Simple Pendulum ODE Using MATLAB

Solving Simple Pendulum ODE Using MATLAB:-

A simple pendulum having a mass m hanging from a string of length L and fixed at a point . When displaced it a little and released, the pendulum will swing back and forth and follow the periodic motion.

The second order ODE use the following equation -

`d^2(theta) / (dt) +(b/m)*d(theta)/(dt) + (g/l) * sin(theta)= 0`

Main function:-

1. Provide all the known input data with time array using \'linspace()\' command and initial condition of pendulum.
2. We use an inbuilt MATLAB solver \'ode45\' which is mainly a function of (t) and (theta). In this, we define this function using \'ode_func( input_ parameters )\' then give the initial condition and time span.
3. The obtained results from this function will be in 2 column of angular displacement and angular velocity along with the given time.
4. Then we plot the obtain data and get two curve of velocity and displacement.
5. From here using the values of 1st column of results as `theta`, we create a \'for- end\' loop from 1 up to the length of \'theta\'.
6. Here provide the initial position and the path equation for the bob and string.
7. Then plot the \'string\' and \'bob\' using initial position and the path equation, after this gave the axis to the pendulum for batter observation.
8. Finally, generate an avi. file to see the animation video and saved that.

clear all
close all 
clc

b= 0.05;
g= 9.8;
l= 1;
m= 1;

% initial conditions
 theta_0 = [0;3];
 
% time points \'time array\'
 t_span= linspace(0,20,200);
 
 % solve ODE & defining the function
 
 [t, results] = ode45(@(t , theta) ode_func(t,theta,b,g,l,m),t_span, theta_0);
 figure(1);
 hold on
 plot(t, results(:,1))
 plot(t, results(:,2))
 xlabel(\'Time\')
 ylabel(\'Plot\')
 
 
 theta = results(:,1);
 ct = 1;
 for i = 1: length(theta)
  THETA = theta(i);
 
  figure(2); 
% initial position
  x0 = 0;
  y0 = 0;
% position at any ponit 
   x1=   l* sin(THETA);
   y1= - l* cos(THETA);
   
    
 % string
  plot([x0 x1], [y0 y1] , \'color\' , \'b\');
  hold on
  plot ([-.5 .5], [0 0], \'linewidth\',5, \'color\', \'r\'); 
 % bob
  plot(x1,y1,\'O\', \'markersize\' ,15, \'markerfacecolor\', \'g\');
  axis([-2 2 -2 2]);
  hold off
  grid on
  
  pause(.03);
  M(ct) = getframe(gcf); 
  ct = ct+1;
  
  
 end
 
 movie(M)
 videofile= VideoWriter(\'ode_main.avi\',\'Uncompressed AVI\');
 open(videofile)
 writeVideo(videofile,M)
 close(videofile)
 

Function Defination:-

function [dtheta_dt] = ode_func(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  

Errors:-

• This error was due to the wrong placing of \'hold on\' command. Hold on should be used after the plot command to avoid this.

• Upper case \'Sin\' is used instead of \'sin\'

• Wrote \'0\' instead of \'o\'.

Link of the animation video:-

https://www.youtube.com/watch?v=648FyWRWoEo