Solving second order ODEs - Simple Pendulum

AIM: Write a MatLab program to simulate the transient behaviour of a simple pendulum and to create an animation of it\'s motion.

SOLUTION:

Coding:

Main program

%Given values
b=0.05;    %damping coefficient
g=9.81;    %gravity in m/s2
m=1;       %mass of the ball in kg
l=1;       %length of pendulum in m

%initial conditions
theta_0=[0;3];

%time points
t_span=linspace(0,20,300);

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

%plotting 
ct=1;
for i=1:length(t)
 
   %initial point or fixed point
   x0=0;
   y0=0;
   
   %calculation of second point
   x1=l*sin(results(i,1));
   y1=-(l*cos(results(i,1)));
   
   figure(1)
   plot([-0.5 0.5],[0 0],\'linewidth\',5,\'color\',\'b\')
   axis([-2 2 -2 2])
   hold on
   plot([x0 x1],[y0 y1],\'linewidth\',2,\'color\',\'r\')
   plot(x1,y1,\'o\',\'markersize\',20,\'markerfacecolor\',\'g\',\'markeredgecolor\',\'b\')
   pause(0.03)
   hold off
   Ani(ct)=getframe(gcf);
   ct=ct+1;
end

%creating animation
movie(Ani)
videofile=VideoWriter(\'Pendulum_animation.avi\',\'Uncompressed AVI\');
open(videofile)
writeVideo(videofile,Ani)
close(videofile)

%plotting of angular velocity and displacement with respect to time
figure(2)
plot(t,results(:,1),\'color\',\'r\')
hold on
plot(t,results(:,2),\'color\',\'g\')
xlabel(\'Time\')
ylabel(\'Angular velocity and Displacement\')

Function program

function [dtheta_dt]=ode_fun(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

Output:

* Animation video

* Time vs Angular displacement and velocity

Report:

* Initially get the values of b,g,l and m of the pendulum.

* Then get the initial velocity and displacement and time span.

* Solving the ODE function of pendulum motion and find out the angular displacement and velocity along the time span of pendulum.

* Plot the graph between time span and angular displacement and velocity.

* Finally creating the animation video of pendulum motion.