Week 3 - Solving second order ODEs

Aim : To write the code on the simple pendulum by solving secound order deffrention equation in matlab

Theory : A pendulum is a body suspended from a fixed point so that it can swing back and forth under the influence of gravity.

• here a mass is attached to the string of length L and this string is fixed at one end and at another end mass M is attached
• G is the gravitetional force acting on the mass M
• displacement D and velocity

there fore from newton's secound law of motion we have

Input : g = 9.81 m/s^2

L = 1m

b = 0.05

M = 1kg

code for pendulum is fallow below:

% writting program to solve the 2nd order ODE's
 clear all
 close all
 clc

 % defining the inputs given in the quesion 
l = 1;              % lenght(L) in M
m = 1 ;             % mass in Kg
b = 0.05 ;          % damping co-efficent
g = 9.81;           % gravity at mass in m/s^2


% definingn the time as given in the question

time_t = linspace(0,20,500);

% defining the angle theta 

theta = [0;3];        % hera in 1st colum dispacement is taken and 2nd colum veocity is taken

% now calling the ode_fun 

[t,result] = ode45(@(t, theta) ode_fun(t,theta,l,m,b,g),time_t,theta);

% ploting the figure 
 figure (1)
 plot(t,result(:,1),'color','r')
 hold on
 plot(t,result(:,2),'color','b')
xlabel('time')
ylabel('velocity[b] & displacemnt[r]')

% creating the animation vedios 
ct = 1;
for i = 1:length(result(:,1))
    x0 = 0;
    y0 = 0;
    x1 = l*sin(result(i,1)+pi);
    y1 = l*cos(result(i,1)+pi);
    figure(2)
    plot([x0 x1],[y0 y1],'-o','markersize',10,'markerfacecolor','r','markerEdgecolor','r','linewidth',4)
    axis([-2 2 -2 2])
    hold on 
    plot([-1 1],[0 0],'color','g','linewidth',4)
    M(ct) = getframe(gcf);
    ct = ct+1;
    hold off
end
movie(M)  % here we are creating the movie of obtained robertic arm
videofile = VideoWriter('pendulum_motion.avi','Uncompressed AVI'); %here we are creating the vedio file for our robertic arm animation  
open(videofile) % here we are opening our vediom file 
writeVideo(videofile,M)
close(videofile)   % after opening the vedio file we have to close it by using fallowing lines 

explanation:

1. 1st we have to start the program with clear all,close all,clc
2. then we have to define the inputs
3. after that we have to declear the ode 45 and ode_fun function
4. then we have to use the vidowritting command to creat th animation vedio

code for farmula fallows below

%writting code to solve the farmula of ode
function[dtheta_dt]=ode_fun(t,theta,l,m,b,g)
theta1=theta(1);
theta2=theta(2);

dtheta1_dt=theta1;
dtheta2_dt=((-b/m)*theta2-(g/l)*sin(theta1));
dtheta_dt=[dtheta1_dt;dtheta2_dt]
end

Animation: https://youtu.be/1OZy0UV-CdE