Week 3 - Solving second order ODEs

% Objective: Solving second order ODEs of a penedulum


clear all
close all
clc


b = 0.05;0
g = 9.81;
l = 1;
m = 0.1

% initial condition

theta_0 = [0;5]

% time span

t_span = linspace(0,10,500)

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

%Plotting
figure(1)
hold on
plot(t,results(:,1))
plot(t,results(:,2))
title('Damped Harmonic motion of a Pendulum')
legend('Angular displacement', 'Angular velocity')
ylabel('Plot')
xlabel('Time(seconds)')

% For loop 

ct =1;
 for i = 1:length(results(:,1))
 x_start = 0     
 y_start = 0
 x_end = l*sin(results(i,1))
 y_end = -l*cos(results(i,1))
 
 %animation
 figure(2)
 plot([-1 1], [0 0], 'linewidth', 5)
 hold on
 plot([x_start x_end],[y_start y_end],'linewidth',3)
 hold on
 
 plot(x_end,y_end,'o','markersize',30 ,'markerFacecolor','b')
 title('Animation of transient behaviour of a Pendulum')
 axis([-1.5 1.5 -1.5 1])
 pause(0.03)
 hold off 
 
 M(ct) = getframe(gcf)
 ct = ct+1
 
 end
 
 
% Animation of the pendulum

 movie(M)
videofile=VideoWriter('Pendulum Animation.avi', 'uncompressed avi')
open(videofile)
writeVideo(videofile,M)
close(videofile)

%ode_func code
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

pendulum animation video drive link

https://drive.google.com/file/d/18P9ldyqWCgek0QQAxwoxvJgAqk9V3h9z/view?usp=sharing

explanation of the code

1. starting the code with the clear codes (clear all, close all , clc).to erase any previous exicutions.
2. state the given values such as damping coefficient,gravity, length of the string, and mass of the ball.
3. state the initial condition of the angle (theta) of the string which is maximum in the initial state.
4. time span is given as an array of matix ranging from 0-10 with 500 values.
5. then the ode function is solved useing a function load file as ode_func
6. ode_func is used to dictate the 2nd order ode equation.
7. plotting is done for the values(t,results) named as figure 1
8. title,legend, axis labels are given to the respective domains.
9. for loop is created for the animation of the pendulum which is named as figure 2.
10. for animation of the figure 2 plot command is used for the signification of the pendulum.
11. the animation of the pendulum is writen as pendulum animation.avi.

errors occured during writing of the code

1. error due to previous exicutions.
2. spelling mistake during codeing.
3. animation block mistrakes such as animation speed, model etc

steps taken to avoid the errors.

1. written clear all code at first to avoid any previous exicutions interferance.
2. spell check of the code.
3. giving correct parameters to avoid errors in animation.