Week 3 - Solving second order ODEs

clear all
close all
clc

% providing the default values provided in the question for execution of the ODE

l=1;
m=1;
b=0.05;
g=9.81;

% l is lenght of pendulum in Meter
% m is mass of ball in Kg, 
% g is  acceleration due to gravityin m/s
% b is the damping coefficient


% initial conditions for ODE execution

theta_0= [0;3];
% the value 0 in first coloumn in the initial diplacement
% the value 3 in second column is the initial velocity 3m/s

% providing a time interval of 20 seconds with 500 time points 
t_span=linspace(0, 20, 500);


% solving ODE
[t, results] = ode45(@(t,theta) ode_func(t,theta,b,g,l,m),t_span,theta_0);

% result stores the displacement and velocity of pendulum in the respective coloumns at time t
% ode45 is a default solver present in Matlab and Octaven
% it takes 3 inputs as the function to be integrated given as ode_func, as first one
% timespan as t_span, as second one and 
% initial condition as theta_0 , as third one.


theta_pend=results(:,1);
Ang_vel= results(:,2);

% theta_pend stores the values of dipacements
%Ang_vel stores values of angular velocity

% loop for animating pendudlum 
ct=1 
for i=1:length(theta_pend)
  theta1=theta_pend(i);
  x0=0;
  y0=0;
  x1=sin(theta1);
  y1= -cos(theta1);
  % -cos is used because if +cos is ued it would create inverse image of pendudlum motion
  
  plot(1.5*[x0 x1], 1.5*[y0,y1],'linewidth',1,'color','r',1.5*x1,1.5*y1,"o", "markersize", 10);

% a facror of 1.5 is multiplied to make to pendulum motion bigger for easy presentation.
 
  grid on ;
  
  axis([-2 2 -2 2]);
  pause(0.000003)
  
   P(ct)= getframe(gcf);
    ct=ct+1;
end

% creating movie file

movie(P)
videofile = VideoWriter('Pendulum_motion.avi','Uncompressed AVI');
open(videofile)
writeVideo(videofile,P)
close(videofile)

Function Used for ODE Calculation

% Function used for solving the ode using ode45 

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

  % dtheta1_dt is the first single oder differential equation and dtheta2_dt is the second single oder differential equation
  % dtheta_dt returns the vaules to the t and result coloumns in the main program
end

AIM :

Developing a program to simulate pendulum motion by solving a second degree ODE

STEPS :

1. Providing the default values provided in the question for execution of the ODE like length, mass, damping coefficient etc.
2. Provide the initial conditions like velocity and displacement of the pendulum for calculation of the equation.
3. Developing a Function that would provide as an input to the ofde45 default solver present in MATLAB and Octave
4. Calling the function inside the main program as an input to the ode45 function to obtain values of time, displacement and velocity from the function.
5. Using a For loop for plotting the position of pendulum at each point of time.
6. Obtaining an animation of the pendulum movement with respect to time
7. Extracting the animation as a movie file.

RESULT: Animation of Pendulation motion has been captured.

Video Links: https://youtu.be/EJH_0cB48ns

https://drive.google.com/file/d/1oJ52u5Xjm2fX4awBL575hVRGkpqgdmg3/view?usp=sharing

Errors:

1. Mistakes in output visualization due to logical errors in codes.
2. Unable to Create Movie File in octave.

Error Resolutions:

1. Logical Errors were rectified using trail and error method and the final solution was derived.
2. Created a Screen-recording using 3^rd party software as recommended my technical support

I have attached the Screenshots for your reference.

Resources:

1.https://skill-lync.com/knowledgebase/understanding-ode45-2

2.https://skill-lync.com/knowledgebase/understanding-anonymous-functions-in-matlab-2

3.https://in.mathworks.com/help/symbolic/simulate-physics-pendulum-swing.html

4.https://octave.org/doc/v4.2.1/Two_002dDimensional-Plots.html

# I have used Octave for running the code thus was unable to obtain a movie file output. So I have submitted the screen recording of the animation as recommended by technical support team. I have also trimmed the screen recording as it forms to be a too big file to upload.