Week 3 - Solving second order ODEs

MOTION OF SIMPLE PENDULAM (ODE45)

AIM :

• To write a program in Matlab to simulate transient behavior of a simple pendulam and cread its animation.

OBJECTIVE :

• To write a program that solves the following ODE.
• This ODE represents the equation of motion of a simple pendulum with damping.

GIVEN :

• • L=1 metre,

  • m=1 kg,

  • b=0.05.

  • g=9.81 m/s2.

  • Simulate the motion between 0-20 sec

  • Angular displacement=0

  • Angular velocity=3 rad/sec

  • Time t=0.

Wher,

• g = gravity in m/s2,

• L = length of the pendulum in m,

• m = mass of the ball in kg,

• b=damping coefficient.

PROGRAM CODE :

% MOTION OF SIMPLE PENDULAM (ODE45)

clear all
close all
clc

%Inputs
l =1; %Length

%Time intervel
t = linspace(0,20,150);

%Initial condition
x0 = [0 3];

%foe diffreantial eqaution using ode45 (matlab input funtion)
[t x] = ode45('oed_func',t,x0);

%plotting for tracing of Angular displacement and Angular velocity
subplot(2,1,1)
plot(t,x(:,1),'b','linewidth',3);
xlabel('time t in sec')
ylabel('Angular displacement');
subplot(2,1,2)
plot(t,x(:,2),'r','linewidth',3);
xlabel('time t in sec')
ylabel('Angular velocity');
grid on;

%Animation
x0 = 0;
y0 = 0;
ct = 1;
for i =l:length(t)
x1 = l*sin(x(i,1));
y1 = -l*cos(x(i,1));

figure(2)
plot([-1 1],[0 0],'linewidth',5,'color','g');
axis([-2 2 -2 2]);
line([x0 x1],[y0 y1],'linewidth',2,'color','b');
hold on;
plot(x0,y0,'O','markersize',3,'markerfacecolor','y');
plot(x1,y1,'O','markersize',20,'markerfacecolor','r');
title('Damping of simple Pendulam');
legend('Support rod','Length of Pendulam','Fixed point','Bod of Pendulam');
grid on;
hold off;
M(ct)=getframe(gcf);
ct=ct+1;
end

% Inserting Frames in the animation to make movie
movie(M);
videofile = VideoWriter('simple_pendulam','uncompressed AVI');
open(videofile);
writeVideo(videofile,M);
close(videofile);

ODE FUNCTION :

function xdot = oed_func(t,x)

b = 0.05; % damping coefficient
m = 1; %mass of the pendulam
g = 9.81; %Accelaration due gravity
l = 1; % length
%Assigning nullmatrix to xdot
xdot = zeros(2,1);
%Rirst element is angular velocity,x(2) position
xdot(1) = x(2);
xdot(2)= (-b/m)*x(2)- (g/l)*sin(x(1));

end

PROCEDURE :

• Program should be started with close all clear all &clc.
• Inorder to solve secound order ode we fiest have to convert into 1st order ode and resuld are shored in 2 different columns.
• function is used ode45 is already had been difined in matlab.
• plotting for tracing of angular displacement and angular velocity.
• Animation to the for loop start to plot even position of pendulam in given time interval when ever the for loop.
• We plot [XY] and assing a marker as per requirement to represent the bob of pendulam .
• Title,gride on is used.
• the plot is stored in counter and then the counter is incremented and the loop closed.
• I had get used get frame command and stored in a veriyable named as M(ct).
• Inserting frame in the animation.
• Close the video file.
• And run the code.

OUTPUT :

ANIMATION VIDEO :

VIDEO LINK :

https://youtu.be/5WGkIYi8uK8

ERRORS :

• No Error

RESULTS :

• To write a program in Matlab to simulate transient behavior of a simple pendulam and cread its animation is successfully complete .