Week 3 - Solving second order ODEs

AIM: To solve the second order odes function using Matlab for simple pendulum and animation video.

Objective : To write a code for following equation of motion of simple pendulum.

Pendulum geometry: Length(l) = 1 meter

Mass(M) = 1 kg.

Damping coiefficient(b) = 0.05

Acceleration due to gravity = 9.81m/s2

Time period = 0-20s

displacement = 0, velocity = 3 rad/sec at t= 0

sub code:

% odes function
function dtheta_dt = ode_funs(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 

Main code:

clear all
close all
clc


%code for odes 
%input

%Damping co-efficient
b = 0.05;
% Acceleration due to gravity(m/m^2)
g = 9.81;
%Mass (kg)
m = 1;
%length(m)
l = 1;

%initial condition
theta_0 = [0;3]
% Time points
t_span = linspace(0,20,200)
%Solving ODES
[t,results] = ode45(@(t,theta) ode_funs(t,theta,b,g,l,m), t_span,theta_0);
%plotting
figure(1)
plot(t,results(:,1))
hold on 
plot(t,results(:,2))
xlabel('Time')
ylabel('angural velocity(rad/s)/displacement(rad)')
legend('displacement','angular velocity')

%Animation 
ct = 1;
for i = 1:length(results(:,1))
    x0 = 0
    y0 = 0
    
x1 = l*sin(results(i,1)+pi);
y1 = l*cos(results(i,1)+pi);

figure(2)
plot([x0,x1],[y0 y1],'-o','MarkerSize',10,'MarkerFaceColor','r','MarkerEdgeColor','r','linewidth',5)
axis([-2 2 -2 2])
hold on
plot([-1 1],[0 0],'linewidth',10,'color','b')
M(ct) = getframe(gcf)
hold off
ct = ct+1;
end

%Movie   & videofile
movie(M)
v_f = VideoWriter('pendulum_1.avi','Uncompressed AVI')
open(v_f)
writeVideo(v_f,M)
close(v_f)

Explanation :

sub code is function program of the second order ODES is simplefied to the first order odes.

Then in main program pendulum fixed parameters are like length,mass, damping coefficient.

t_span command in which 200 value are taken between 0 to 20 times period.

odes45 command is used that solve ordinary or partial differential equations using a numerical method.

plot command is used to plot graph between velocity , displacement vs time.

for animation command used are movie,videoFile,writeVideo,close video .

PLOT:

VIDEO LINK: https://youtu.be/iQEbW_Xad9I