Week 3 - Solving second order ODEs

Objective

write a program that solves the given ODE, this ODE represents the equation of motion of a simple pendulum with damping

code

• % solve the ODE that represents the equation of motion of a simple pendulum with damping
  close all
  clear all
  clc
  
  % inputs
  
  L = 1;    % L = length of the pendulum in m
  m = 1;    % m = mass of the ball in kg
  b = 0.05; % b=damping coefficient
  g = 9.81; % g = gravity in m/s2
  
  % inetial displacement and initial time
  theta_o = [0;3];
  
  % time array
  t_span = linspace(0,20,1000);
  
  % integration of ode
  [t,result] = ode45(@(time,theta) pendulum_function(theta,b,g,L,m), t_span, theta_o);
  
  
  subplot(5,6,[19,20,21,25,26,27])
  
  plot(t,result(:,1), 'r')
  hold on
  plot(t,result(:,2), 'b')
  axis([0 20 -5 15]);
  xlabel('time');
  ylabel('y axis');
  legend('angular displacement','angular velocity')
  title('Damping penedulum');
  
  
  
  subplot(5,6,[23,24,29,30])
  
  plot(result(:,1), result(:,2))
  xlabel('angular displacement');
  ylabel('angular velocity');
  
  
  
   ct = 1;
  for i = 1 :  length(result(:,1))
   
      X0 = 0;
      Y0 = 0;
      
      
      X = L*sin(result(i,1));
      Y = -L*cos(result(i,1));
      
       subplot(5,6,[3,4,9,10])
         
          plot([-1.5 1.5], [0 0]);
          hold on
          line([X0 X],[Y0 Y]);      
          plot(X,Y, '.r','markersize', 50 );     
          axis([-1.5 1.5 -1.5 0]);
          hold off
  
      f(ct) = getframe;
      ct = ct+1;
      
      
  end
  
  movie(f)
  videofile = VideoWriter('damping_pandulum_a.avi');
  open(videofile)
  writeVideo(videofile,f)
  close(videofile)

Code Explanation

• first inetial condetions applied
• split second order ode into two first order ordinary differential equations and created a function
• 

•  function [dtheta_dt] = pendulum_function(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

• then using ode45 first order equations are integrated
• making animation of pendulum we required (X0,Y0) and (X,Y)
• 
• then using "getframe" created animation

Error faced

• stucked at this animatio theoary (instructor sir. jithin help me to overcome that)

Output

• 
• video file (https://drive.google.com/drive/folders/1DOPfrjDh-YQnOToVBMdK89TB_cCQBM_V?usp=sharing)

Output explanation

• created graphs using subplots and then save it
• video save as avi and then compress it using VLC player and then save it as mp4 file