Project 2 - MATLAB - second order ODE - Simple Pendulum.

 AIM:  Make a Mat Lab code to solve and generate plot for Simple Pendulum.

 OBJECTIVE:

                     Make a Mat Lab code to solve and generate plot for Simple Pendulum. The code should create to simulate the motion of a pendulum by solving the equation that represent

The damping pendulum.

 SOLUTION:

                      A Mat Lab code was written for Simple Pendulum. To find the velocity and displacement at the different state points. Further, the code generates a plot for it. With respect to the input variables given and also we generate the output video.

 Explanation:

                      The aim of the program is to solve and generate plot for Simple Pendulum.

ODE is used for transits the equation of behaviour of simple system, the simple example is the pendulum. The way the pendulum is moves it depend on the newton’s low. When we wright the low the second order ODE (ordinary differential equation) it define the position of ball w.r.t. time.  The ODE represent the equation of motion of a simple pendulum damping,

Here in above equation,

                               g = 9.81 - gravity in m/s^2

                               l = 1 - length of the pendulum in meter

                               m =1 - mass of ball in kg

                               b =0.05 - damping coefficient in kg

Now we simulate the motion between 0 – 20 sec, for angular displacement is = 0.

Angular velocity – 2 rad/sec^2 at time = 0.

                 Fig:  SIMPLE PENDULUM

By solving this eq., we get value of theta 1 & 2 for simple pendulum. Using the ode45 function in the Mat-Lab to solve this ODE’s.

Procedure –

 %ODE represent the equation of motion of simple pendulum with damping

%the given teams

%damping coefficient in kg

b = 0.05;

%gravity in m/s^2

g = 9.81;

%Length of pendulum in m

l = 1;

%Mass of ball in kg

M = 1;

%Input conditions

theta_0 = [0;2.5];

% time

t_span = linspace(0,20,250);

 above parameters are the input parameters for pendulum condition

 For calculating ODE equation using function ode45 -

 The matlab allows to solution of first order ODE’s using the inbuilt function called ‘ode45’.

So the given second order ODE will reduce to two first order ODE in order to be solver using MATLAB. The given ODEE represent the variation of the position of pendulum w.r.t. time.

 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];

end

 To find out ODE & plot  –

 %to solve ODE

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

plot(t,results(:,1))

hold on

grid on

plot(t,results(:,2));

ylabel('time in second')

xlabel('plot')

 To make animation   –

 %to make the animation we have to use for looping

%counter variable for tracking the counter variable

 ct = 1;

 for i = 1 : length(results(:,1))

    x0 = 0;

    y0 = 0;

    x1 = 1*sin(results(i,1));

    y1 = -1*cos(results(i,1));

    figure(2)

    %to make dead fix support

    plot([-1 1], [0 0], 'linewidth',15, 'color','b')

    axis([-2 2 -2 2])

    hold on

    %for plotting string possition

    line([x0 x1], [y0 y1], 'linewidth', 5,'color','r')

    %plot ball position

    plot(x1, y1, 'o','markersize', 25, 'markerfacecolor', 'y')

    hold off

    m(ct) = getframe(gcf)

    %store current frame

    ct = ct+1;

end

 To make movie output –

 To save the animation of pendulum moment while time period will be saved by following program

 %to create video output

 movie(m)

videofile = VideoWriter('pendulum_simulation.avi', 'Uncompressed AVI');

 %to open video

open(videofile);

 %write frame to video file

writeVideo(videofile, m)

 %close video file

close(videofile)

 THE MAIN PROGRAM - 

%THE SIMPLE PROGRAM TO DEFINE THE SECOND ORFER ODE FOR PENDuLUM. 
%ODE - ORDNARY DIFFRENTIAL EQUATION

close all
clear all
clc

%ODE represent the equation of motion of simple pendulum with damping
%the given tearms 
%damping coefficient in kg 
b = 0.05;
%gravity in m/s^2
g = 9.81;
%Length of pendumum in m
l = 1;
%Mass of ball in kg
M = 1;
%Input conditions
theta_0 = [0;2.5];
% time 
t_span = linspace(0,20,250);

%to solve ODE
[t, results] = ode45(@(t,theta) ode_func(t, theta, b,g,l,M),t_span, theta_0);
plot(t,results(:,1))
hold on
grid on
plot(t,results(:,2));
ylabel('time in second')
xlabel('plot')


%to make the animation 

%counter variable for tracking the counter variable 
ct = 1;

for i = 1 : length(results(:,1))
    
    x0 = 0;
    y0 = 0;
    x1 = 1*sin(results(i,1));
    y1 = -1*cos(results(i,1));
    figure(2)
    %to make dead fix support 
    plot([-1 1], [0 0], 'linewidth',15, 'color','b')
    axis([-2 2 -2 2])
    hold on 
    %for plotting string possition 
    line([x0 x1], [y0 y1], 'linewidth', 5,'color','r')
    %plot ball position 
    plot(x1, y1, 'o','markersize', 25, 'markerfacecolor', 'y')
    hold off 
    m(ct) = getframe(gcf)
    %store current frame
    ct = ct+1;
    
end


%to creat video output 
movie(m)
videofile = VideoWriter('pendulum_simulation.avi', 'Uncompressed AVI');
%to open video 
open(videofile);
%write frame to video file 
writeVideo(videofile, m)
%close video file 
close(videofile)


function 


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];

end

YOUTUBE LINK FOR VIDEO -

https://youtu.be/l-S9kKJJ16c

Conclusion   –

 The effect of dampings on pendulum.

The displacement of the pendulum keep reduced while progressing thiem due to effect of damper the velocity of pendulum keep decrese with time.

The program has successfully run. The plot time w.r.t the velocity & displacement.

There are so many errors occurs wile programming, are attached in attachments & the most of errors are occurs while typing program. Its only reducing by practising for typing the program.

YOUTUBE LINK FOR VIDEO -

https://youtu.be/l-S9kKJJ16c

                              fig; the final plot.