File:Spring resonance.gif

function main()
 
% colors
   red      = [0.867    0.06    0.14];
   blue     = [0        129     205]/256;
   green    = [0        200     70]/256;
   black    = [0        0       0];
   white    = [1        1       1]*0.99;
   cardinal = [196      30      58]/256;
   cerulean = [0        123     167]/256;
   denim    = [21       96      189]/256;
   cobalt   = [0        71      171]/256;
   pblue    = [0        49      83]/256;
   teracotta= [226      114     91]/256;
   tene     = [205      87      0]/256;
   wall_color   = pblue;
   spring_color = cobalt;
   mass_color   = tene;
   exc_color=cardinal;
   a=0.65; bmass_color   = a*mass_color+(1-a)*black;
   % linewidth and fontsize
   lw=2;
   fs=20;
 
   ww = 0.5;  % wall width
   ms = 0.25; % the size of the mass        
   sw=0.1;    % spring width
   curls = 5;
   exc_size=0.05;
   plot_width=1.5; % width of plots
   
   K_osz = 0.05; % excitation amplitude
   omega =1; % excitation frequency
   omega0=1; % eigen frequency
   gamma=0.02; % damping factor
   
   filename='spring_resonance.gif';
   
   frames=100;

 
   options = odeset('RelTol',1e-4,'AbsTol',1e-4);
   [T,YODE] = ode45(@(t,y) dampedoszi(t,y,K_osz, omega, omega0, gamma),[0 20*pi],[0 0],options);
   figure(2)
   plot(T,YODE(:,1));

   A = 0.2; % the amplitude of spring oscillations
   B = -1; % the y coordinate of the base state (the origin is higher, at the wall)
 
   %  Each of the small lines has length l
   l = 0.05;
 
   N = length(T);  % times per oscillation 
   No = 1; % number of oscillations
   for f = 1:frames
      i=floor(length(T)/frames*f);  
      % set up the plotting window
      figure(1); clf; hold on; axis equal; axis off;
 
 
      t = T(i); % current time
      
      POSW=K_osz*sin(omega*t); % position of exciter with cos-excitation is a sine!
      
      H= B+YODE(i);      % position of the mass
      %H=K/sqrt((1-omega).^2+(2*gamma*omega).^2)*cos(
      
      
 
      % plot the spring from Start to End
      Start = [0, POSW]; End = [0, POSW+H];
      [X, Y]=do_plot_spring(Start, End, curls, sw);
      plot(X, Y, 'linewidth', lw, 'color', spring_color); 
 
      % Here we cheat. We modify the point B so that the mass is attached exactly at the end of the
      % spring. This should not be necessary. I am too lazy to to the exact calculation.
      K = length(X); End(1) = X(K); End(2) = Y(K);
 
      % plot the wall from which the spring is hanging
      plot_wall(-ww/2, ww/2, l, lw, wall_color);
 
      % plot the mass at the end of the spring
      X=[-ms/2 ms/2 ms/2 -ms/2 -ms/2 ms/2]+End(1); Y=[0 0 -ms -ms 0 0]+End(2);
      H=fill(X, Y, mass_color, 'EdgeColor', bmass_color, 'linewidth', lw);
 
      % plot exciter
      rectangle('Position',[0-exc_size/2,POSW-exc_size/2,exc_size,exc_size],  'FaceColor',exc_color)
      
          % the bounding box
          Sx = -0.4*ww;  Sy = B-max(abs(YODE(:,1)))-ms-0.05;
          Lx = ww+l+plot_width; Ly=l+K_osz;
          axis([Sx, Lx, Sy, Ly]);

          % plot amplitude time course
          plot(ww+T(1:i)./max(T).*plot_width, B+YODE(1:i,1), 'b-');
          line([ww ww+plot_width], [B B], 'Color', black);
          plot(ww+(T(1:i)-1)./(max(T)+1).*plot_width, K_osz*cos(omega*T(1:i)), 'r-');
          line([ww ww+plot_width], [0 0], 'Color', black);
          
