Week 3 - Solving second order ODEs

https://youtu.be/-DPWa7PK0-8

• code

%solving second order ODEs of a Pendulum
clear all
close all
clc
%inputs
%damping coefficient
b = 0.05
%gravity
g = 9.81
%length of the spring
l = 1
%mass of the ball
m= 1

%initial condition
%angular displacement is 0 and angular velocity is 3

theta_0 =[0;3]
%time span of pendulum osillates between 0-20sec

t_span=linspace(0,20,500)
% solving ODE
[t,results] = ode45(@(t,theta)ode_func(t,theta,b,g,l,m),t_span,theta_0);

%plotting
figure(1)
hold on
plot(t,results(:,1))
plot(t,results(:,2))
title('Damped harmonic motion of a pendulum')
legend('Angular displacement ','Angular velocity')
ylabel('plot')
xlabel('Time (seconds)')

%for loop
ct =1
for i=1:length(results(:,1))
x_start =0
y_start =0
x_end=1*sin(results(i,1))
y_end = -1*cos(results(i,1))
%plotting the animation
figure(2)
plot([-1 1],[0 0],'linewidth',5)
hold on
plot([x_start x_end],[y_start y_end],'linewidth',3)
hold on

plot(x_end,y_end,'o','markersize',20,'markerFacecolor','y')
title('Animation of transient behaviour of a pendulum')
axis([-1.5 1.5 -1.5 1])
pause(0.03)
hold off
M(ct)=getframe(gcf)
ct =ct+1
end
%Animation of the pendulum
movie(M)
videofile=VideoWriter('Pendulum movement Animation.avi','uncompressed avi')
open(videofile)
writeVideo(videofile,M)
close(videofile)

• function code

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

• output of the code

• damping of pendulum

• Animation link of pendulum :https://youtu.be/-DPWa7PK0-8

• explanation

1. start your code with close all ,clear all and clc
2. define inputs such as gravity-g ,mass of ball -m lenght of pendulum string-l etc.
3. define initial condition of the pendulum i.e angular displacement,time span and angular velocity.
4. use ode45 which is used to solve equtioin by numerical method.
5. also creat function file of eqution having same name file name as fution name
6. etablish for loop
7. for loop is used where all values of time span are assigned to variable 'i' one value at a time.
8. calculate the code output
9. plot the graph
10. define axis
11. use pause command to pause a code with required duration
12. use ct
13. use M commad to caputre current frame every time
14. use ct+1
15. creat animation file using VideoWriter