Week 4.1 - Solving second order ODEs

Aim: To describe the transient behavior of a simple pendulum using ODE.

EQUATION:

In the above equation,

g = gravity in m/s2,

L = length of the pendulum in m,

m = mass of the ball in kg,

b=damping coefficient.

where,

L=1 metre,

m=1 kg,

b=0.05.

g=9.81 m/s2.

CODE:

clear all
close all
clc
%input

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


% initail conditiion
theta_0 = [0;5];

% time points
t_span = linspace(0,10,500);

% solve ode

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


plot(t,results(:,1))
hold on
plot(t,results(:,2))
ylabel('plot')
xlabel('time')
 
 %ANIMATION
 X0=0;
 Y0=0;

ct=1;%counter actually represents each and every frame
 
 for i=1:length(t_span)
     X1=l*sin(results(i,1));
     Y1=-l*cos(results(i,1));



   plot([-1 1],[0 0],'linewidth',6,'color','g');
   hold on
   plot(X1,Y1,'O','markersize',20,'markerfacecolor','r');
   plot([X0 X1],[Y0 Y1]);

    axis([-2 2 -2 2]);
grid on 

     title ('Damping of simple Pendulum');
     legend ('Support rod','bob','Fixed point');
    
     hold off;
    
    M(ct)=getframe(gcf);
     ct=ct+1;
 end
 
 
 movie(M);
 videofile=VideoWriter('simple_ pendulum.avi','Uncompressed AVI');
 open (videofile);
 writeVideo (videofile, M);
 close (videofile);

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/1)*sin(theta1);
dtheta_dt = [dtheta1_dt; dtheta2_dt];
end

CODE EXPLAINATION:

1) write the input data such as

2) initial condition

theta_0 = [0;5];

where the inital displacement of pendulum is zero

and the initial velocity of pendulum is 5 m/s

time array

t_span = linspace(0,10,500);

it goes on 0 to 10 second in which 500 values in between.

3)solve ode

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

main inputs are t and theta

input parameter passing b,g,l,m.

then adding iinitial condition. t span

ode function is defined separatly

4) ploting the the animation in graph

5) using for loop comand for animation

creating a lenth bar with width and lenght in horizontal

from middle adding a string vertically downward.

add a bob or ball at bottom tip of the string.

6) creating a movie effect avi

video writting command.

the simple pendulum link youtube:

https://www.youtube.com/watch?v=FmrtvCE0bqY