompass a tremendous range of scale, ranging from geological scales observable from satellites in the wake of Eastern Atlantic Ocean islands, down to individual blades of grass, and down to our tiniest free reed tongues. They probably also occur in outer space. Notice here that the wake of the free reed tongue doesn’t extend very far (taking “far” to mean in comparison to W) before any turbulent eddies approach/hit the slot. I’m assuming the eddy interaction about the immediate region of the tongue surface is key and that perhaps the slot allows eddies to pass through easily enough not to greatly disturb the picture. The attached figure shows the St vs Re plot attained by Strouhal, around the year 1878. This is a cleaned up plot, eliminating the large number of data points and error spread bars. We make use of this information in our Free Reed Tongue Tip Offset theory. The Strouhal Number is given by St = F*W/V, where F is the frequency of eddy formation, W is tongue tip width, and V is air velocity past the tip. Notice that it’s the ratio of two times, the time an air particle moving past the tip remains in the vicinity of the tip (W/V) and the period time of vortex formation, since F = 1/T, where T is the period. This ratio has physical meaning. For instance, if the time an air particle spends near the tip is small compared to the period, its view, or experience of a forming vortex will be as though the vortex is stationary. Unless it’s trapped in the whirl of the vortex. The vortex, being a whirlpool just like a hurricane, experiences very fast tangentially moving air, while the entire structure moves at a relatively slow rate (e.g., the “eye”). Such a picture explains why St values in the figure are not much above 0.2, even though the motion is unsteady. Normally, such low time ratios in periodic fluid motion would lead to a conclusion that individual fluid packets experience very little unsteady (time dependent) changes, allowing one to view the overall motion as “quasi-steady.” But in this case, the washing machine regurgitation of the packets in the vortex - like Hurricane Harvey did to Houston - keeps the packets in the region of unsteadiness, leading to the conclusion that vortex formation is inherently unsteady. Without vortices, air flow produced by constant pressure difference would cause steady flow. Hence, vortices convert a steady flow into a periodic unsteady flow, which is necessary to start the unsteady vibration of the tongue. The Reynold’s Number is given by Re = W*V/nu, where nu is the kinematic viscosity of the fluid. As I understand it, this expression was first introduced by George Stokes, an Irishman, but it was made popular by Osborne Reynolds, another Irishman. The Reynolds Number is an extremely important parameter in fluid flow, showing up in all kinds of disparate flow conditions. It is usually interpreted as a ratio of forces, inertial forces to viscous forces, and I recommend that interested people check out the wiki page (https://en.wikipedia.org/wiki/Reynolds_number) that explains the progression from these forces to the expression above. This page is useful also because it shows a cartoon video on the vortex street behind a bluff body in fluid flow. It's a beautiful play produced by Nature, and from it, one cannot but help not to make the connection between vortices and periodic structural forces. (This is an example of the www at its finest, and incidentally, wiki asks all of its users for a measly $3 donation per year.) With that background, the gist of the Free Reed Tongue Tip Offset Theory is to first make two important assumptions. 1) the minimum bellows pressure to start the eddies is that static pressure that will hold the tongue tip into the entrance plane of the slot. 2) the dissipation (friction) in the system is small enough to allow a very sharp and pronounced resonance when the tongue is excited by an external periodic force very close to its natural frequency. That last complicated sentence can be greatly simplified by stating that a “high Q” is assumed. Anyone who has studied vibrations or electric circuits should know what that means. Physically speaking, what we are doing here is to assume that, in order to start speaking, the tongue needs to be excited by a periodic fluid force that is near its (first mode) resonance frequency, and that periodic force is the force supplied by formation of periodic eddies in the wake of the air flow passing over the tongue tip. When those frequencies match, we have excitation. Strictly speaking, we are not even assuming a vortex street. We are only using the data from Strouhal, in which he has correlated discernible periodic wake forces, as expressed by F in his expression for St (above). We know that, for Re from about 47 to many thousands, as in the case with free reeds, vortex streets are the most likely outcome, but that’s an added understanding to the starting process details. Continuing, one then looks up the Young’s Modulus (E) and bulk density (rho) for the tongue material, decides on the geometry of the tongue, using W, L, and t, the tongue thickness, and assumes a starting value for a, the tip offset. The rectangular cross area moment of inertia is calculated