Week 3 - Solving second order ODEs

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

Governing Equation :

second order differential equation

`(d^2 θ)/(dt^2 ) +b/m (dθ)/dt +g/l(sinθ)=0`

where,

g = gravity in m/s2,

L = length of the pendulum in m,

m = mass of the ball in kg,

b=damping coefficient.

Objective :

1. write a program that solves the simple pendulam ODE.
2. plot the animation of simple pendulum.

Steps of code :

1. write a function code so that we can check any value of influencing parameter
2. start coding main program with close all clear all clc.
3. define the inbput parameters fpor the equation
4. define the condition for displacement and velocity
5. then call the function solve with ode45
6. plot the graph
7. to obtain the pendulum motion use the loop . define the loop to get different value
8. plot bob by marker command and pendulum roop
9. for animation use getframe video write movie command

code:

function

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

main code

   close all
   clear all
   clc
   
   % Inputs 
   b = 0.05;
   g = 9.81;
   l = 1;
   m = 1;
   
   % initial condition
   theta_0 = [0;3];
   
   % TIME POINT 
   t_span = linspace(0,20,200);
   
   % solve ode 
   [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))
   ylabel('plot')
   xlabel('time')
   
   % animation
   figure(2)
   ct = 1;
   
   for i = 1:length(t_span)
       
       x0 = 0;
       y0 = 0;
       
       x1 = l*sin(results(i,1));
       y1 = -l*cos(results(i,1));
       
       % ploting horizontal line
       plot([-1 1],[0 0],'linewidth',3,"Color",'k')
       hold on
       % ploting the rope 
       plot([x0 x1],[y0 y1],'linewidth',3,'Color','r')
       hold on
       %ploting bob
       plot(x1,y1,'o','markerfacecolor','b','markersize',10)
      
       grid on
       axis([-2 2 -2 2])
       pause(0.005)
       hold off
       M(ct) = getframe(gcf);
       ct = ct+1;
   end
   
     movie(M)
     videofile = VideoWriter('o.avi','Uncompressed avi');
     open(videofile)
     writeVideo(videofile,M)
     close(videofile)
   

Result:

graph

screenshot of output

Animation youtube link :

https://www.youtube.com/watch?v=rJlYQEcUTPI

Conclusion:

written program to simulate the behavior of simple pendulum amd created the animation