Solving second order ODE and Animation of Simple Pendulum
In Engineering, ODE is used to describe the transient behavior of a system. A simple example is a pendulum The way the pendulum moves depends on the Newtons second law. When this law is written down, we get a second order Ordinary Differential Equation that describes the position of the "ball" w.r.t time.
The program written here solves the following ODE.
This ODE represents the equation of motion of a simple pendulum with damping. In the above equation
- g = gravity in m/s2,
- L = length of the pendulum in m,
- m = mass of the ball in kg,
- b=damping coefficient.
Constants used L=1 metre, m=1 kg, b=0.5. g=9.81 m/s2. Simulate the motion between 0-20 sec, for angular displacement=0,angular velocity=3 rad/sec2 at time t=0.1.
clear all
close all
clc
%constants
b=0.5;
g=9.81;
m=1;
l=1;
%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);
x0=0;
y0=0;
THETA_1=results(:,1);
ct=1
for i=1:length(t_span)
r=THETA_1(i);
x1=1*sin(r);
y1=-1*cos(r);
plot([x0 x1],[y0 y1],'linewidth',1,'color','bla')
viscircles([x1 y1],0.05)
rectangle('position',[-0.4 0 0.8 0.2],'facecolor','c')
title('simple pendulum')
line([0 0],[-3 0],'color','blue','linestyle','--')
axis([-1 1 -1.4 0.2])
M(ct)=getframe(gcf)
ct=ct+1
end
movie(M)
videofile=VideoWriter('pendulum_animation.avi')
open(videofile)
writeVideo(videofile,M)
close(videofile)
%displacement and velocity against time
figure(2)
plot(t,results(:,1),'color','r')
hold on
plot(t,results(:,2))
grid on
xlabel('time (in sec)')
legend('displacement','velocity')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
The plot of Displacement and Velocity is below .
The link for the animation of the pendulum is below.