Week 3 - Solving second order ODEs

AIM: to create a program in matlab to solve the 2^nd order ode of motion of a simple pendulam with damping.

THEORY :

In matheamatical an ordinary diffrential equation (ode) is adiffrential equatons contaning one or more function of one indipendent variable and the derivatives of those function.

Equation for second order

MATLAB PROGRAM:

clear all

close all

clc

b=0.05; % damping coefficient.

g=9.81; % acceleration due to gravity in

l=1; % length of the pendulam in M.

m=1; % mass of pendulam in KG.

%initial condition

theta_0=[0;3];

%time points

t_span=linspace(0,20,100);

%solve ode

[t results]=ode45(@(t,theta) ode_func(t,theta,b,g,l,m),t_span,theta_0);

%plotting

figure(1)

A=plot(t,results(:,1))

hold on

B=plot(t,results(:,2))

disp=results(:,1);

ct=1;

for i=1:length(t_span);

x0=0;

y0=0;

x1=l*sin(disp(i));

y1=-l*cos(disp(i));

figure(2)

plot([-2 2],[0 0],'linewidth',5)

hold on

plot([x0 x1],[y0 y1],'-o','markersize',3,'linewidth',5,'color','r');

plot(x1,y1,'-o','markersize',20,'markerfacecolor','k');

axis([-1.5 1.5 -1.5 1.5])

pause(0.03)

hold off

M(ct)=getframe(gcf)

ct=ct+1;

end

movie(M)

v=VideoWriter('pendulam.avi');

open(v)

writeVideo(v,M)

close(v)

Results;

1. As seen in the graph , the pereodic curve of angular displacment and angular velocity reduce W.R Time . Thisb is manily due to the damping factor.This is the motion of simple pendulam is called as damped simple arnic motion.
2. From the animation of motion of the simple pendulam , it can be conculded ,as the damping coefficient is increase ,the motion of the pendulam . Also increase in damping coefficient reduce the time taken by pendulam to reach equliliburium position .