Week 3 - Solving second order ODEs

• Solving the second order ode to first order ode of a simple pendulum and to plot displacement, velocity vs time and to animate the motion of pendulum with given conditions.

clear all
close all
clc

% Given parameters
% Damping Coefiicient
b = 0.05;
% Gravitational constant (m/s^2)
g = 9.81;
%length of the pendulum (m)
l = 1;
% mass of the ball (kg)
m = 1;

% Initial conditions
theta_0 = [0 3];

t_span = linspace(0,20,300);

%solving ode- ode45 command is used to solve ode
[t, results] = ode45(@(t,theta) ode_func(t, theta,b,g,l,m),t_span,theta_0);
figure(1)
plot(t,results(:,1))
hold on
plot(t, results(:,2))
xlabel('Time')
ylabel('Plot')

% Annimation
result2 = results(:,1);
ct = 1;
for i = 1:length(result2)
    position = result2(i);
    x= 0;
    y = 0;
    x1 = l*sin(position);
    y1 = -l*cos(position);
    figure(2)
    plot([x x1],[y y1],'-o','MarkerSize',15,'markerfacecolor','g','markeredgecolor', 'g','Linewidth',3)
    axis([-2 2 -2 2])
    hold on
    plot([-1 1],[0 0],'Linewidth',5)
    M(ct) = getframe(gcf);
    hold off
    ct = ct+1;
end
movie(M)
videofile = VideoWriter('solving_ode.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/l)*sin(theta1);
    dtheta_dt = [dtheta1_dt;dtheta2_dt];
end

• In the above program initial parameters are given such as damping coefficient, pendulum length and mass of the ball.
• time period is 0 to 20 and taken 300 values in between, given angular displacement = 0, angular velocity = 3 rad/sec at time t = 0 sec
• using the ode45 command to solve ode, convert second order ode to first order ode by creating a function using function command, we obtain main components time t and angular displacement theta.
• plot command is used to plot velocity and displacement on y-axis and time on x-axis.
• using for loop we have obtained the position at different time and it is plotted.
• Movie command is used to create an annimation and marker command to create sphere.

https://youtu.be/SrbClC-aTQ8