Week 3 - Solving second order ODEs

AIM:

To write a program in MATLAB/OCTAVE to simulate the motion of a simple pendulum.

EQUATION/FORMULA USED:

`(d^2theta)/dt^2 + b/m (d theta)/dt + g/L sintheta = 0`

`theta = theta1`

`(d theta)/dt = (d theta1)/dt = theta2`

`(d^2theta)/dt = (d^2theta1)/dt^2 = d/dt((d theta1)/dt) = (d theta2)/dt`

`d/dt((d theta1)/dt) + b/m (d theta1)/dt + g/L sintheta = 0`

`(d theta2)/dt + b/m theta2 + g/L sintheta = 0`

g = gravity in m/s2

L = length of the pendulum in m

m = mass of the ball in kg

b=damping coefficient

OBJECTIVE:

1. To solve the second-order differential equation and plot the graph.
2. To understand the movement of a simple pendulum by creating an animation.

THEORY:

Ordinary Differential Equations(ODE) are a set of equations which can explain the motion of many mechanical systems which include springs and dampers.

A simple pendulum is an ideal pendulum consisting of a point mass suspended by a weightless inextensible perfectly flexible thread and free to vibrate without friction.

BODY:

Code:

1. Function:

function [dtheta_dt] = pendulum_damped_ode(theta,t)

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

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

2. Main code:

clear all
close all
clc


% Intial condition
theta_0 = [0;3];
l = 1;

% time points
t_span = linspace(0,20,500);


% solve ODE
[results] = lsode('pendulum_damped_ode', theta_0, t_span);

%plotting
plot(t_span,results(:,1))
hold on
plot(t_span,results(:,2))
xlabel('time')
ylabel('plot')
hold off

figure(2)
plot(results(:,1),results(:,2))
xlabel('angular displacement')
ylabel('angular velocity')

%simulation of simple pendulum
ct=1;
for i=1:length(results(:,1));
 theta3=results(:,1);
 THETA3=theta3(i);
 
 x0=0;
 y0=0;
 
 x1=l*sin(THETA3);
 y1=-l*cos(THETA3);
 
  % horizontal support for string
 plot([-0.5 0.5],[0 0],'linewidth',2,'color','k');      
 hold on
 grid on 
 
  % string
 plot([x0,x1],[y0,y1],'linewidth',1,'color','r');       
 hold on
 
  % bob tied at the end of string
 plot(x1,y1,'-o','markersize',30,'markerfacecolor','k');
 axis([-1.5 1.5 -1.5 1]);
 
   file_text = sprintf("armanimation%05d.png",ct)
      saveas(gca, file_text)
 pause(0.03);
 hold off
 
 ct=ct+1;
 
 end 

STEP-WISE EXPLANATION OF THE CODE:

• This program is written and executed using octave.
• Starting with all the general command e. clear all, close all & clc.
• Inputting all the code comment at all the required location starting with %.
• Then writing the function code for solving the 2^nd Also inputting all the given data.
• Then the initial condition and time points are inputted in the main code.
• The output from the function code is inputted in the main code in order to get the result.
• Then the graph is plotted.
• Now the code is written for the stimulation of simple pendulum.
• Code is written for horizontal support for the string.
• Code is written for the string.
• Code is written for bob tied at the end of the string.
• Note: Since the program is executed in octave therefore the process of stitching the images and saving the output video is little different than that in MATLAB.

ERROR FACED WHILE PROGRAMMING:

• Since the tutorial video was explained in MATLAB. The first error was that I was not knowing the process of stitching the image and saving the output in the form of video is different in case of the octave. Also in the video, I was not aware of this change.
• The second error was made while saving the file I was selecting a wrong file format due to a minor error in my code i.e. %05d screenshot is attached below.
• The third error was in CMD I could not able to change the directory.

STEP TAKEN TO OVERCOME THOSE ERROR:

• The first error was rectified with the help of my instructor over a call.
• The second error was fixed by amending the code.
• The third error was rectified by adding cd before pasting my file location in CDM.

SCREENSHOT OF ERROR:

SECOND ERROR:

THIRD ERROR:

OUTPUT:

STIMULATION OF SIMPLE PENDULUM:

RESULT:

After rectifying all the error, we have got the output plot and the video of the output plot.