Week 4.1 - Solving second order ODEs

YOUTUBE LINK FOR ANIMATION

https://youtu.be/NxZ4eJsbXEo

MATLAB CODE

clear all
close all
clc

% inputs
l= 1; % length of the pendulam
m=1; % Mass in Kg
b=0.05; % Dampning Co-efficient
g=9.81; % Acc due to gravity in m/s^2

% Initial Inputs
theta_0 =[0;3];  % theta_0 is the intial condition corrosponding starting time.

% time span
t_span = linspace(0,20,500);

% solve ode
% For solving ode we use inbuilt matlab function name ode45 which has a
% syntax ode45(@(odefunc,timespan)
[t,thetad_v]=ode45(@(t,theta) ode_fnc(t,theta,b,g,l,m),t_span,theta_0);
% Here t and theta are the dependent and indipendent variable of our ode
% system
A_D = thetad_v(:,1); 
A_V = thetad_v(:,2);

%Animation
% Giving counter
ct=1;
for i= 1:length(A_D)
    w=A_D(i);
    x0=0; %Initial position of the Pendulum string, which is attached to the support
    y0=0; %Initial position of the Pendulum string, which is attached to the support
    
    x1=l*cos(w)
    y1=l*sin(w)
    
    % Plotting
    plot([y0 y1],-[x0 x1],'linewidth',2,'color','b')
    hold on
    plot(y1,-x1,'-o','Markersize',10)
    hold on
    plot([-1 1],[0 0]) %This can be considered as a wall or the support holding the Pendulum.
    hold off 
    axis([-2 2 -2 2])
    pause(0.003)
    M(ct)=getframe(gcf);
    ct=ct+1;
end
movie(M)
Videofile = VideoWriter('Pendulum_Animation.avi', 'uncompressed AVI');
open(Videofile) %This is the 1st step  to open the video file
writeVideo(Videofile,M)  %this command helps to write the frames(Which is stored in M) into this video file
close(Videofile) 

figure
plot(t,thetad_v(:,1))
hold on
plot(t,thetad_v(:,2))
ylabel('Angular disp and velocity')
xlabel('time')
    




                  FUNCTION CODE



function[dtheta_dt]= ode_fnc(t,theta,b,g,l,m)

theta1 = theta(1) % here we define theta 1 
theta2 = theta(2) % here we define theta 2
dtheta1_dt= theta2 % here we defined first ode
dtheta2_dt= -(b/m)*theta2-(g/l)*sin(theta1); % this is 2nd ode
dtheta_dt=[dtheta1_dt;dtheta2_dt]; % here we combined them into an array and semicolon means this is going to be coloum  matrix.
end