          % plot resonance curve
          omeg=0.05:0.01:2;
          phase=atan2(-2.*gamma.*omeg, (omega0.^2-omeg.^2));
          amplitude=K./sqrt((omega0^2-omeg.^2).^2+(2*gamma*omeg).^2);
          plot(ww+omeg./max(omeg).*plot_width, B/2+B/3*phase/abs(max(phase)-min(phase)), 'g-')
          plot(ww+omeg./max(omeg).*plot_width, B/2-B/3*amplitude/abs(max(amplitude)-min(amplitude)), 'r-')
          line([ww ww+plot_width], [B/2 B/2], 'Color', black);
          rx=ww+omega/max(omeg).*plot_width;
          line([rx rx], [B/2 B/2-B/3], 'Color', cardinal)
          
 
      frame=getframe;
      [im,map1] = rgb2ind(frame.cdata,32,'nodither');
      if f==1
          map=map1;
          imwrite(im, map, filename, 'gif', 'WriteMode', 'overwrite', 'DelayTime', 0.05, 'LoopCount', Inf);
      else
          im= rgb2ind(frame.cdata,map);
          imwrite(im, map, filename, 'gif', 'WriteMode', 'append', 'DelayTime', 0.05);
      end
      
          
      disp(sprintf('Spring_frame%d', 1000+f)); %show the frame number we are at
 
      pause(0.1);
 
   end
 
function dy = dampedoszi(t,y, K, omega, omega0, gamma);
    dy = zeros(2,1);    % a column vector
    dy(1) = y(2);
    dy(2) = K*cos(omega*t)-2*gamma*y(2)-omega0^2*y(1);
 
function dy = damper(t,y, K, omega, omega0, gamma);
    dy = zeros(2,1);    % a column vector
    dy(1) = y(2);
    dy(2) = K*cos(omega*t);


 
function [X, Y]=do_plot_spring(A, B, curls, sw);
%  plot a 3D spring, then project it onto 2D. theta controls the angle of projection.
%  The string starts at A and ends at B
 
   % will rotate by theta when projecting from 1D to 2D
   theta=pi/6;
   Npoints = 500;
 
   % spring length
   D = sqrt((A(1)-B(1))^2+(A(2)-B(2))^2);
 
   X=linspace(0, 1, Npoints);
 
   XX = linspace(-pi/2, 2*pi*curls+pi/2, Npoints);
   Y=-sw*cos(XX);
   Z=sw*sin(XX);
 
%  b gives the length of the small straight segments at the ends
%  of the spring (to which the wall and the mass are attached)
   b= 0.05; 
 
% stretch the spring in X to make it of length D - 2*b
   N = length(X);
   X = (D-2*b)*(X-X(1))/(X(N)-X(1));
 
% shift by b to the right and add the two small segments of length b
   X=[0, X+b X(N)+2*b]; Y=[Y(1) Y Y(N)]; Z=[Z(1) Z Z(N)]; 
 
   % project the 3D spring to 2D
   M=[cos(theta) sin(theta); -sin(theta) cos(theta)];
   N=length(X);
   for i=1:N;
      V=M*[X(i), Z(i)]';
      X(i)=V(1); Z(i)=V(2);
   end
 
%  shift the spring to start from 0
   X = X-X(1);
 
% now that we have the horisontal spring (X, Y) of length D,
% rotate and translate it to go from A to B
   Theta = atan2(B(2)-A(2), B(1)-A(1));
   M=[cos(Theta) -sin(Theta); sin(Theta) cos(Theta)];
 
   N=length(X);
   for i=1:N;
      V=M*[X(i), Y(i)]'+A';
      X(i)=V(1); Y(i)=V(2);
   end
 
function plot_wall(S, E, l, lw, wall_color)
 
%  Plot a wall from S to E.
   no=20; spacing=(E-S)/(no-1);
 
   plot([S, E], [0, 0], 'linewidth', 1.8*lw, 'color', wall_color);