ElasticTRAP.m

function ElasticTRAP(T,dt,theta_init,thetadot_init,r_init,rdot_init,label)
%
%Elastic(T,dt,powers)
%----------------------------------------------------------------------------------- 
%Plots how the angles theta1 theta2 and their derivatives vary in time for
%the ELASTIC pendulum using the Trapezoidal scheme - assuming L=1, k=10, m=1
%Animated the elastic pendulum system with desired parameters using the Trapezoidal Rule
% @ Sohan Dharmaraja MIT JUN2007


powers = 1;
N=T/dt;
k=10;
m=1;
L=1;

timevec(1)=0;
thetavec(1)=theta_init;
thetadotvec(1)=thetadot_init;
rvec(1)=r_init;
rdotvec(1)=rdot_init;

xold=[theta_init; thetadot_init; r_init; rdot_init];
xnew=xold;


for Showimp=1:powers
    dt=dt^Showimp;
    N=T/dt;
    iter=1;
    
    timevec(1)=0;
    thetavec(1)=theta_init;
    thetadotvec(1)=thetadot_init;
    rvec(1)=r_init;
    rdotvec(1)=rdot_init;
    xold=[theta_init; thetadot_init; r_init; rdot_init];
    xnew=xold;

    f1=inline('iternew1 - (dt/2)*iternew2 - orig1 - (dt/2)*orig2','dt','orig1','orig2','iternew1','iternew2');
    f2=inline('iternew2 - (dt/2)*((2*iternew4*iternew2)/((-1)*iternew3) + ((9.81/((-1)*iternew3))*sin(iternew1))) - orig2 - (dt/2)*((2*orig4*orig2)/((-1)*orig3) + (((9.81/(-1*orig3)))*sin(orig1)))','dt','orig1','orig2','orig3','orig4','iternew1','iternew2','iternew3','iternew4');
    f3=inline('iternew3 - (dt/2)*iternew4 - orig3 - (dt/2)*orig4','dt','orig3','orig4','iternew3','iternew4');
    f4=inline('iternew4 - (dt/2)*((9.81*cos(iternew1) - (k/(m))*(iternew3-L) + iternew3*((iternew2)^2))) - orig4 - (dt/2)*((9.81*cos(orig1) - (k/(m))*(orig3-L) + orig3*((orig2)^2)))','dt','orig1','orig2','orig3','orig4','iternew1','iternew2','iternew3','iternew4','k','L','m');

for iter=1:N

xinit=xnew;

for j=1:5
    theta=xnew(1,1);
    V=xnew(2,1);
    r=xnew(3,1);
    W=xnew(4,1);
    
    J = [1, -1*(dt/2), 0, 0; (9.81/r)*cos(theta)*(dt/2), 1+((W/r)*dt), -1*(dt/2)*((9.81/(r^2))*sin(theta)+ (2*W*V)/(r^2)), V*(dt/r); 0, 0, 1, (-1)*(dt/2); (9.81*sin(theta))*(dt/2), (-1)*r*V*dt, (-1)*(V^2)*(dt/2)+(k/(2*m))*dt, 1];
    
    RHSvec(1,1)=f1(dt,xinit(1,1),xinit(2,1),xold(1,1),xold(2,1));
    RHSvec(2,1)=f2(dt,xinit(1,1),xinit(2,1),xinit(3,1),xinit(4,1),xold(1,1),xold(2,1),xold(3,1),xold(4,1));
    RHSvec(3,1)=f3(dt,xinit(3,1),xinit(4,1),xold(3,1),xold(4,1));
    RHSvec(4,1)=f4(dt,xinit(1,1),xinit(2,1),xinit(3,1),xinit(4,1),xold(1,1),xold(2,1),xold(3,1),xold(4,1),k,L,m);
    
    deltax=inv(J)*((-1)*RHSvec);
    det(inv(J));
    norm(deltax);
    xnew=xold+deltax;
    xold=xnew;

end


thetavec(iter+1)=xnew(1,1);
thetadotvec(iter+1)=xnew(2,1);
rvec(iter+1)=xnew(3,1);
rdotvec(iter+1)=xnew(4,1);
timevec(iter+1)=iter*dt;

end

subplot(2,2,1)
plot(timevec,thetavec,'LineWidth',2)
title('theta vs time','FontWeight','bold')

hold on

subplot(2,2,2)
plot(timevec,thetadotvec,'LineWidth',2)
title('theta dot vs time','FontWeight','bold')
hold on

subplot(2,2,3)
plot(timevec,rvec,'LineWidth',2)
title('r vs time','FontWeight','bold')
hold on

subplot(2,2,4)
plot(timevec,rdotvec,'LineWidth',2)
title('r dot vs time','FontWeight','bold')
hold on

store=[thetavec', thetadotvec'];
end

% now plot the exact solution using Matlab
% ------------------------------------------------------------------------------------------------- change this value below for spring constant 
%------------------------------------------------------------------------------------------------- ((k/m))*(R - Lnatural) change this value below for spring constant 
g = inline('[ y(2); (2*y(2)*y(4))/((-1*y(3))) + (9.81 *sin(y(1)))/(-1*y(3));y(4); 9.81*cos(y(1)) - ((10)*(y(3)-1)) + y(3)*((y(2))^2)]', 't', 'y');
[t,y]=ode45(g,[0,T],[theta_init; thetadot_init; r_init; rdot_init]);
% Can do - [t,y]=ode45(g,[0:0.1:2*pi],[theta_init;0]); if diff time steps required

subplot(2,2,1)
plot(t,y(:,1),'Color','red', 'LineWidth',2)
hold on

subplot(2,2,2)
plot(t,y(:,2),'Color','red', 'LineWidth',2)
hold on

subplot(2,2,3)
plot(t,y(:,3),'Color','red', 'LineWidth',2)
hold on

subplot(2,2,4)
plot(t,y(:,4),'Color','red', 'LineWidth',2)
hold on
hold off
 
initinfo = [num2str(T), ' ', num2str(2*dt), ' ', num2str(theta_init),' ',num2str(thetadot_init), ' ', num2str(r_init),' ', num2str(rdot_init)];
saveas(gcf,[num2str(label) ' Trap - ' initinfo '.png']);close;