as I = W*t^3/12. Assumption (1) then allows calculation of the start pressure difference Ps (push or pull of the bellows is inconsequential) in terms of the assumed a and geometry, using a well-known beam formula for a cantilever, fixed at one end, free at the other (where the tip is), and uniformly loaded (as in the case here, with a constant pressure difference). The formula is Ps = a*E*I/(W*L^4). From the calculated start pressure, Ps, one calculates the resulting air velocity for the air moving about the tongue tip, as follows: V = (2*Ps/rho)^0.5. This air flow of course occurs before the tongue covers the slot, and it’s obtained from energy considerations (Bernoulli), not dependent on the particular geometries, passages, and streamlines. Next, calculate the Reynolds Number: Re = W*V/nu Here, we use Strouhal’s correlation in the attached figure, between Strouhal Number, St, and Reynolds Number, Re. The fact that it’s in graphical form breaks complete automation of the calculation. I did make curve fits for this correlation, but I used Excel, and there I’m limited in the number of functional forms. Notice that the graph is a semi-log plot, so the polynomial fits offered cannot cover the entire range of Re. I thus split the graph into three fits, all using a 6th order polynomial. I will return to this shortly. Once we find the appropriate Strouhal Number value for the Reynolds Number calculated above, we can calculate the frequency of periodic eddy formation from: F = St*V/W, which is derived from the definition of St, above. With F now calculated, one compares this value to the actual vibration frequency of the tongue. If the calculated F and the actual frequency differ too much, we then assume another offset, a, and redo the calculation until we get reasonable agreement. Physically, this comparison completes the application of the model, which states that the starting eddy frequency is very near the actual vibration frequency. Returning to the graph issue, I have put in an Excel spreadsheet the entire calculation for a. There’s an input section (material properties, geometry, etc.), a calculated parameter section (I, Ps, V, Re, etc.), a logic (decision) section in which the user selects the proper curve fit for the calculated Re with a simple click, and a final comparative section for the two frequencies. I would gladly email this spreadsheet to anyone who’d like it. A small request I make is that anyone who receives it to please not share it with others, because I’d like to know who the interested people are. Or, if you do send it to someone, please let me know who it is. For that, I thank you. Though regardless, I won’t contact my lawyer over it. As a finer point of discussion, the two assumptions listed above are perhaps a bit contradictory. The second requires a large resonance response, yet the first chooses a static start pressure, Ps, able to statically deflect the tip an amount equal to the offset, a. Considering the fact that any mechanical resonant system acted upon by a periodic force produces vibration amplitudes much larger than the amplitude of deflection that force would produce in a strictly static process, one might suspect that the calculated Ps is far larger than the true Ps. But I think it’s more complicated than that. First off, there is no real Ps operating on the system. It’s a fictitious quantity necessary to estimate a start air flow velocity, and since pressure is the only mechanism here to cause air flow, we look for the minimum pressure that can be defined by the given parameters. The actual external periodic force on the tongue that causes motion is that due to the dynamics of eddy formation. This force is different from a Ps pressure force. The eddy dynamics are of course ultimately the result of the applied static bellows pressure, Pb, but there isn’t a one-to-one identification. For me, at least now, I think the idea may give decent results. Of course, we can add complexity to the model by defining a modified start pressure, Ps’ = k*Ps, where Ps is as before and k is some number less than unity that would have to be inputted. I’d be surprised if anyone would be interested enough in this modification to try it, because it means for the maker to keep track of how well the model works for different assumed values of k. I clarify this for purposes of completeness. From calculations, it seems this model predicts at least realistic values for a, though I plan to investigate it experimentally. Trouble is that my workshops are a mess now, in the middle of major clean up. My guess is that makers – like myself – are so familiar with setting a value for a that it’s done almost by second nature, with simple start trials to check out the setting. But I’m not a maker, and if there does appear that such a theory can be useful in any aspect of this chore, I’d appreciate knowing about it. For me, it was just fun enough putting the pieces together. It gave me a better physical feel for underlying physics, and as far as I know, it’s entirely my own. For instance, I have not come across any hint of it in academic literature on acoustics and vibration. Best regards, Tom www.bluesbox.biz
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