TRANSIENT BEHAVIOR OF SIMPLE PENDULUM SIMULATION | MATLAB

AIM: write a matlab program to simulate the transient behavior of a simple pendulum and create animation of the motion

Given conditions:

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.

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.

THEORY:

A simple pendulum consists of a concentrated mass known as the ‘bob’, which is free to swing from side to side at the end of a light cord or rod. When bob is given a small angular displacement and released, the pendulum bob appears to describe simple harmonic motion (SHM). It doesn’t travel along a perfectly straight line, but for small angular displacements it is approximately straight, with a central equilibrium position.

If the mass is given a small displacement (r) from its equilibrium position and then released, it will oscillate about the equilibrium position with amplitude ‘r’ as shown in figure 1 (a). At some instant when the bob is at distance ‘x’ from its equilibrium position, and accelerating towards it, the forces acting on it will be as shown in figure 1 (b). It is assumed that the bob oscillates with free or natural vibrations, i.e. that the air resistance is negligible and that there are no energy losses at the point of suspension.

code:

function code:

% Function to calculate angular displacement and angular velocity of the pendulum over its time span

function [dtheta_dt] = ode_func(t,theta,b,g,l,m)
% initialisation for calculation
theta1 = theta(1);    % initial angular displacement
theta2 = theta(2);    % initial angular velocity

% calculating the differential
dtheta1_dt = theta2;
dtheta2_dt = -(b/m)*theta2 - (g/l)*sin(theta1);

% Assigning the result to 'd_theta'
dtheta_dt = [dtheta1_dt;dtheta2_dt];

end

MAIN PROGRAM:

%  Program to simulate the transient behaviour of a simple pendulum and to 
% create an animation of it's motion using second order Ordinary Differential
% Equation in MATLAB

clear all
close all
clc

b = 0.05; % damping constant
g = 9.81; % aceeleration due to gravity in m^2/s
l = 1; % length of the pendulum in meter
m = 0.1;  % mass of the pendulu in kg

%initial condition
theta_0 = [0;3]; % angular displacement and angular velocity at time t=0

 
%time points
t_span = linspace(0,20,500); % creating a time span of 20 secs


%solving ode

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

%output
figure(1)
plot(t,results(:,1))
hold on
plot(t,results(:,2))
legend('Angular displacement','angular velocity');
title('Transient behaviour of Simple pendulum');
ylabel('plot')
xlabel('time')
hold off

%creating the animation
figure(2)
ct = 1;
for i = 1:length(results(:,1))
    x1 = 0;
    y1 = 0;
    
    x2 =  l*sin(results(i,1));
    y2 = -(l*cos(results(i,1)));
    
   % hang pendulum
    plot([-1 1],[0 0],'linewidth',5,'color','k')
    hold on
    
    %hanging the line
    plot([0 0],[-1.3 0],'.')
    hold on
    
    plot([x1 x2],[y1 y2],'linewidth',4,'color','b')
    hold on
    
    plot(x2,y2,'.','markersize',50,'color','r')
    hold on
    
    plot(x2,y2,'.')
    hold off
    
    title('simple pendulum');
    text(-0.5,-1,['Time = ', num2str(t_span(i)),'sec'],'fontsize',15);
    text(-1,-1.2,['Angular velocity = ', num2str(results(i,2)),'rad/sec'],'fontsize',15);
    
    axis([-1.5 1.5 -1.5 0.5])
    grid on
    
     F(ct) = getframe(gcf);
    ct = ct+1;
   
end


%creating the movie 
movie(F)
videofile = VideoWriter('pendulum_animation.avi','uncompressed avi');
open(videofile)
writeVideo(videofile,F)
close(videofile)

code Explanation:

• First step is to write code to define the formula
• In the main program
• Given the initial condition and time span of the pendulum
• using the ode45 command to solve the ode formula.
• Then plotting the graph between Angular displacement and Angular velocity wrt time
• using the animation code to simulate the transient behavior of simple pendulum.
• creating the video file
• close and save the video file

OUTPUTS:

GRAPH:

SIMULATION VIDEO: