Solving second order ODEs (MATLAB)

AIM:

objective is to write a program that solves the given ODE. This ODE represents the equation of motion of a simple pendulum with damping

governing equations:

CODE:

the following functions are used in this code

ode45- this is a preexisting MATLAB solver that is used to solve second order differential equations using the range kutta method. we need to create a function that defines the given 2nd order ode, this function is then passed on to the solver along with the time and the initial conditions. it returns the results as an array and the time array

clear all
close all
clc

b=0.05;
g=9.81;
l=1;
m=1;

% initial conditions 
theta_0=[0;3];

t_span=linspace(0,20,500);
%solve ode 

[t,result]=ode45(@(t,theta) ode_func(t,theta,b,g,l,m),t_span,theta_0);
figure(1)
plot(t,result(:,1))
hold on 
plot(t,result(:,2))
xlabel('plot')
ylabel('time')

f=1;
for i=1:length(t)
    %initial position
    x=0;
    y=0;
    %final position
    x1 = l*sin(result(i,1));
    y1 = -l*cos(result(i,1));
    
    %plotting
    figure(2)
    plot([-1,1],[0,0],'linewidth',5,'color','r')    %for the base line
    hold on
    plot([x,x1],[y,y1],'linewidth',5,'color','b')   %for the pendulum line
  
    plot(x1,y1,'o','markers',20,'markerfacecolor','b')   %for the pendulum ball
    grid on
    axis([-2 2 -2 1])
    pause (0.03)
    M(f)= getframe()
     
    f=f+1;
    hold off
   
end

movie(M) % to play the array of frames in 'm' as animation 
 %to create a video file of the animation
 videofile= VideoWriter('Pendulum.avi','Uncompressed AVI' );
 open(videofile)
 writeVideo(videofile,M)
 close(videofile)

the function ode_func is defined as shown below

%the ode_func function 
function[dtheta_dt]= ode_func(t,theta,b,g,l,m)
% b=0.05;
% g=9.81;
% l=1;
% m=0.1;

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

Results:

the video was generated as shown in the link given, the graph of the velocity vs displacement is attached below

video link:https: //drive.google.com/file/d/1CBmWd8nvKwoZn-FzyzrJJj2h2K0u8gNm/view?usp=sharing