Week 4.1 - Solving second order ODEs

Youtube Video Link: https://youtu.be/zHxwsAE0BaQ

CODE:

clear all
close all
clc

%INPUTS

g = 9.8;      %Acceleration due to gravity, m/s^2
l = 1;        %Length of the pendulum, m
m = 1;        %Mass of the pendulum, kg
p1 = 0;       %Initial position in radians
v1 = 3;       %Initial velocity in rad/sec
t1 = 20;      %Time span for result calculation, sec
c = 0.05;     %Damping coefficient

%INITIAL CONDITIONS
theta_0=[p1;v1];

%CREATING 500 TIME POINTS IN THE GIVEN TIME PERIOD
t_span=linspace(0,t1,500);

%Solve ODE
[t,results]=ode45(@(t,theta) simplependulum_equn(g,l,c,theta,t),t_span,theta_0);

%Configure the video writer
VideoFile = VideoWriter('simple_pendulum_anil.avi','Uncompressed AVI');
open(VideoFile)

%Start plotting the results to create a video
for k = 1:500
    
    x1=0;
    y1=0;
    x2=l*sin(results(k,1));
    y2=-l*cos(results(k,1));
    x=-2*l;
    y=2*l;
    z=-2*l;
    plot([0,0],[z,0],'--',[x1,x2],[y1,y2],x2,y2,'go','color','k',...
    'linewidth',2,'markersize',20,'markerfacecolor','g','markeredgecolor','g');
    axis([x y z 0]);
    daspect([1 1 1]);%To ensure same scales along both the axes.
    xlabel('Length in meter');
    ylabel('Length in meter');
    title([num2str(t_span(k)),'s'])
    pause_t=t1/500;
    pause(pause_t);
    M = getframe(gcf);
    writeVideo(VideoFile,M)
    
end

close(VideoFile)

%THE FOLLOWING FUNCTION PROVIDES THE SYSTEM OF FIRST ORDER DIFFERENTIAL EQUATIONS TO
%THE ODE45 SOLVER
function [dtheta_dt]= simplependulum_equn(g,l,c,theta,t)

    w=g/l;
    theta1=theta(1);
    theta2=theta(2);
    dtheta1_dt=theta2;
    dtheta2_dt=-w*sin(theta1)-(c/l)*theta2;
    dtheta_dt=[dtheta1_dt;dtheta2_dt];

end

EXPLANATION:

First, all the input variables are assigned respective values as specified in the problem.

The function simplependulum_equn available at the end of the code takes the input variables along with dependent variables (theta_1 and theta_2) and returns the ODE in the form of a system of first-order differential equations.

Finally, the respective arguments (function handle to simplependulum_equn function , time span and initial condition vector) are passed to the ode45 solver to finally obtain the solution of the differential equation in t and results matrices

The solver returns angle (theta_1) and angular velocity(theta_2) as the output in the results vector at each time point corresponding to the value stored in the t vector.

The screenshot of the execution is attached below.

SCREENSHOT: