Week 3 - Solving second order ODEs

clear all
close all
clc



%initial values
l=1                             %length of pendulum
m=1                             %mass of pendulum
b=0.05                          %dampimg coefficient
g=9.81                          % gravity

theta_0=[0;3];                  %initial values of angular displacement and velocity
t_span= linspace(0,20,200);     %time limit

%using matlab inbuilt function ode45
[t results]= ode45(@(t,theta)  ode_function(t,theta,b,g,l,m),t_span,theta_0);

plot(t,results(:,1)) %ploting graps between time and displacement
hold on
plot(t,results(:,2))

xlabel('plot')
ylabel('time')


ct=1
for i=1:length(results)   %using for loop to get different orientations of pendulum
    
    x1=l*sin(results(i,1));% determining  co-oedinates for pendulum position
    y1=l*cos(results(i,1));
    
    %ploting co-ordinates for pendulum.
    plot([-1,1],[0,0],[0,-x1],[0,-y1],'s','markers',15,'markerfacecolor','g')
    line([0,-x1],[0,-y1],'linewidth',5)
    axis([-2 2 -2 2]);
    pause(0.04)
    
    M(ct)=getframe(gcf)  % assigning all frame into M
    ct=ct+1;
end
% creating movie of M
movie(M);
videofile=VideoWriter('ode_pendulum','uncompressed AVI');
open(videofile);
writeVideo(videofile,M);
close(videofile);

    
    
    



%function which converts second order differential equation to first order
%differentional equation.

function [dtheta_dt]=ode_function(t,theta,b,g,l,m)
theta1=theta(1)
theta2=theta(2)
dtheta1_dt=theta2;
dtheta2_dt= -(b/m)*theta2-(g/l)*sin(theta1);
dtheta_dt=[dtheta1_dt;dtheta2_dt];
end



%explation:

% First i have assigned initial values for t,theta,b,g,l,m, angular
% displacement and velocity.
%using ode45 inbult function or solver to solve our 2nd order
%differentional equation and assigning the values to t and results then we
%have used plot commands to plot graps between time and displacement.
%Then we have used for loop to generate co-ordinates for each values in the
%results and by using plot command we created different plots and stored
%all the frames in M.
%Creating movie of M 
%Writing the function which converts 2nd order diff equation to 1st order
%diff equation,this function is not a main function but it is called in
%main function.

https://youtu.be/Gy7cfPtX0QQ