Simulation Of a Simple Pendulum using MATLAB's ODE45

The objective of this challenge is to solve a second-order ODE which brings out the behaviour of a simple pendulum's movement. The second-order ODE is simplified to two first-order ODE to perform the motion.

The given equation is as follows:

DATA REQUIRED FOR SOLVING:

The data required for solving the ODE is

Length of the Pendulum, L = 1m

mass of the Pendulum, m = 1kg

Damping Coefficient, b = 0.05

Acceleration due to gravity, g = 9.81m/s.

BREAKING THE EQUATION:

θ=θ1θ=θ1

dθdt=dθ1dt=θ2dθdt=dθ1dt=θ2

dθ1dt=θ2dθ1dt=θ2

d2θ1dt2=dθ2dtd2θ1dt2=dθ2dtt = theta 2`

dθ2dt=−bmθ2−gLsinθ1dθ2dt=-bmθ2-gLsinθ1

Here, θ1=θ2θ1=θ2

In the code, theta gives us two rows and 100 columns. These are the values of Displacement and Velocity respectively.

CREATING THE MOVIE:

• In the video, you could see that in axis (0,0) is fixed from which the pendulum simulates.
• Then the values of x1 and y1 are stored in the same plot for each value of displacement
• Hence by capturing all the values for displacement, we get a movie.

MAIN CODE FOR SIMULATING SIMPLE PENDULUM:

clear all
close all
clc

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

%TIME LIMIT
t_span = linspace(1,20,100);

%INITIAL CONDITIONS
theta_1 = [0 3];

[t, theta] = ode45(@(t, theta) ode_pend(t, theta, b,g,L,m), t_span, theta_1);


figure(1)

plot(t, theta(:,1), 'color' , 'r');
hold on
plot(t,theta(:,2), 'color', 'k');
hold on  
grid on
xlabel('Time');
ylabel ('angular velocity and Displacement');
legend ('angular Displacement','Angular Velocity');

%STICHING IMAGES FOR MOVIE GENERATION
ct = 1;
for i = 1:length(theta(:,1))
    x0 = 0;
    y0 = 0;
    x1 = L*sin(theta(i,1));
    y1 = -L*cos(theta(i,1));
    
    figure(2)
    plot([x0 x1], [y0 y1] , 'linewidth', 2, 'color' ,'g')
    hold on
    plot(x1,y1)
    hold off
    grid on
    pause(0.01)
    axis ([-2 2 -2 0])
    M(ct) = getframe(gcf);
    ct = ct+1;
    
end


%CREATING A MOVIE
movie(M)
videofile = VideoWriter('forward_kinematics.avi' , 'uncompressed AVI');
open(videofile)
writeVideo(videofile,M)
close(videofile)

FUNCTION CODE:

function [dthetadt] = ode_pend(t, theta, b,g,L,m)

theta1 = theta(1)
theta2 = theta(2)

dtheta1dt = theta2;
dtheta2dt = -(b/m)*theta2 - (g/L)*sin(theta1);

dthetadt = [dtheta1dt;dtheta2dt];

end

RESULTS:

Graph comparing Angular velocity and Angular acceleration:

CONCLUSIONS:

From the graph of Angular velocity and displacement, we could see the performance of the pendulum with respect to time from 0-20s.

The angular velocity and displacement decrease with respect to time.

The images of each value of x1 and y1 are stitched and the movie is created which gives a harmonic motion.

VIDEO OUTPUT: