Oscillating Pendulum With Solving second order ODEs

For solving this challenge first create a function according to the Pendulum\'s ODE.

`(d^2theta)/(dt^2) + b/m(d theta)/dt + g/l sin theta = 0`

where:

`b =`damping coefficient`

`m =`mass of ball in kg`

`l =`length of the pendulum in m`

`g =`gravity in m/s^2`

Function is

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

After creating ODE function, Now write the programme.

% Writing a code to oscillate the pendulum.
clear all
close all
clc

%inputs
b = 0.05;   %Damping Coefficient
g = 9.81;   %Gravity m/s^2
m = 1;      %Mass of the Pendulum in kg
l = 1;      %Length of the string attached to pendulum in meters

%initial conditions

theta_0 = [0;3];  %[angular displacement=0;angular velocity=3]   

%time points
t_span = linspace(0,20,500); %pendulum oscillates between 0-20 seconds

%solve ODE

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

%plotting the graph
figure(1)

subplot(2,1,1)
p1 = plot (t,result(:,1),\'linewidth\',2,\'color\',\'b\');
xlabel(\'\\bf{Time}\');
ylabel(\'\\bf{Theta [\\theta]}\');
title(\'Angular Displacement [{\\theta}] v/s Time\');

hold on 
subplot(2,1,2)
p2 = plot(t,result(:,2),\'linewidth\',2,\'color\',\'g\');
xlabel(\'\\bf{Time}\');
ylabel(\'\\bf{Velocity}\');
title(\'Angular Velocity v/s Time\');
legend([p1(1),p2(1)],\'angular displacement\',\'angular velocity\')

%pendulum simulation using ode
theta = result(:,1);
ct=1;
%%creating a for loop for oscillating the pendulum

for i=1:length(result)
    THETA = result(i,1);
    
    %initial boundary conditon
    x0=0;
    y0=0;

    %Defining x and y coordinates
    x1=  l*sin(THETA);
    y1= -l*cos(THETA);
    
    figure(2)
    
    hold on
    plot([x0 x1],[y0 y1],\'linewidth\',4);
    plot(x1,y1,\'-o\',\'markersize\',30,\'markeredgecolor\',\'r\',\'markerfacecolor\',[0.6 0.6 0.6]);
    plot(x0,y0,\'-*\',\'markersize\',3,\'markeredgecolor\',\'r\',\'markerfacecolor\',[0.6 0.6 0.6]);
    title(\'Oscillating Pendulum\');
    axis([-2 2 -2 0]);
    pause(0.005);
    
    
    M(ct)=getframe(gcf);
    ct=ct+1;
    clf()         %clf() code is  used for clearing the current graphic image
end
%%creating the animation and movie(avi)

movie(M);
videofile = VideoWriter(\'second_ode_pendulum.avi\',\'uncompressed AVI\');
open(videofile)
writeVideo(videofile,M)
close(videofile)

Here are the programme output-