Project

Solving second order ODEs IN SIMPLE PENDULUM

RRavindhar M3 Apr 2022

AIM:

To write a MATLAB program which solve the following ODE. This ODE represents the equation of motion of a simple pendulum with damping. that will simulate the pendulum motion.

THEORY OF ODE :(Ordinary differential equation)

An ordinary differential equation (or ODE) is an equation involving derivatives of an unknown quantity with respect to a single variable.

The purpose of this exercise is to enhance your understanding of linear second order homogeneous differential equations through a modeling application involving a Simple Pendulum which is simply a mass swinging back and forth on a string. Like the mass on a spring application, this model problem is representative of a wide variety of problems described by second order linear differential equations.

SOLUTION:

THE Pendulum oscillation it's like a Simple Harmonic Motion. That describe hown an object's velocity oscillates proportional to the amount of displacement equilibrium can be used to describe the equation of a pendulum.
In pendulum motion ,Three Forces Work Directly On The Pendulum

Mass ON The BOB
Gravity
Tension In The String

Mass and gravity both work vertically downeards.since the pendulum doesn't move up or down , the vertical component of the string tension cancels out the mass and gravity.

GIVEN:

L=1 metre,

m=1 kg,

b=0.05.

g=9.81 m/s2.

EQUATION USED:

`{d^2theta}/{dt^2}+b/m {d theta}/{dt}+g/lsintheta=0`

where,

`theta=theta1`

`{d theta}/{dt}={d theta_1}/dt=theta_2` (1St ODE)

`{d^2theta }/dt^2={d^2theta_1}/{dt}=d/dt({d theta_1}/{dt})=d/dt(theta_2)`

BY Rewriting the main equation we get,

`{d theta_2}/dt+b/m(theta_2)+g/lsin theta_1=0=> {d theta_2}/dt=-b/m (theta_2)-g/lsin theta_1` (2nd ODE)

BY PUT THE ABOVE TWO ODEs IN THE MATRIX FROM WE GET:

`d/dt[theta_1*theta_2]=[theta_2*-(b/m)theta_2-(g/l)sin theta_1]`

MAIN PROGRAM:

clear all 
close all
clc

b=0.05; % THERE IS A Damping Coefficient.
g=9.81; % THERE IS GRAVITY (m/s^2) 
l=1;    % THERE IS LENGTH OF THE Pendulum (m)
m=0.1;    % THERE IS MASS OF THE BALL (kg)

%INITIAL CONDITION
theta_0=[0;3]; % THERE IS A INITIAN CONDITION OF PENDULUM AND THE FIRST VALUE OF DISPLACEMENT & Second One Is ANGULARVELOCITY

%TIME POINTS
t_span=linspace(0,20,500); %THERE IS ARRAY IT'S CONTAIN 500 VALUES OF 0 TO 20

% SOLVE ODE
[t,results] = ode45(@(t,theta)ode_function(t,theta,b,g,l,m),t_span,theta_0); % THERE IS ANONYMOUS FOUNCTION THAT IS ODE45 IS USED SOLVE OUR ODE

figure(1) % THERE IS A PLOT COMMANT TO ANGULAR DISPLACEMENT AND VELOCITY 

plot(t,results(:,1)) % COMMAND TO  PLOT THE ANGULAR DISPLACEMENT
hold on              % THERE IS A PREIVES PLOT HOLDING TO CURRENT PLOT COMMAND

plot(t,results(:,2)) % COMMAND TO PLOT THE ANGULAR VELOCITY 
xlabel('time')       % THERE IS CALLED X AXIS OF TIME 
ylabel('velocity')   % THERE IS CALLED Y AXIS OF VELOCITY
legend("angular displacement","angular velicity")
title("Simple Harmonic motion")

ct=1; % THERE IS A COUNTER  VARIABLR THE VALUE IS 1
for i = 1:length(t_span)
x_0=0; % THERE IS A VALUE OF X_0
y_0=0; % THERE IS A VALUE OF Y_0

x_1 = l*sin(results(i,1));  % THER IS A CALCULATE THE VALUE OF Angular displacement
y_1 = -l*cosd(results(i,1));% THERE IS A CALULATE THE VALUE OF Angular velocity

figure(2)

plot([x_0 x_1],[y_0 y_1],'linewidth',3,'Color','b') % THERE IS A PLOT COMMAND ON STRING

hold on   %THERE IS A PREIVES PLOT HOLDING TO CURRENT PLOT COMMAND

plot([0.8 -0.8],[0 0],'linewidth',2,'color','g') % THERE IS COMMAND ON THE STRING FIXED POINT
plot(x_0,y_0,'.','Markersize',30,'MarkerFaceColor','b')% THERE IS COMMAND ON STRING FIXED POINT SIZE MARK
plot(x_1,y_1,'.','markersize',75,'MarkerFaceColor','g') % THIS COMMAND HELP US CREATE A SPHERE IN MATLAB 
axis([-1 1 -1.3 1])  %there is a Range of x and y axis
grid on

hold off

M(ct)=getframe(gcf);  % M is array its carry the number of figure
ct=ct+1;   % ct is a counter variable its over write the counter variable
end
         

movie(M)   % there is a plot movie command its carry the (M-string)
videofile = VideoWriter('pendulam_4ode.avi','Uncompressed AVI'); % there is video writeing command on (file name)
open(videofile) % OPEN A VIDEO COMMAND 
writeVideo(videofile,M); % THERE IS (M)movie string to WRITE VIDEO COMMAND 
close(videofile) % CLOSE THE VIDEO FILE COMMAND

THIS FUNCTION HELP US BREAK UP THE 2order ODE INTO FIRST ORDER ODE's

alredy we know

`{d theta_1}/dt=theta_2`

`{d theta_2}/dt=[-b/mtheta_2-g/lsin theta_1]`

ODE FUNCTION:

function[dtheta_dt] = ode_function(t,theta,b,g,l,m)

theta1 = theta(1) % THERE IS VALUE OF ANGULAR DISPLACEMENT
theta2 = theta(2) % THERE IS VALUE OF ANGULAR VELOCITY

dtheta1_dt = theta2; % THERE IS A 1st ORDER ODE
dtheta2_dt = -(b/m)*theta2-(g/l)*sin(theta1); % THERE IS A 2st ORDER ODE
dtheta_dt = [dtheta1_dt;dtheta2_dt];
end

[dtheta_dt] - is soluction vector

And ,there is contian the two values