A program in Matlab Octave that will simulate the pendulum motion

AIM :

• SOLVING SECOND ORDER ODE.

OBJECTIVE:

• Write a program in Matlab\\Octave that will simulate the pendulum motion.
• write a program that solves the following ODE. This ODE represents the equation of motion of a simple pendulum with damping

THEORY:

• WHAT IS SIMPLE PENDULUM?

A pendulum is a weight suspended from a pivot so that it can swing freely. When a pendulum is displaced sideways from its resting, equilibrium position, it is subject to a restoring force due to gravity that will accelerate it back toward the equilibrium position.

• THEORY OF SIMPLE PENDULUM :

A simple pendulum consists of a mass m hanging at the end of a string of length L. The
period of a pendulum or any oscillatory motion is the time required for one complete
cycle, that is, the time to go back and forth once.

• GOVERNING EQUATION :

second-order ODE representing the equation of a simple pendulum motion.

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.
• To write a program in Matlab\\Octave that will simulate the pendulum motion, just like the one shown in the start of this challenge.
• We are using,
• L=1 metre,
• m=1 kg,
• b=0.05.
• g=9.81 m/s2.

MATLAB CODE:

clear all
close all 
clc

%input
b=0.05;
g=9.8;
l=1;
m=1;

%initial conditions
initial=[0,3];

%time points
time=linspace(0,150,500);

%solve ode
[t,results]=ode45(@(t,theta) ode(t,theta,b,g,l,m),time,initial);

%plot
C=1

for a=1:length(results(:,1))
        x1=-l*sind(results(a,1))
        y1=-l*cosd(results(a,1))
     
        plot([0,x1],[0,y1],\'linewidth\',1)
        hold on
        plot(x1,y1,\'O\',\'color\',\'r\')
        
        axis([-.2,.2,-1.1,0])
        pause(.006)
         
        
        %storing frames in m
        K(C)=getframe(gcf)
        C=C+1
  
end


%creatig videofile for m
movie(K)
videofile=VideoWriter(\'forwardkinamatics.avi\',\'uncompressed AVI\')
open(videofile)
writeVideo(videofile,K)
close(videofile)



close all
plot(t,results(:,1))
hold on
plot(t,results(:,2))
axis([0,50,-8,8])
title(\"displacement vs velocity\")
xlabel(\'time\')
ylabel(\'plot\')
legend(\'displacement\',\'velocity\')

MATLAB CODE OF FUNCTION \'ode.m\':

function [result]= ode(t,theta,b,g,l,m)

    theta1=theta(1);
    theta2=theta(2);
    dtheta1_dt=theta2;
    dtheta2_dt=-(b/m)*theta2-(g/l)*sind(theta1);
    result=[dtheta1_dt;dtheta2_dt];

end

RESULTS:

• For better understanding, I am providing time=150 secs.
• Animation 01:
• https://youtu.be/B9KsNxtBJh8

• Animation 02:
• https://youtu.be/w9nz76nCBF8

ERRORS AND REMEDIES :

• error due to wrong initial value of the variable

solved by assigning value equal to 1.

REMARKS:

• The water-marks on the animation video files are due to the use of external software to reduce its size.