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| - | == Introduction ==
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| - | This page contains all the MATLAB files necessary to run the tuning analysis we did. In order to run the code, copy the text from each section into its own MATLAB file (named for the title of the section). Put all these files in the same directory, then run them in MATLAB. These functions were written in MATLAB 2007b with the symbolic math toolbox, and may or may not run in other versions. The files are completely automated, no user input is necessary.
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| - | To run the analysis, simply open MATLAB, and run the bridge_analysis.m file. All other files are helper functions.
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| - | == bridge_analysis.m ==
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| - | % Jeremy Ozer
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| - | % November 8th, 2008
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| - | % Engineering Design II
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| - |
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| - | close all;
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| - | clear all;
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| - | clc;
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| - |
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| - | global stiffness k ten0 def;
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| - |
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| - | stiffness = 2e11;
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| - | k = [46.2/5.63 49.0/3.99 57.4/2.21 59.3/3.46 62.5/2.74 65.7/2.53]*1000; % Spring constants in N/m
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| - | ten0 = [46.2 49.0 57.4 59.3 62.5 65.7]; % Neutral string tensions
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| - | def = [5.63 3.99 2.21 3.46 2.74 2.53] / 1000; % Neutral string deflections
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| - |
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| - | theta = linspace(0,5,1000)*pi/180; % Create set of angles we are evaluating between 0 and 2 degrees
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| - |
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| - | % Shifting Saddle Analysis
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| - |
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| - | m = sin(theta)*0.0254;
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| - | ot = offtune(theta);
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| - | figure;
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| - | plot(1000*m, ot);
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| - | title('Comparison of Saddle Slippage to Tune Shift');
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| - | xlabel('Saddle offset (mm)');
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| - | ylabel('Frequency Error (Cents)');
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| - | legend('e','B','G','D','A','E');
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| - |
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| - |
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| - | % Now twisting Analysis
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| - |
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| - | theta = linspace(0,90,1000)*pi/180;
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| - | dia = [229 279 406 610 813 1067]*10^-6;
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| - |
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| - | l0 = 25.5*0.0254;
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| - | for i=1:length(theta)
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| - | ltwist(i,:) = l0+pi*dia*theta(i);
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| - | end
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| - | % figure;
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| - | % plot(theta, ltwist);
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| - |
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| - | delta = ltwist-l0;
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| - | for(j=1:length(theta))
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| - | for i=1:6
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| - | t(j,i) = ten0(i) + k(i)*delta(j);
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| - | end
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| - | end
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| - |
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| - | % figure;
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| - | % plot(theta,t);
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| - |
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| - | ml = [1.67 3.16 5.88 10.84 20.33 38.10]*10^-4;
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| - | ml = ml.*[1 1 1 0.902 0.886 0.867];
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| - |
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| - | for(j=1:length(theta))
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| - | for i=1:6
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| - | tunes(j,i) = 1/2*(t(j,i)./(ml(i)*l0)).^0.5;
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| - | end
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| - | end
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| - |
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| - | % figure;
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| - | % plot(theta,tunes);
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| - |
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| - | f0 = tune(0);
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| - |
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| - | sharp = [349.23 261.63 207.65 155.56 116.54 87.31];
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| - | for j=1:length(theta)
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| - | for i=1:6
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| - | n(j,i) = 1200*log2(tunes(j,i)/f0(i));
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| - | end
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| - | end
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| - | cents = n;
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| - |
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| - |
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| - | nx = [0 100];
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| - | n = [100 100];
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| - | figure;
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| - | hold on;
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| - | plot(theta*180/pi, cents);
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| - | title('Comparison of Ball End Rotation to String Tune');
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| - | xlabel('Ball End Rotation (degrees)');
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| - | ylabel('Frequency Error (Cents)');
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| - | legend('e','B','G','D','A','E');
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| - | plot(nx,n,'r--',nx,2*n,'r--',nx,3*n,'r--');
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| - | axis([0 90 0 400]);
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| - | hold off;
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| - |
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| - |
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| - | % Prototype 2 Analysis
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| - | a = atan(0.0015/0.75);
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| - | lextra = 0.2*sin(a)*0.0254
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| - | l0 = 25.5*0.0254;
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| - | t = zeros(1);
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| - | delta = lextra;
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| - | for i=1:6
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| - | t(i) = ten0(i) + k(i)*delta;
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| - | end
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| - | t
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| - | % figure;
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| - | % plot([0 a], [ten0;t]);
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| - |
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| - | tunes = zeros(1);
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| - | for i=1:6
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| - | tunes(i) = 1/2*(t(i)./(ml(i)*(l0+delta))).^0.5;
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| - | end
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| - | % figure;
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| - | % plot([0 a], [tune(0); tunes]);
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| - |
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| - | f0 = tune(0);
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| - | sharp = [349.23 261.63 207.65 155.56 116.54 87.31];
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| - | for j=1:length(theta)
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| - | for i=1:6
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| - | n(i) = 1200*log2(tunes(i)/f0(i));
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| - | end
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| - | end
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| - | cents = n;
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| - |
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| - |
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| - | nx = [0 100];
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| - | n = [100 100];
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| - | figure;
