{"id":282,"date":"2016-01-16T16:22:07","date_gmt":"2016-01-16T23:22:07","guid":{"rendered":"https:\/\/supermediocre.org\/?p=282"},"modified":"2022-03-05T09:07:13","modified_gmt":"2022-03-05T16:07:13","slug":"using-the-math-to-shape-bars","status":"publish","type":"post","link":"https:\/\/supermediocre.org\/index.php\/2016\/01\/16\/using-the-math-to-shape-bars\/","title":{"rendered":"Putting the Math to Work"},"content":{"rendered":"

Finding the beam theory papers was just a baby step to designing my own bars. What I needed was software to apply the math. It may have been possible to write the software from the math in the papers, but software like that can be very difficult to write and debug, especially given my\u00a0very tenuous grasp on the mechanical engineering concepts that underlie it. My breakthrough moment was when I found a masters thesis on the web that was written by\u00a0Mingming Zhao. The paper, called “Automatic multi-modal tuning of idiophone bars,” utilized the standard beam theory to compute the bar shapes, but most importantly, the appendix contained the actual Matlab code that applied the beam theory!<\/p>\n

Comments in the code indicated that is was a modified version of software\u00a0written by Dr Rodney Entwistle. So I looked up the\u00a0Entwistle paper too, which is cited below along with a web link. This paper provided some excellent background and context for the Zhao\u00a0paper.<\/p>\n

I started by copying\u00a0the Matlab functions from\u00a0the PDF file of the Zhao thesis to discrete files. To confirm that I had copied the code correctly,\u00a0my\u00a0first task was to verify that I could repeat the results from the paper. The following is a code snippet that was modified from code in appendix G.1 of the paper. It demonstrates the convergent nature of the algorithm. It takes as arguments the target frequency (300 Hz), the desired ratios for the 2nd and 3rd partials (4.0 and 9.8), the undercut functional form (cubic) and a\u00a0seed<\/em>\u00a0for the iterative algorithm. Here is the run and the program output (captured in Matlab comments).<\/p>\n

bar_props.E = 17.96E9;\r\nbar_props.G = 6.75E9; \r\nbar_props.rho =  837.187565; \r\nbar_props.width = 0.06; \r\nbar_props.d0 = 0.01915; \r\nbar_props.bar_length = 0.3; \r\nclose all;\r\n[test_coeffs{1},test_reqs{1}]=main('ploton',true,'bar_props',bar_props,'selected_frequency',300,'overtone_ratios',[4;9.8],'profile','cubic', 'coeffs',[6;0.3;0.003]);\r\n%E=1.796e+10 G=6.750e+09 rho=837.2 width=0.06000 d0=0.01915 length=0.30000 profile=cubic\r\n%seed_coeffs = [6.000000000000; 0.300000000000; 0.003000000000]\r\n%=== Loop number: 1\r\n%   freq_new = [129.86691; 4.53455; 12.17812]\r\n%   freq_new = [129.86691; 4.53455; 12.17812]\r\n%   freq_new = [129.86705; 4.53455; 12.17811]\r\n% delta freq = [170.13310; -0.53455; -2.37811]\r\n%=== Loop number: 2\r\n%   freq_new = [319.67484; 4.13737; 10.36343]\r\n%   freq_new = [319.67484; 4.13737; 10.36343]\r\n%   freq_new = [319.67518; 4.13737; 10.36342]\r\n% delta freq = [-19.67482; -0.13737; -0.56343]\r\n%=== Loop number: 3\r\n%   freq_new = [299.01530; 3.97747; 9.66143]\r\n%   freq_new = [299.01530; 3.97747; 9.66143]\r\n%   freq_new = [299.01562; 3.97746; 9.66142]\r\n% delta freq = [0.98471; 0.02253; 0.13857]\r\n%=== Loop number: 4\r\n%   freq_new = [299.96624; 3.99930; 9.79651]\r\n%   freq_new = [299.96623; 3.99930; 9.79651]\r\n%   freq_new = [299.96656; 3.99930; 9.79650]\r\n% delta freq = [0.03378; 0.00070; 0.00349]\r\n%=== Loop number: 5\r\n%   freq_new = [299.99992; 4.00000; 9.80000]\r\n%   freq_new = [299.99992; 4.00000; 9.80000]\r\n%   freq_new = [300.00024; 4.00000; 9.80000]\r\n% delta freq = [0.00009; -0.00000; -0.00000]\r\n%final freqs = [299.99991; 4.00000; 9.80000]\r\n%coeffs = [10.308557568566; 0.322686204614; 0.006247284721]\r\n<\/pre>\n
In the highlighted line at the end, you can see that the algorithm converged to the desired final frequency and ratios in just a handful of iterations. It also provided the \u00a0coefficients to the cubic that provided the desired overtones.<\/p>\n
The graphical output from this run is at the top of this post. The two figures show the computed bar shape with the bar lying upside down (i.e., with the undercut at the top of the plot). The top figure has an aspect ration of 1:1, and the bottom figure zooms the y axis to better show the curve shape. There are multiple profile because the figure redraws the computed profile after each iteration of the converging algorithm. The line corresponding to the most broad undercut came from the final iteration of the code.<\/p>\n
This was pretty exciting! I finally had working code that could predict modal frequencies as a function of a specified bar shape.<\/p>\n
Using the Code To Design My Bars<\/h1>\n
As previously stated, I had decided to design my bars to conform to the “double minimum” shape that I had observed on most of the commercial xylophones. To gain experience, I set out with the following plan in mind:<\/p>\n
    \n
  1. Use the software to design a “double-minimum” bar.<\/li>\n
  2. Fabricate a “test bar” out of some inexpensive wood (i.e., not Rosewood) that conformed to the computed\u00a0shape.<\/li>\n
  3. Measure the frequency modes of the test bar to evaluate the accuracy of the predictions.<\/li>\n<\/ol>\n
    Step 1 – Designing My Test Bar<\/h2>\n
    I chose a cubic function for the undercut profile. The choice was motivated by the fact that the cubic profile could realize the\u00a0desired double-minimum shape, but the parabola would have limited the shape to a simple arch shape. The math that computes the bar shape required information about the material type. The code needs the following parameters:<\/p>\n