Challenge

Week 3 - Solving second order ODEs

MMaheshwaran Senthil8 May 2022

1. Aim :

To write a program that solves the following ODE `(d^2∅)/(d^2 t)+b/m (d∅)/dt+g/l sin⁡∅= 0`
To write a Matlab program that will simulate the pendulum motion

2. Formulas used :

`∅= ∅_1`

`(d∅)/dt= (d∅_1)/dt=∅_2`

`(d∅_2)/dt+b/m ∅_2+g/l sin ∅_1⁡= 0`

`(d∅_2)/dt⁡= -b/m ∅_2-g/l sin ∅_1`

`d/dt [■(∅_1@∅_2 )]= [■(∅_2 &@-b/m ∅_2-g/l sin ∅_1&)]`

3. Objective of the project:

Animate the pendulum motion
To show the graph how angular displacement and angular velocity changes with respect to time

4. Body of content:

Matlab is used to write the program
Stating the inputs as given in the Question
Stating the initial contions as given
Used the ODE 45 is used to solve the differential equation with the help of the function

function ode_func is used to solve the given qiven equation using the formulas

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

plotting the graph of angular displacement and angular velocity changes with respect to time
Using the for loop to excute the pendulum motion with the time
plotting the pendulum motion

create the animation with videoWriter as folows

movie(M)

videofile = VideoWriter('pendulum motion.avi','Uncompressed AVI')

open(videofile)

writeVideo(videofile,M)

close(videofile)

the whole program is in below

clear all
close all
clc

% inputs
b = 0.05;
m = 1;
l = 1;
g = 9.81;

% Intial conditions
theta_0 = [0,3];
t_span = linspace(0,20,700);

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

%Plotting displacement and velocity varying w.r.t time
figure(1)
hold on
plot (t,results (:,1))
plot (t,results (:,2))
xlabel ('Time')
ylabel ('Displacement & velocity')

%For loop excution
theta1 = results (:,1);
a=1;
for i = 1:length(theta1)
    THETA = theta1(i);
    x0 = 0;
    y0 = 0;
    x1 = l*sin(THETA);
    y1 = -l*cos(THETA);
    % Plotting for the swinging pendulum
    figure (2)
    plot ([0.5 -0.5],[0 0],'linewidth',6)
    hold on
    plot (x1,y1,'o',MarkerSize=20,MarkerFaceColor='r')
    plot ([x0 x1],[y0 y1],'color','r')
    hold off
    axis ([-1 1 -1.5 0.5]);
    pause(0.005)
    M(a) = getframe(gcf);
    a=a+1;
end
%creating the animation
movie(M)
videofile = VideoWriter('pendulum motion.avi','Uncompressed AVI')
open(videofile)
writeVideo(videofile,M)
close(videofile)

5. Results :

the graph is attached in the image
various errors faced during the execution are also been attached
Movie is attached in the drivehttps://drive.google.com/file/d/1580L5yN7SFTptUW0k8SqidCNr6k_Wrv1/view?usp=sharing

Week 3 - Solving second order ODEs

Ship work like this
in your own course.

Talk to a career counsellor

Ship work like thisin your own course.

Talk to a career counsellor

Ship work like this
in your own course.