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

b = damping coefficient.

m = mass of the pendulam in kg.

This diffrential equation is 2^nd order equation

Derivation for code simplify :

Angular displacment=theta 1

And angular velocity=

Substituing these in the 2^nd order ode for a simple pendulam .

^{This is a 1st order differntial equation in the form of a matrix,}

=

^{writing these equation in the form of a matrix ,}

^Where,

Angular displacment of the pendulam,

= Angular velocity of the pendulam,

By the solving equation in matlab , we can get the value angular velocity and angular displacment W.R To Time.

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.

ODE FUNCTION

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

%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)

plot(t,results(:,1))

hold on

plot(t,results(:,2))

title('bf second order ode')

xlabel('bf Time')

ylabel('bf Velocity')

legend('Displacement','Angular velocity')

grid on

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.

ERRORS:-

YOU TUBE LINK

https://youtu.be/h7F4vya94yg

so i submitted all the things which is required.and i hope this should be satisfy you and i will get the full marks over here.

THANK YOU

SKILL LYNC