Create single_stiff_string.m

Author: mhamilt

schemes/stiff_string/single_stiff_string.m

@@ -0,0 +1,254 @@
+clearvars, 
+close all
+
+%%%%% flags
+
+plot_on = 0;                % in-loop plotting on (1) or off (0)
+itype = 1;                  % type of input: 1: struck, 2: plucked
+bctype = 1;                 % boundary condition type: 1: simply supported, 2: clamped
+outtype = 1;                % output type: 1: string displacement, 2: string velocity
+
+%%%%% parameters
+
+% physical string parameters (see Q7a)
+L(1:6) = 0.648;                 % guitar scale length(m)
+rho(1:6)= 7750;                 % Density of steel (kg/m^3)
+% Strings:     E        B        G       D       A     E
+% gauges:   .046     .036     .026    .017    .013   .01
+r   = [1.1684e-3 9.144e-4 6.604e-4 4.3e-4 3.3e-4 2.5e-4];   % string radius (m)
+
+% Young's modulus (Pa) taken from
+% http://srjcstaff.santarosa.edu/~yataiiya/E45/PROJECTS/guitar%20strings.pdf
+% considering the lowest three strings to be wound and the top three
+% single-wire
+E   = [6.4e10 6.4e10 6.4e10 1.88e11 1.88e11 1.88e11] ;
+freq= [82.41 110 146.83 196 246.94 329.63]; % Note frequencies
+
+
+
+% T60 (s) was left to 5 s. Although longer T60 were found by analysing
+% decays of plucked guitar strings, 5 s produced nicer sound. 
+T60(1:6) = 5;
+
+% I/O
+SR = 44100;                 % sample rate (Hz)
+Tf = 5;                     % duration of simulation (s)
+
+xi(1:6) = 0.3;              % coordinate of excitation (normalised, 0-1)
+famp = [1 1 1 1 1 1];                        % peak amplitude of excitation (N)
+dur = [0.001 0.001 0.001 0.001 0.001 0.001]; % duration of excitation (s)
+exc_st = [0 0.5 1 1.5 2 2.5];                % start time of excitation (s)
+xo(1:6) = 0.2;              % coordinate of output (normalised, 0-1)
+window_dur = 0.01;          % duration of fade-out window (s). See Q6.
+
+%%%%% Error checking
+
+if any([r, rho, T60, L, SR, Tf, famp, window_dur]<=0)
+    error('Parameters must be positive');
+end
+if any(E<0)
+    error('Young''s modulus must be non-negative');
+end
+if any(exc_st<0 | exc_st>Tf)
+    error('Start time of excitation are beyond the simulation time')
+end
+if any([xi, xo]>=1 | [xi, xo]<=0)
+    error('Excitation and output coordinates must be between 0 and 1')
+end
+if any([all(plot_on~=[0 1]) all(itype~=[1 2])...
+        all(bctype~=[1 2]) all(outtype~=[1 2])])
+    error('Flag values are incorrect')
+end
+if any(exc_st+dur>Tf)
+    error('Excitation duration exceeds simulation duration')
+end
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+%%%%% derived parameters
+A = pi*r.^2;                    % string cross-sectional area
+I = 0.25*pi*r.^4;               % string moment of intertia
+K = sqrt(E.*I./(rho.*A));       % stiffness constant
+
+% Tension (N) using stiff string frequency equation (7.21) from NSS
+T = (4*L.^2.*freq.^2+K.^2*pi^2).*rho.*A;
+
+c = sqrt(T./(rho.*A));        % wave speed
+
+sig = 6*log(10)./T60;         % loss parameter
+
+%%%%% grid
+
+k = 1/SR;                   % time step
+hmin = sqrt(0.5*(c.^2.*k^2+sqrt(c.^4.*k^4+16*K.^2.*k^2)));    % minimal grid spacing
+
+N = floor(L./hmin);          % number of segments (N+1 is number of grid points)
+
+%%%%% rest of error checking
+if any(N>10000)
+    error('Too many grid points.')
+end
+
+h = L./N;                    % adjusted grid spacing
+
+lambda = c*k./h;             % Courant number
+mu = K*k./h.^2;               % numerical stiffness constant
+
+%%%%% I/O
+
+Nf = floor(SR*Tf);          % number of time steps
+
+li = floor(xi.*N);         % grid index of excitation
+lo = floor(xo.*N);         % grid index of output
+
+Nstr = length(T);            % number of strings