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| - | hold on;
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| - | plot([0 180/pi*a], [zeros(1,6);cents]);
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| - | title('Comparison of Ball End Rotation to String Tune');
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| - | xlabel('Ball End Rotation (degrees)');
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| - | ylabel('Frequency Error (Cents)');
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| - | legend('e','B','G','D','A','E');
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| - | % plot(nx,n,'r--',nx,2*n,'r--',nx,3*n,'r--');
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| - | % axis([0 90 0 400]);
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| - | hold off;
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| - |
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| - |
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| - | == eten.m ==
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| - | function tension = eten(theta)
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| - | global stiffness k ten0 def;
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| - |
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| - |
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| - | % orig_length = string_length(0);
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| - | % lengths = string_length(theta);
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| - | % delta = orig_length - lengths;
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| - | % t = zeros(length(theta),6);
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| - | % for(j=1:length(theta))
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| - | % for i=1:6
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| - | % if delta(j) < def(i)
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| - | % t(j,i) = ten0(i)-slope(i).*delta(j);
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| - | % else
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| - | % t(j,i) = 0;
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| - | % end
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| - | % end
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| - | % end
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| - | % tension = t;
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| - |
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| - | l0 = string_length(0);
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| - | l = string_length(theta);
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| - | delta = l-l0;
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| - | for(j=1:length(theta))
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| - | for i=1:6
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| - | t(j,i) = ten0(i) + k(i)*delta(j);
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| - | end
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| - | end
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| - |
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| - | tension = t;
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| - |
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| - |
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| - | == hands.m ==
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| - | function fhand = hands(theta)
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| - | lhand = 0.127;
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| - | lstring = 0.01346962;
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| - | lspring = 0.0254;
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| - |
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| - | fspring = spring(theta);
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| - | tensions = eten(theta);
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| - | fstring = tensions(:,1)+tensions(:,2)+tensions(:,3)+tensions(:,4)+tensions(:,5)+tensions(:,6);
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| - | fstring = fstring';
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| - | fhand = (-fstring.*lstring+fspring.*lspring)./lhand;
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| - |
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| - |
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| - |
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| - | == offtune.m ==
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| - | function cents = offtune(theta)
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| - |
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| - | f0 = tune(0);
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| - | f = tune(theta);
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| - |
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| - | % Frequencies from http://www.phy.mtu.edu/~suits/notefreqs.html
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| - | sharp = [349.23 261.63 207.65 155.56 116.54 87.31];
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| - | for j=1:length(theta)
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| - | for i=1:6
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| - | n(j,i) = 1200*log2(f(j,i)/f0(i));
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| - | end
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| - | end
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| - | cents = n;
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| - |
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| - |
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| - | == spring.m ==
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| - | function tension = spring(theta)
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| - | hstring = 0.01346962;
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| - | hspring = 0.0254;
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| - | kspring = 7100.59;
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| - | initial_stretch = 0.0254;
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| - |
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| - | tension = kspring.*spring_stretch(theta)+kspring*initial_stretch;
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| - |
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| - |
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| - | % stens = eten(theta);
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| - | % stenst = zeros(1);
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| - | % for i=1:length(theta)
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| - | % stenst(i) = sum(stens(i,:));
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| - | % end
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| - | %
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| - | % tension = stenst*hstring/hspring;
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| - |
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| - |
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| - | == spring_stretch.m ==
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| - | function stretch = spring_stretch(theta)
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| - | stretch = 0.0254*sin(theta);
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| - |
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| - |
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| - | == string_length.m ==
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| - | function lengths = string_length(theta)
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| - |
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| - | l0 = 25.5;
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| - | m = sin(theta); % Find horizontal offset
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| - | l = (m.^2+l0^2).^0.5+(1-cos(theta));
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| - | lengths = 0.0254*l; % Convert inches to meters
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| - |
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| - |
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| - | == tune.m ==
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| - | function frequency = tune(theta)
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| - | orig_length = string_length(0);
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| - | lengths = string_length(theta);
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| - | delta = lengths - orig_length;
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| - | l = string_length(theta);
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| - | T = eten(theta);
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| - | density = 7800;
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| - | % a = [4.10 6.13 12.9 29.2 51.9 89.4]*10^-8;
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| - | % m = a*7800;
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| - | m = [1.67 3.16 5.88 10.84 20.33 38.10]*10^-4;
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| - | m = m.*[1 1 1 0.902 0.886 0.867];
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| - | % length(T(1,:))
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| - | % length(m)
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| - | % length(l(1))
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| - | % length(.5.*(T(1,:)./(m.*l(1))).^0.5)
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| - |
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| - | f = zeros(length(theta),6);
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| - | for i=1:length(theta)
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| - | f(i,:) = 1/2.*(T(i,:)./(m.*l(i))).^0.5;
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| - | end
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| - | frequency = f;
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