Week 3 - Solving second order ODEs

Generating Second Order ODE function

Youtude link is provide "https://youtu.be/16SAazQsiyg"

clear 
close all
clc

%Inputs 
b=0.2; %damping co_eff
g=9.81; %gravitional m/s^2
m=1; %mass of ball in kg
l=1; %length in metre

%Initial values of angular disp and anugular velocity
theta_0=[0;3];

%time taken in between the points
t_span=linspace(0,20,600);

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

%plotting
figure(1)
hold on
plot(t,results(:,1))
plot(t,results(:,2))
title('Simple pendulum by Sathishkumar')
xlabel('Time (sec)')
ylabel("Angular velocity/Displacement")
legend('Displacement','Angular velocity')

%Animating for pendulum 
amplitude=results(:,1); %Amplitude values in matrix for loop
p=1;
for i=1:length(amplitude)
    theta_ang=amplitude(i);
    figure(2)
    title('Simple Pendulum by Sathishkumar')
    
    % coordinates of starting point of pendulum
    x0=0;
    y0=0;
   
    %another end of pendulum(ending_point)
    x1=l*sin(theta_ang);
    y1=-l*cos(theta_ang);

    %plot
    
    plot([-1.1,1.1],[0,0],'color','g','Linewidth',6);
    hold on
    plot([x0 x1],[y0 y1],'color','b','LineWidth',3);
    plot(x1,y1,'O','markers',20,'markerfacecolor',[1 1 0])
    hold off
    grid on
    axis([-1.5 1.5 -1.5 0])
    xlabel('Motion of the pendulum')
    ylabel('Fixture of the string')
    pause(0.002)
    M(p)=getframe(gcf);
    p=p+1;
end

%Movie
movie(M)
videofile1 = VideoWriter('simple pendulum.avi','Uncompressed AVI');
open(videofile1)
writeVideo(videofile1,M)
close(videofile1)

Inputs

To run the motion of pendulum basic inputs are needed such as

Damper coeff (b) 0.2;

Gravitional force = 9.81;

Mass of ball in KG = 1;

Length of string = 1 metre;

Initial Value for angular dispalcement(rad) and velocity(rad/sec):

theta value is given as input tends from 0 to 3

time taken for swing upto 20sec for 600 points

By creating ODE function to run the program and to input the given in that function

To plot the damping figure using 't,results' giving the suitable title for the motion and giving the name for x axis and y axis

Next is to creat a plot for Animating the Motion of Pendulum

Using the 'For' loop command

i=1:length(amplitude) is used for that figure(2)

To start up the motion the coordinates are given x0,x1,y0,y1

Formula used for x1,y1:

x1=l*sin(theta_ang);
y1=-l*cos(theta_ang);

To graphical representation for

x axis tends from -1.1 to 1.1

y axis tends from 0 to 0

mainly 'Hold On' command used to hold the two different inputs

Axis commands for pendulum from [-1.5 1.5 -1.5 0]

Label for X axis and Y axis are given

xlabel ('Motion of the pendulum')
ylabel ('Fixture of the string')

Command prompt

p=1; and p=p+1 is used for continue for 20 sec

By the end 'movie' command is used

The output of the figure 1 and figure 2 are attached

Error occured during the programming

Error 1 :

line 6 : b=0.05 by using b as 0.05 vibration can't be arrested so that changed into 0.2 is used

Error 2 :

line 31 : amplitude=results(:,1) in this using result i have used Result.