+
+
+%%%% Adjusting li and lo in case they are out of range and calculating
+%%%% indeces for I/O in the case of concatenated u
+
+mli=zeros(1,Nstr);      % Input for matrix form
+mlo=zeros(1,Nstr);      % Output for matrix form
+
+for s=1:Nstr
+    if any([li(s),lo(s)]<2 | [li(s), lo(s)]>N(s))
+        warning('Param:outRange',...
+            ['Excitation and/or output is outside the range of h to L-h.' ...
+            '\n Forcing it to the range.'])
+    end
+    if li(s)<2
+        li(s)=2;
+    elseif li(s)>N(s)
+        li(s)=N(s);
+    end
+    if lo(s)<2
+        lo(s)=2;
+    elseif lo(s)>N(s)
+        lo(s)=N(s);
+    end
+    
+    % Adding the summed number of grid points of the previous strings and
+    % subtracting s number of points, since the 1st point of each string is
+    % missing
+    mli(s)=li(s)+sum(N(1:s-1))-s;
+    mlo(s)=lo(s)+sum(N(1:s-1))-s;
+end
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+% create force signal
+
+f = zeros(Nf,Nstr);                     % input force signal
+durint = floor(dur*SR);                 % duration of force signal, in samples
+exc_st_int = floor(exc_st*SR)+1;        % start time index for excitation
+
+
+% Computing forcing terms for each string
+for s=1:Nstr
+    f(exc_st_int(s):exc_st_int(s)+durint(s)-1,s)=famp(s)*0.5.*...
+        (1-cos(2/itype*pi*(0:durint(s)-1)/durint(s)));  
+end
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+%%%%% scheme coefficients
+out = zeros(Nf, 1);                % output
+tic
+% loop over the strings
+AA=cell(1,Nstr);
+BB=cell(1,Nstr);
+CC=cell(1,Nstr);
+
+% Finding matrices for each string
+for s=1:Nstr
+    % interior
+    I = speye(N(s) - 1);
+    Dxx = fidimat(N(s)-1,'xx', 1);
+    Dxxxx = fidimat(N(s)-1,'xxxx', 1);
+        
+    a = 1/(1+sig(s)*k);
+    BB{s} = a*(2*I + (lambda(s)^2*Dxx) + (-mu(s)^2) * Dxxxx);
+    CC{s} = a*(-(1-sig(s)*k)*I);         
+end
+
+% Concatenating them into block diagonal matrices
+AA=blkdiag(AA{:});
+BB=blkdiag(BB{:});
+CC=blkdiag(CC{:});
+
+
+% input
+d0 = 1./(1+sig*k).*(k^2./(h.*rho.*A));
+
+%%%%% initialise scheme variables
+
+u2 = zeros(sum(N)-Nstr,1);      % state
+u1 = u2;                        % state
+u = u2;                         % state
+
+% plot
+if(plot_on==1)
+    % draw current state
+    fig=figure;
+    for s=1:Nstr
+        subplot(Nstr,1,s)
+        pl(s)=plot((0:N(s)-2)'*h(s), u(1-s+1+sum(N(1:s-1)):sum(N(1:s))-s), 'k');
+        axis([0 L(s)-h(s) -0.0005 0.0005])
+        % set data source (to be later used for refreshing data)
+        set(pl(s),'YDataSource',['u(' num2str(1-s+1+sum(N(1:s-1)))...
+            ':' num2str(sum(N(1:s))-s) ')'])
+        ylabel(['String ' num2str(s)])
+    end
+    drawnow
+end
+    
+%%%%% main loop
+for n=1:Nf
+    
+    u=(BB*u1+CC*u2);
+    
+    % send in input
+    u(mli) = u(mli)+(d0.*f(n,:))';
+    
+    % read output
+    if(outtype==1)
+        out(n)= sum(u(mlo));
+    end
+    if(outtype==2)
+        out(n) = sum((u(mlo)-u1(mlo))/k);
+    end
+    
+    % plot
+    if(plot_on==1)
+        refreshdata(fig)
+        drawnow
+    end
+    
+    % shift state
+    u2 = u1;
+    u1 = u;
+    
+end
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+%%%% windowing the end
+nWin=floor(SR*window_dur);
+out(end-nWin:end)=0.5*(1+cos(pi*(0:nWin)/nWin))'.*out(end-nWin:end);
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+toc
+%%%%% play sound
+soundsc(out,SR);
+
+%%%%% plot spectrum
+
+figure(1)
+outfft = 10*log10(abs(fft(out)));
+plot((0:SR/Nf:SR*(1-1/Nf))', outfft, 'k')
+xlabel('Frequency (Hz)')
+ylabel('Magnitude')
+title('Frequency Spectrum of Output Signal')
+set(gca, 'FontSize', 12)
+xlim([0 SR/2])
+