Week 3 - Solving second order ODEs

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

Writing a program in Matlab that will simulate the pendulum motion

useing,

L=1 metre,

m=1 kg,

b=0.05.

g=9.81 m/s2.

Simulate the motion between 0-20 sec, for angular displacement=0,angular velocity=3 rad/sec at time t=0.

in this above equation

b=dampping co-efficent

m=mass of the ball (kg)

g=gravity (m/s^2)

L=length of the pendulam (m)

NEWTON'S SECOND LAW:-

"The amount of the acceleration of the body is propostional to the net force , and inversely propostional to the mass of the body". is called newtons second law.

F=m*a

where

F=net force

m= mass of the body

a= acceleration.

simple pendulam:-

A mass suspended from a light string that can oscillate when displaced from its rest position.

(d^2θ/dt^2)+dθ/dt+c=0 is a second order ode

(d^2θ/dt^2)+(b/m)dθ/dt+(g/L)sinθ=0 ----------------------------------------------(1)

where θ=θ1;

dθ/dt=dθ1/dt -------------------------------------------------------------------------------(2)

now we assume dθ1/dt=θ2 ------------------------------------------------(3)

d^2θ/dt^2=d^2θ1/dt^2

d^2θ1/dt^2=d/dt(dθ1/dt)==>dθ2/dt

d^2θ2/dt^2=dθ2/dt ----------------------------------------------------------------------(4)

now we can substitute the (2) and (4)eguations in eg.(1)

d^2θ2/dt+(b/m)dθ1/dt+(g/L)sinθ=0 ------------------------------------------------------(5)

from equqtion (3) we have

dθ1/dt=θ2 ; -------------------------------------------------------(3)

dθ2/dt=-(b/m)θ2-(g/L)sinθ1 -----------------------------------------------------------(6)

from the given data we have

angular dispalcement =0

angular velocity =3 m/s

time= 0

theta_0=[0:3];

t_span=linspace(0,20,300)

now we get 't' and output by using

function [dtheta_dt]=ode_func_m(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 program is

%inputs

b=0.05;
g=9.81;
L=1;
m=0.1;

%initial condition

theta_0=[0;3];

%time plots

t_span=linspace(0,20,300);

%solving ode

[t,results]=ode45(@(t,theta) ode_func_m(t,theta,b,g,L,m),t_span,theta_0);
figure(1)

plot(t,results(:,1));
hold on
plot(t,results(:,2))
hold off
xlabel('time')
ylabel('velocity')
figure(2)
plot(results(:,1),results(:,2))
theta2=results(:,1)
ct=1
for i=1:length(theta2)
THETA=theta2(i)
x0=0
y0=0

x1=L*sin(THETA)
y1=-L*cos(THETA)

figure(3)

plot([-0.5 0.5],[0 0],'linewidth',3,'color','b')
hold on
%plotting
plot([x0 x1],[y0 y1],'linewidth',3,'color','b')
plot(x1,y1,'o','markersize',20,'color','b','MarkerFaceColor','r')

hold off

axis([-2 2 -1.5 1.5])
M(ct)=getframe(gcf);
pause(0.02)
ct=ct+1
end
movie(M)
videofile=VideoWriter('second_order_ode.avi','uncompressed AVI')
open(videofile)
writeVideo(videofile,M)
close(videofile)

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