Week 3 - Solving second order ODEs

Report--

Steps

1. A separate function is written for solving the differential equation.
2. pendulum parameters are initialized.
3. To obtain the motion equation, time and angular displacement are mentioned.
4. the results are obtained by solving the differential equation using ODE45.
5. for plotting the graph of pendulum movement, vertical and horizontal positions of the endpoint of the pendulum are calculated by the formula shown.x-coordinate=sin(w*t

   %code for function
   
   function [dtheta_dt]=ode_func(t_span,theta)%objective function
   b=0.05;
   g=9.81;
   l=1;
   m=1;
    
   theta1=theta(1);     %dtheta/dt
   theta2=theta(2);     %dtheta^2/(dt)^2
   dtheta_dt(1)=theta2;%angular velocity
   dtheta_dt(2)=-(b/m)*theta2-(g/l)*sind(theta1);
   dtheta_dt=dtheta_dt(2)+dtheta_dt(1)*(b/m)+sind(theta)*(g/l);
   end
   
   
   
   %main
   
   
   b=0.05; 
   g=9.81;
   l=1; %length of string
   m=1; %mass of thr bob
   w=3; %frequency
   
   
   
   
   theta_0=linspace(-90,90,20);
   %time
   
   
   
   %Fixed point of the pendulum.
   d0=0;
   v0=1;
   %Solving differential equation.
   
   t_span=linspace(0,20,20);
   [t,theta]=ode45('ode_func',t_span,theta_0);
   tic
   ct=1;
   
   %for loop for plotting in one direction
   for i=length(t(:,1))
   
       d=-l*sind(3*t(:,1));    %x coordinate
       v=l*(1-cosd(3*t(:,1))); %y coordinate
       
       
       for i=1:length(t(:,1)) 
           plot([0 d(i)],[1 v(i)],'LineWidth',3,'color','b')
           hold on
           scatter(d(i),v(i),'o','LineWidth',10)
           hold off
           axis([-2 2 -2 2])
           pause(0.1)
           S(ct)=getframe(gcf);
           ct = ct+1;   
       end
       
   end
   
   %for loop for plotting in another direction.
   ct=1;
   
   for i=1:length(t(:,1))
       d=l*sind(3*t(:,1));     %x coordinate
       v=l*(1-cosd(3*t(:,1))); %y coordinate
       
       
       for i=1:length(theta) 
           plot([0 d(i)],[1 v(i)],'LineWidth',3,'color','b')
           hold on
           scatter(d(i),v(i),'o','LineWidth',10)
           hold off
           axis([-2 2 -2 2])
           pause(0.1)
           S(ct)=getframe(gcf);
           ct = ct+1;
            
       end   
   
   end
   
   
   time=toc
   movie(S) 
   videofile=VideoWriter('pendulum')  %creating a video file
   open(videofile)
   writeVideo(videofile,S)
   close(videofile)

   ), y-coordinate=L(1-cos(w*t)
6. for every value of theta obtained from the equation x and y coordinates are calculated using the above formulae and stored.-----(first for loop)
7. Later for every value of x and y graph is plotted.----(second for loop)
8. to create an animation look,ct is used.
9. Avi video file is created by using 'videowriter'.

ERRORS--

1. Index exceeds the number of array elements. the index must not exceed.
2. Array indices must be positive or local values.
3. Error in odearguments.
4. Dot indexing is not supported for this type of variable.
5. plot was not coming correctly.

Steps were taken to overcome errors--

1. Error(1&2) was due to a number of array elements, a number of expected results didn't match with the input. By correcting the matrix error was corrected.
2. Error due to arguments in the ode function and it was correctedby giving the correct arguments.
3. There was a dot operator in the formulas, by removing that dot operator error was corrected.
4. Plot was not meeting the expectations so by making more changes to the code it was corrected.

https://youtu.be/4G9jDYHA5Ho