second oder ordinary diiferential equation

INTRODUCTION:

Among ordinary differential equations, linear differential equations play a prominent role for several reasons. Most elementry and special functions that are encountered in physics and applied mathematics are solutions of linear differential equations ( harmonic motion). When physical phenomena are modeled with non-linear equations, they are generally approximated by linear differential equations for an easier solution.

ODE for a simple pendulam:

m = mass at the swinging end of the pendulum (kg)

g = acceleration due to gravity (m/s²)

L = length from the swivel point to the center of mass (meters)

b=damping coefficient

θ= angle between the string position to the string position at rest

t = time (seconds)

CODE WITH STEPWISE EXPLANATION

1. Creating function:

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

theta1=theta(1)
theta2=theta(2)
dtheta1_dt=theta_2
dtheta2_dt=-(b/m)*theta2-(g/l)*sin(theta1)
dtheta_dt=[dtheta1_dt;dtheta2_dt]

end

2.second order ODE for pendulam motion.

%Second order ODE for pendulam motion 

%step1: close all previous work 
clear all
close all
clc

% step2 :inputs

%Damping Coefficient
b = 0.05; 

%Gravity m/s^2  
g = 9.81;

%Mass of the Pendulum in kg   
m = 1; 

%Length of the string in meters     
l = 1;      

%step3 :initial conditions

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

%step4 :time points
t_span = linspace(0,20,1000); %pendulum oscillates between 0-20 seconds

%step4: solving second order ODE by calling the function

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

%step5 :plotting the graph
figure(1)
%subplot to represent two curves in a single figure window

subplot(2,1,1)
p1 = plot (t,result(:,1),\'linewidth\',2,\'color\',\'b\')
xlabel(\'Time\')
ylabel(\'Theta\')
title(\'Angular Displacement v/s Time\')
legend([p1(1)],\'angular displacement\')

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


%step6 :pendulum simulation using second order ode

theta = result(:,1)  %it means theta is equal to values in cloumn 1 of result matrix
%counter to counts events in a loop     
       ct=1;

%step7 : for loop for oscillating the pendulum

for i=1:length(result)
    THETA = result(i,1);
    
%step8 :initial boundary conditon
    x0=0;
    y0=0;

%step9: Defining x and y coordinates at different coordinates
    x1=  l*sin(THETA);
    y1= -l*cos(THETA);
    
    
%step10: to represent motion of a simple pendulam
    figure(2)
    hold on
    plot([x0 x1],[y0 y1],\'linewidth\',4);
    plot(x1,y1,\'-o\',\'markersize\',40,\'markeredgecolor\',\'g\',\'markerfacecolor\',[0.8 0.8 
    0.8]);
    plot(x0,y0,\'-*\',\'markersize\',5,\'markeredgecolor\',\'k\',\'markerfacecolor\',[0.8 0.8 
    0.8]);
    axis([-1.5 2 -1.5 0]);
    pause(0.003);
    
%defining frame
   M(ct)=getframe(gcf)
%incrementing counter
    ct=ct+1;
% clearing the current graphic image 
     clf()         


%step12: creating the animation 

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

ERRORS FACED:

while coding in loop i put : intead of loop intiger i so i get error as shown in snapshot:

so to overcome it I put i instead of : and code works effectively.

OUTPUT CURVES:

ANIMATION LINK:

https://youtu.be/FCZt9407C9s