📄 rksquare.mht

📁 it is a very essential matlab code.
💻 MHT
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Date: Tue, 12 May 2009 09:54:49 -0700
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<BODY><PRE>% Runge-Kutta solution of second order O.D.E.
% damped spring-mass system with applied square wave forcing function
%
%  ODE is:   m*a + c*v + k*x =3D f
%
% where f is a square wave of specified amplitude from time t1 to t2

clear;

% define function file smdrk4.m (spring-mass-damper Runge-Kutta 4th =
order)    =20
%
%   function a =3D smdrk4(x,v,f,c,k,m)
%   a =3D (f - c*v - k*x)/m;

% define constants

m =3D 1;         % mass
c =3D 1;         % damping
k =3D 6;         % spring stiffness

t0 =3D 0;        % initial time
tmax =3D 40;     % total time
dt =3D 0.1;      % time step

t1 =3D 2;        % start time of square wave
t2 =3D 14;       % end time of square wave
fsquare =3D 10;  % square wave amplitude

% initial conditions

x =3D 0;         % position
v =3D 0;         % velocity
time =3D t0;

fprintf(['Solution of Spring-Mass-Damper with Square ',...
         'Wave Forcing Function\n']);
fprintf('Using manual 4th order Runge-Kutta and Matlab''s ode45\n');

% start Runge-Kutta solution

i =3D 0;
while time &lt; tmax
  time =3D time + dt;
  ti =3D time - dt;     % initial time for this time step
  xi =3D x;             % initial position for this time step

  if (ti &gt;=3D t1) &amp; ( ti &lt;=3D t2)       % evaluate forcing =
function
    f =3D fsquare;                    % amplitude
  else
    f =3D 0;
  end

  k1x =3D v;
  k1v =3D smdrk4(x, v,f,c,k,m);

  k2x =3D v + k1v * dt/2;
  k2v =3D smdrk4(x + k1x * dt/2, v + k1v * dt/2,f,c,k,m);

  k3x =3D v + k2v * dt/2;
  k3v =3D smdrk4(x + k2x * dt/2, v + k2v * dt/2,f,c,k,m);

  k4x =3D v + k3v * dt;
  k4v =3D smdrk4(x + k3x * dt, v + k3v * dt,f,c,k,m);

  x =3D x + (k1x + 2*k2x + 2*k3x + k4x) * dt/6;
  v =3D v + (k1v + 2*k2v + 2*k3v + k4v) * dt/6;

  i =3D i + 1;
  position(i) =3D x;
  timev(i) =3D time;
end;

% the one line Matlab solution; note NEW function: smdode45

[time45,results45] =3D ode45('smdode45',t0,tmax,[0 0]);

% view results

force =3D zeros(1,length(timev));  % create force vector for viewing
for i =3D t1/dt:t2/dt
  force(i) =3D 1;=20
end

% plot the curves

plot(timev,position,time45,results45(:,1),'o',timev,force,'-');
title('Runge-Kutta solution of spring-mass-damper system');
legend('4th order Runge-Kutta','Matlab''s ode45','forcing function');
axis([t0 tmax -1 3]);

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