The 2/9th point, or the 22.4% nodal point.

[Ed Pegg]

If a beam vibrated, as in a vibrophone, the best supports are nodal points ~2/9th from the ends of the beam, or .224 L and .776 L, as is found in many engineering books.

To be more precise, first calculate lambda

lambda =
 y /. FindRoot[Cosh[y] Cos[y] == 1, {y, 4.73004},
   AccuracyGoal -> 1000, PrecisionGoal -> 1000,
   WorkingPrecision -> 1000, MaxIterations -> 500]  

Next, calculate K  

K = (Sinh[lambda] + Sin[lambda])/(Cosh[lambda] + Cos[lambda]);  

And then find roots for the Euler-Bernoulli beam equation.  

FindRoot[
 Cosh[lambda eta] + Cos[lambda eta] -
   K (Sinh[lambda eta] + Sin[lambda eta]) == 0, {eta, .224},
 AccuracyGoal -> 1000, PrecisionGoal -> 1000,
 WorkingPrecision -> 1000, MaxIterations -> 500]   

eta = 0.2243236424234744578146402798001316771763471524285779499856298206541344025526173300210928259436452551198091427012539178809015932808568872627336708311606071130999  

That is not the exact value, but upping lambda to 8000 digits doesn't change eta to this precision.  

Is a more accurate value for the 2/9th nodal point interesting enough for OEIS?  Does anyone have any idea how to find a closed form?  

[Charles Greathouse]

I definitely find this interesting enough for inclusion. Could you also cite one of the many engineering books you referred to when you do?

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[Hugo Pfoertner]

A "cons" sequence is definitely justified for this. To my knowledge, no closed-form solution is known.

http://www.varg.unsw.edu.au/Assets/link%20pdfs/Beam_vibration.pdf

Related sequence: https://oeis.org/A076414, A076415, A076416.

[Ed Pegg]

Good to see the other references. In the previous email, I found a value for eta.  

lambda =y/.FindRoot[Cosh[y] Cos[y]==1,{y,4.73004},AccuracyGoal->8000,PrecisionGoal->8000,WorkingPrecision->8000,MaxIterations->1000]; 

K=(Sinh[lambda]+Sin[lambda])/(Cosh[lambda]+Cos[lambda]);

eta1=eta/.FindRoot[Cosh[lambda eta]+Cos[lambda eta]-K (Sinh[lambda eta]+Sin[lambda eta])==0,{eta,.224},AccuracyGoal->8000,PrecisionGoal->8000,WorkingPrecision->8000,MaxIterations->500]

  I also found another new value for eta, based on the ending line in the following source:

https://mechanicsandmachines.com/?p=330     

alpha=(Sin[lambda]-Sinh[lambda])/(Cosh[lambda]-Cos[lambda]); 

eta2=eta/.FindRoot[Sin[lambda eta]+Sinh[lambda eta]+alpha (Cos[lambda eta]+Cosh[lambda eta])==0,{eta,.224},AccuracyGoal->8000,PrecisionGoal->8000,WorkingPrecision->8000,MaxIterations->500]  

eta1  
0.22432364242347445781464027980013167717634715242857794998562982065413440255261733002109282594364525511980914270125391788090159328085688726273367083116060711309991914099175721278053826750868479424141829505521487956762782143945920142098157685192871095478638283753902127915047031

eta2
0.22415752270235765921355959342838932344676201036896058784800066924960837095046619743404124870741451102910551908945436037008465278428273918787929486397182693498667999111062578444056912398708599673626051813621591372068039333208654033665482168228245029866293784410787670723398066

Since  https://mechanicsandmachines.com/?p=330   is published as an answer with more accuracy, I'd trust that value more. But both are very similar combinations of normal and hyperbolic sines and cosines.  

Which do people like more: eta1 or eta2?  

A few other sources for .224 L 
https://en.wikipedia.org/wiki/Vibraphone "For a uniform bar, the nodal points are located 22.4% from each end of the bar." 
https://www.fpl.fs.usda.gov/documnts/pdf1997/murph97a.pdf  
https://lambdasys.com/uploads/info/LEMI-13-Theory.pdf   
Ultrasonics: Fundamentals, Technologies, and Applications (4th ed.).
    Table 3.2 gives “Vibration Characteristics of a Free–Free Uniform Bar in Flexure”
https://physics.case.edu/about/history/antique-physics-instruments/tone-bars-2/ 
   "More complicated analysis shows that nodes for the fundamental mode occur about one quarter of the way in from each end; in reality, 22.4% from the ends."

A few places that call it the 2/9th point

William Sethares – “Tuning, Timbre, Spectrum, Scale”
http://www.r-5.org/files/books/rx-music/tuning/William_A_Sethares-Tuning_Timbre_Spectrum_Scale-EN.pdf  
"As other partials require nonzero excursions at the 2/9 point, they rapidly die away."
Hold the bar (or pipe) at roughly 2/9 of its length, tap it, and listen closely. 

“African music and its use in the school: an investigation”
“The best sound is usually obtained at a point approximately two-ninths of the length from the end… Sand will gather in a straight line exactly over the two nodes

PercussionClinic marimba build article – discusses bar length and “the 2/9 point” for drilling/suspension.

Straight Dope xylophone thread (2017) – OP explicitly says: “The formula they said to use was 2/9 inside from each end…”, i.e., put the supports at 2/9 L from both ends.  

Dennis Havlena’s xylophone page – classic DIY page; he tells builders to find or use approximately 2/9 of the bar length for node/suspension.  

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[Ed Pegg]

I've put in  https://oeis.org/draft/A389416  as a start on eta2  

Should I also do eta1, in case it might be right?  Or just trust that  https://mechanicsandmachines.com/?p=330  picked the right one?

[Ed Pegg]

Published.  https://oeis.org/A389416  

[Allan Wechsler]

As with all constants, I'm in favor of also adding the denominators of the standard continued fraction.

-- Allan

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[Ed Pegg]

Here's some terms
{0, 4, 2, 5, 1, 14, 3, 21, 2, 1, 2, 1, 13, 1, 14, 1, 3, 2, 1, 4, 1, 17, 1, 1, 1, 3, 3, 1, 33, 1, 21, 3, 5, 1, 1, 28229, 1, 3, 1, 1, 12, 1, 2, 3, 4, 2, 4, 1, 3, 1, 1, 2, 3, 2, 36, 4, 4, 2, 1, 18, 2, 2, 1, 2, 4, 5, 1, 20, 1, 1, 1, 2, 2, 1, 4, 4, 2, 4, 4, 1, 1, 1, 6, 2, 1, 3, 1, 26, 2, 4, 15, 1, 1, 1, 5, 6, 1, 20, 1, 5, 5, 2, 5, 1, 6, 12, 2, 1, 1, 4, 1, 1, 2, 1, 1, 1, 1, 2, 1, 11, 1, 1, 1, 70, 1, 3, 1, 34, 1, 1, 1, 3, 1, 64, 3, 2728, 3, 18, 4, 1, 6, 2, 3, 9, 1, 17, 1, 2, 3, 56, 1, 4, 1, 1, 1, 126, 20, 1, 2, 20, 1, 1, 2, 1, 1, 4, 2, 80, 9, 1, 3}

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