Animation of a simple pendulum using MATLAB

Objective - To understand the motion of simple pendulum using MATLAB.

Explanation -  A simple pendulum consists of a mass m, string of length l which is attached to the mass, where one end of string is fixed and the other end is freely suspended where the mass is attached.

When the ball(mass) in equilibrium is subjected to an external force, displacement takes place and ball oscillates to and fro along the vertical plane. This motion can be described in mathematical form and code can be written in MATLAB to get the actual representation of motion.

A simple pendulum is the best example of ODE which describes the transient behaviour of a system.

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. 

Equation - This  below second order ODE represents the equation of motion of a simple pendulum with damping.

where, b = damping co-effecient

           L  = length of the pendulum string in m

          m = mass of the ball in kg

           g = gravitational force in m/sec^2

Steps Involved -

1. To solve this above second order ODE equation, the second order ODE is simplified to first order ODE equation.
2. A program for function is written, where it is called by a function definition(ode_func).
3. By substituting the necessary inputs to the function definition, an inbuit solver ode45 is used to call the function operation.
4. Plot command is used where it is used to display the rate of change of angular displacement & angular velocity w.r.t time(figure 1). 
5. Plotting is done and finally movie command is written to show the animation.

% Program which describes the solving of second order equation 

clear all
close all
clc

% inputs
l = 1;
m = 1;
b = 0.05;
g = 9.81;

% initial condition
theta_01 = [0;3];

% time condition
t_span = linspace(0,20,100);

% solve
[t,results] = ode45(@(t,theta) ode_func(t,theta,b,g,l,m),t_span,theta_01);

% plotting
plot(t,results(:,1))
hold on
plot(t,results(:,2))
xlabel('time')
ylabel('plot d & v')


% program to animate the simple pendulum

l1 = 1;

ct=1;
for i = 1:length(t_span)

x = 0;
y = 0;

x1 = l1*sind(results(i,1));
y1 = -l1*cosd(results(i,1));

figure(2)

plot([x x1],[y y1],'color','r','linewidth',3)
hold on
axis([-0.1 0.1 -1.4, 0.5])
plot(x1,y1,'-o','markers',25,'markerfacecolor','g')
pause(0.3)
M(ct) = getframe(gcf);
hold off
ct = ct+1;
end

movie(M)
videofile = VideoWriter('ode_func.avi','Uncompressed AVI');
open(videofile)
writeVideo(videofile,M)
close(videofile)

% function defintion for the pendulum ode

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];

Figure1 - Plot showing the rate of change of angular displacement & angular velocity w.r.t time.

 Figure 2 showing the piece of animation

Inference - The time t is independent variable, where as the theta is dependent variable.

With varying the damping co-effecient values we can obtain a smooth curves of angular displacement & angular velocity with respect to time.

youtube video link -youtu.be/ouPFScAnOFA